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GRAPHS: THEORY AND ALGORITHMS GRAPHS: THEORY AND ALGORITHMS K. T H U L A S I R A M A N Μ. N. S. S W A M Y Concordia U...

Author: K. Thulasiraman | M. N. S. Swamy

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GRAPHS: THEORY AND ALGORITHMS

GRAPHS: THEORY AND ALGORITHMS K. T H U L A S I R A M A N Μ. N. S. S W A M Y Concordia University Montreal, Canada

A Wiley-lnterscience Publication JOHN WILEY & SONS, INC. New York /

Chichester

/

Brisbane /

Toronto /

Singapore

In recognition of the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc., to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end. Copyright © 1992 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Cataloging in Publication Data:

Thulasiraman, K. Graphs: theory and algorithms / K. Thulasiraman and M.N.S. Swamy. p. cm. "A Wiley-Interscience publication." Includes bibliographical references and index. ISBN 0-471-51356-3 1. Graph theory. 2. Electric networks. 3. Algorithms. I. Swamy, Μ. N. S. II. Title. QA166.T58 91-34930 1992 511'.5-dc20 CIP 10 9 8 7 6 5 4 3 2 1

Dedicated to Our Parents and Teachers

^ g ^ f t f e i Star ^

U

from Vishwasara Tantra

Salutations to the Guru who with the collyrium stick of knowledge has opened the eyes of one blinded by the disease of ignorance.

CONTENTS

PREFACE 1

BASIC CONCEPTS

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2

xiii

Some Basic Definitions / 1 Subgraphs and Complements / 4 Walks, Trails, Paths, and Circuits / 7 Connectedness and Components of a Graph / 9 Operations on Graphs / l l Special Graphs / 16 Cut-Vertices and Separable Graphs / 19 Isomorphism and 2-Isomorphism / 22 Further Reading / 25 Exercises / 26 References / 29

TREES, CUTSETS, AND CIRCUITS

2.1 2.2 2.3 2.4 2.5 2.6

1

31

Trees, Spanning Trees, and Cospanning Trees / 31 Λ-Trees, Spanning Λ-Trees, and Forests / 38 Rank and Nullity / 41 Fundamental Circuits / 41 Cutsets / 42 Cuts / 43 vli

Vlii

CONTENTS

2.7 2.8 2.9 2.10 2.11

3

EULERIAN AND HAMILTONIAN GRAPHS

3.1 3.2 3.3 3.4 3.5 4

Eulerian Graphs / 57 Hamiltonian Graphs / 62 Further Reading / 67 Exercises / 68 References / 70

GRAPHS AND VECTOR SPACES

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

5

Fundamental Cutsets / 46 Spanning Trees, Circuits, and Cutsets / 48 Further Reading / 51 Exercises / 51 References / 54

Groups and Fields / 72 Vector Spaces / 74 Vector Space of a Graph / 80 Dimensions of Circuit and Cutset Subspaces / 86 Relationship between Circuit and Cutset Subspaces / Orthogonality of Circuit and Cutset Subspaces / 90 Further Reading / 93 Exercises / 94 References / 96

DIRECTED GRAPHS

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Basic Definitions and Concepts / 97 Graphs and Relations / 104 Directed Trees or Arborescences / 105 Directed Eulerian Graphs / 110 Directed Spanning Trees and Directed Euler Trails / Directed Hamiltonian Graphs / 115 Acyclic Directed Graphs / 118 Tournaments / 119 Further Reading / 121 Exercises / 121 References / 124

CONTENTS

6

MATRICES OF A GRAPH

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 7

7.4 7.5 7.6 7.7 7.8 8

179

Planar Graphs / 179 Euler's Formula / 182 Kuratowski's Theorem and Other Characterizations of Planarity / 186 Dual Graphs / 188 Planarity and Duality / 193 Further Reading / 196 Exercises / 196 References / 198

CONNECTIVITY AND MATCHING

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

126

Incidence Matrix / 126 Cut Matrix / 130 Circuit Matrix / 133 Orthogonality Relation / 136 Submatrices of Cut, Incidence, and Circuit Matrices / 139 Unimodular Matrices / 145 The Number of Spanning Trees / 147 The Number of Spanning 2-Trees / 151 The Number of Directed Spanning Trees in a Directed Graph / 155 Adjacency Matrix / 159 The Coates and Mason Graphs / 163 Further Reading / 172 Exercises / 173 References / 176

PLANARITY AND DUALITY

7.1 7.2 7.3

iX

Connectivity or Vertex Connectivity / 200 Edge Connectivity / 207 Graphs with Prescribed Degrees / 209 Menger's Theorem / 213 Matchings / 215 Matchings in Bipartite Graphs / 217 Matchings in General Graphs / 224 Further Reading / 230 Exercises / 231 References / 234

200

X

9

CONTENTS

COVERING AND COLORING

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

Independent Sets and Vertex Covers / 236 Edge Covers / 243 Edge Coloring and Chromatic Index / 245 Vertex Coloring and Chromatic Number / 251 Chromatic Polynomials / 253 The Four-Color Problem / 257 Further Reading / 258 Exercises / 259 References / 262

MATROIDS

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 11

265

Basic Definitions / 266 Fundamental Properties / 268 Equivalent Axiom Systems / 272 Matroid Duality and Graphoids / 276 Restriction, Contraction, and Minors of a Matroid / 282 Representability of a Matroid / 285 Binary Matroids / 287 Orientable Matroids / 292 Matroids and the Greedy Algorithm / 294 Further Reading / 298 Exercises / 299 References / 303

GRAPH ALGORITHMS

11.1 11.2 11.3 11.4 11.5

236

Transitive Closure / 307 Shortest Paths / 314 Minimum Weight Spanning Tree / 324 Optimum Branchings / 327 Perfect Matching, Optimal Assignment, and Timetable Scheduling / 332 11.6 The Chinese Postman Problem / 342 11.7 Depth-First Search / 346 11.8 Biconnectivity and Strong Connectivity / 354 11.9 Reducibility of a Program Graph / 361 11.10 si-Numbering of a Graph / 370 11.11 Planarity Testing / 373

306

CONTENTS

11.12 11.13 11.14 12

Further Reading / 379 Exercises / 380 References / 382

FLOWS IN NETWORKS

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14

Xi

390

The Maximum Flow Problem / 391 Maximum Flow Minimum Cut Theorem / 392 Ford-Fulkerson Labeling Algorithm / 396 Edmonds and Karp Modification of the Labeling Algorithm / 400 Dinic Maximum Flow Algorithm / 404 Maximal Flow in a Layered Network: The MPM Algorithm / 408 Preflow Push Algorithm: Goldberg and Tarjan / 411 Maximum Flow in 0-1 Networks / 422 Maximum Matching in Bipartite Graphs / 426 Menger's Theorems and Connectivities / 427 NP-Completeness / 433 Further Reading / 436 Exercises / 437 References / 439

AUTHOR INDEX

445

SUBJECT INDEX

451

PREFACE

In the past two decades graph theory has come to stay as a powerful analytical tool in the understanding and solution of large complex problems that arise in the study of engineering, computer, and communication systems. While its origin is traced to Euler's solution in 1735 of the Konigsberg bridge problem, its first application to a problem in physical science did not occur until 1847, when Kirchhoff developed the theory of trees for its application in the study of electrical networks. The elegance with which the graph of an electrical network captures the structural relationships between the voltage and current variables of the network has led to equally elegant contributions to electrical network theory. One such condition is Tellegen's theorem, the application of which in the computation of network sensitivities is now well recognized. The theory of network flows developed by Ford and Fulkerson in 1956 was the first major application of graph theory to operations research. This theory provides the main link between graph theory and operations research and continues to be a fascinating topic of further research. Computer and communication systems are among the recent additions to the growing list of application areas of graph theory. Motivated by applications in the design of interconnection networks for these systems, in recent years there has been a great deal of interest in the design of graphs having specified topological properties such as distance, connectivity, and regularity. Fascinated by the challenges encountered in the design of efficient algorithms for graph problems, theoretical computer scientists have developed in the past two decades a large number of interesting and deep graph algorithms adding to the richness of graph theory. Theoretical computer scientists have also identified the class of graph problems for which "efficient" algorithms are not likely to xiii

XiV

PREFACE

exist, giving birth to the theory of NP-Completeness. This is indeed a significant contribution of computer science to graph theory. Every time a new area of application of graph theory emerged, the need arose for the introduction and study of new concepts or a further study of several known concepts. This continuous interaction has immensely contributed to the recent explosion of graph theory, which was fairly dormant for more than a century after its origin. Thus graph theory is now a vast subject with several fascinating branches of its own: enumerative graph theory, extremal graph theory, random graph theory, algorithmic graph theory, and so on. As its name implies, this book is on graph theory and graph algorithms. It is addressed to students in engineering, computer science, and mathematics. Our choice of topics has been motivated by their relevance to applications. Thus we attempt to provide a unified and an in-depth treatment of those topics in graph theory and graph algorithms that we believe to be fundamental in nature and that occur in most applications. Broadly speaking, the book may be considered as consisting of two parts dealing with graph theory and graph algorithms in that order. In the first ten chapters we discuss the theory of graphs. The topics discussed include trees, circuits, cutsets, Hamiltonian and Eulerian graphs, directed graphs, matrices of a graph, planarity, connectivity, matching, and coloring. We have also included an introduction to matroid theory. Among the matroid topics presented are Minty's self-dual axiom system, which makes obvious the duality between circuits and cutsets of a graph, the arc coloring lemma, the greedy algorithm, and its intimate relationship with matroids. The last two chapters of the book deal with graph algorithms. In Chapter 11 we discuss several algorithms which are basic in the sense that they serve as building blocks in designing more complex algorithms. In most cases the algorithms of Chapter 11 are based on results and concepts presented in earlier chapters. In certain cases we also introduce and discuss new concepts such as branching and graph reducibility. In Chapter 12 we develop the theory of network flows. We start with the maximum flow minimum cut theorem of Ford and Fulkerson and then proceed to develop several algorithms for the maximum flow problem, culminating with the recent work of Goldberg and Tarjan. In this chapter we also show how the network flow technique can be used to develop connectivity and matching algorithms as well as prove Menger's theorems on connectivities. We conclude Chapter 12 with a brief introduction to the theory of NP-Completeness. While developing the algorithms of Chapters 11 and 12 we pay particular attention to the proof of correctness and complexity analysis of the algorithms. The book can be used to organize different courses to suit the needs of different groups of students. The first ten chapters contain adequate material for a one-semester course on graph theory at the senior or beginning graduate level. The authors have taught for several years a course on graph

PREFACE

XV

theory with system applications based on the first seven chapters and a selection of topics from the remaining chapters. The last two chapters and appropriate background material selected from the other chapters can serve as the core of a course on algorithmic graph theory. These two chapters can also serve as supplemental material for a general course on design and analysis of algorithms. Several colleagues and students have assisted us in the writing of this book. Raghu Prasad Chalasani, Concordia University; Joseph Cheriyan, Cornell University; Anindya Das, University of Montreal; Andrew Goldberg, Stanford University; R. Jayakumar, Concordia University; V. Krishnamoorthy, Anna University, Madras (India); and N. Srinivasan, University of Madras deserve special thanks. We are greatful to Anindya Das, Joseph Cheriyan, and Andrew Goldberg for their careful reading of the last chapter of the book and drawing our attention to recent developments on the maximum flow problem. It is a pleasure to thank the following organizations for their support to our research leading to the preparation of the book: Natural Sciences and Engineering Research Council of Canada; Fonds pour la Formation de Chercheurs et Γ Aide a la Recherche (FCAR), Quebec; Bell Northern Research Laboratory, Ottawa; Centre de Recherche Informatique de Montreal, Montreal; German National Science Foundation; and the Japan Society for Promotion of Science. Finally we thank our wives—Santha Thulasiraman and Leela Swamy— and our children for their patience and understanding during the entire period of our efforts. K. THULASIRAMAN Μ. N. S. SWAMY

It is probably fair to say, and has been said before by many others, that graph theory began with Euler's solution in 1735 of the class of problems suggested to him by the Konigsberg bridge puzzle. But had it not started with Euler, it would have started with Kirchhoff in 1847, who was motivated by the study of electrical networks; had it not started with Kirchhoff, it would have started with Cayley in 1857, who was motivated by certain applications to organic chemistry, or perhaps it would have started earlier with the four-color map problem, which was posed to De Morgan by Guthrie around 1850. And had it not started with any of the individuals named above, it would almost surely have started with someone else, at some other time. For one has only to look around to see "real-world graphs" in abundance, either in nature (trees, for example) or in the works of man (transportation networks, for example). Surely someone at some time would have passed from some real-world object, situation, or problem to the abstraction we call graphs, and graph theory would have been born.

D. R. Fulkerson (From Preface to Studies in Graph Theory, Part II, The Mathematical Association of America, 1975)

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 1

BASIC CONCEPTS

We begin our study with an introduction in this chapter to several basic concepts in the theory of graphs. A few results involving these concepts will be established. These results, while illustrating the concepts, will also serve to introduce the reader to certain techniques commonly used in proving theorems in graph theory.

1.1

SOME BASIC DEFINITIONS

A graph G = (V, E) consists of two sets: a finite set V of elements called vertices and a finite set Ε of elements called edges. Each edge is identified with a pair of vertices. If the edges of a graph G are identified with ordered pairs of vertices, then G is called a directed or an oriented graph. Otherwise G is called an undirected or a nonoriented graph. Our discussions in the first four chapters of this book are concerned with undirected graphs. We use the symbols v ,v ,v ,... to represent the vertices and the symbols e,, e , e , . . . to represent the edges of a graph. The vertices v, and v associated with an edge e, are called the end vertices of e . The edge e, is then denoted as e, = (υ,, υ,). Note that while the elements of Ε are distinct, more than one edge in Ε may have the same pair of end vertices. All edges having the same pair of end vertices are called parallel edges. Further, the end vertices of an edge need not be distinct. If e, = (ν,,ν,), then the edge e, is called a self-loop at vertex v,. A graph is called a simple graph if it has no parallel edges or self-loops. A graph G is of order η if its vertex set has η elements. l

2

t

2

3

3

t

1

2

BASIC CONCEPTS

A graph with no edges is called an empty graph. A graph with no vertices (and hence no edges) is called a null graph. Pictorially a graph can be represented by a diagram in which a vertex is represented by a dot or a circle and an edge is represented by a line segment connecting the dots or the circles, which represent the end vertices of the edge. For example, if V={v ,v ,v ,v ,v ,v } 1

2

3

4

5

6

and Ε — {e^,

e*2,

e , e^, c } , 5

3

such that *\ =

(ν ,ν ), χ

2

e = (v ,

υ ),

(^i,

v ),

3

e

5

= 4

6

2

then the graph G = (Κ, E) is represented as in Fig. 1.1. In this graph e and e are parallel edges and e is a self-loop. An edge is said to be incident on its end vertices. Two vertices are adjacent if they are the end vertices of some edge. If two edges have a common end vertex, then these edges are said to be adjacent. For example, in the graph of Fig. 1.1, edge e, is incident on vertices u, and v ; w, and i> are two adjacent vertices, while e, and e are two adjacent edges. The number of edges incident on a vertex y, is called the degree of the vertex, and it is denoted by d(u,). Sometimes the degree of a vertex is also x

4

5

2

4

Figure 1.1. Graph G = (V,E).

2

V= {v v , v„ v , v , υ }; Ε = {*„ e , e , e„ e }. u

2

4

s

6

2

3

5

SOME BASIC DEFINITIONS

3

referred to as its valency. A vertex of degree 1 is called a pendant vertex. The only edge incident on a pendant vertex is called a pendant edge. A vertex of degree 0 is called an isolated vertex. By definition, a self-loop at a vertex v, contributes 2 to the degree of υ,. 5(G) and A(G) denote, respectively, the minimum and maximum degrees in G. In the graph G of Fig. 1.1 d(v,) = 3 , d(v ) = 2 , 2

d(v ) = 0 , 3

d(v ) = 1 , 4

d(v ) = 3 , 5

d(v ) = 1 . 6

Note that u is an isolated vertex, u and v are pendant vertices, and e is a pendant edge. For G it can be verified that the sum of the degrees of the vertices is equal to 10, whereas the number of edges is equal to 5. Thus the sum of the degrees of the vertices of G is equal to twice the number of edges of G and hence an even number. It may be further verified that in G the number of vertices of odd degree is also even. These interesting results are not peculiar to the graph of Fig. 1.1. In fact, they are true for all graphs as the following theorems show. 3

4

6

2

Theorem 1.1. The sum of the degrees of the vertices of a graph G is equal to 2m, where m is the number of edges of G. Proof. Since each edge is incident on two vertices, it contributes 2 to the sum of the degrees of the graph G. Hence all the edges together contribute 2m to the sum of the degrees of G. • Theorem 1.2. The number of vertices of odd degree in any graph is even. Proof. Let the number of vertices in a graph G be equal to n. Let, without any loss of generality, the degrees of the first r vertices υ , , v ,..., v be even and those of the remaining η - r vertices be odd. Then 2

Σ <*(!>,) = Σ <*(«,)+ Σ 1=1

ι - l

d( ). Vl

r

(i.i)

i=r+l

By Theorem 1.1, the sum on the left-hand side of (1.1) is even. The first sum on the right-hand side is also even because each term in this sum is even. Hence the second sum on the right-hand side should be even. Since

4

BASIC CONCEPTS

each term in this sum is odd, it is necessary that there be an even number of terms in this sum. In other words, η - r, the number of vertices of odd degree, should be even. •

1.2

SUBGRAPHS AND COMPLEMENTS

Consider a graph G = (V, E). G' = (V, E') is a subgraph of G if V and E' are, respectively, subsets of V and Ε such that an edge (υ,, υ) is an E' only if v, and v are in V. G' will be called a proper subgraph of G if either E' is a proper subset of Ε or V is a proper subset of V. If all the vertices of a graph G are present in a subgraph G' of G, then G' is called a spanning subgraph of G. For example, consider the graph G shown in Fig. 1.2a. The graph G' shown in Fig. 1.2ft is a subgraph of G. Its vertex set is {u,, v , v , v ). In fact, it is a proper subgraph of G. The graph G" of Fig. 1.2c is a spanning subgraph of G. Some of the vertices in a subgraph may be isolated vertices. For example, the graph G'" shown in Fig. 1.2d is a subgraph of G with an isolated vertex. l

2

4

5

Figure 1.2. A graph and some of its subgraphs, (a) Graph G. (Z>) Subgraph G'. (c) Subgraph G". (d) Subgraph G'".

SUBGRAPHS AND COMPLEMENTS

5

If a subgraph G' = (V, Ε ' ) of a graph G has no isolated vertices, then it can be seen from the definition of a subgraph that every vertex in V is the end vertex of some edge in Ε'. Thus in such a case, E' uniquely specifies V and hence the subgraph G'. The subgraph G' is then called the induced subgraph of G on the edge set E' (or simply edge-induced subgraph of G) and is denoted as (Ε'). Note that the vertex set V of (Ε') is the smallest subset of V containing all the end vertices of the edges in Ε'. The subgraphs G' and G" of Fig. 1.2ft and c are edge-induced subgraphs of the graph G of Fig. 1.2a, whereas G'" shown in Fig. \.2d is not an edge-induced subgraph. Next we define a vertex-induced subgraph. Let V be a subset of the vertex set V of a graph G = (V, E). Then the subgraph G' = (V, Ε') is the induced subgraph of G on the vertex set V (or simply vertex-induced subgraph of G) if E' is a subset of Ε such that edge (υ,, Vj) is in E' if and only if v, and u are in V. In other words, if v, and v, are in V, then every edge in Ε having υ, and υ as its end vertices should be in Ε'. Note that, in this case, V completely specifies E' and thus the subgraph G'. Hence the vertex-induced subgraph G' = ( V , Ε') is denoted simply as (V). As an example, the graph shown in Fig. 1.3 is a vertexinduced subgraph of the graph G of Fig. 1.2a. Note that the edge set E' of the vertex-induced subgraph on the vertex set V is the largest subset of Ε such that the end vertices of all of its edges are in V. Unless otherwise stated, an induced subgraph will refer to a vertexinduced subgraph. A subgraph G' of a graph G is said to be a maximal subgraph of G with respect to some property Ρ if G' has the property Ρ and G' is not a proper subgraph of any other subgraph of G having the property P. A subgraph G' of a graph G is said to be a minimal subgraph of G with respect to some property PUG' has the property Ρ and no subgraph of G having the property Ρ is a proper subgraph of G'. Maximal and minimal subsets of a set with respect to a property are defined in a similar manner. For example, the vertex set V of an edge-induced subgraph (Ε') of a graph G = (V, E) is a minimal subset of V containing the end vertices of all ;

Figure 1.3. A vertex-induced subgraph » of the graph G of Fig. 1.2a. 5

6

BASIC CONCEPTS

the edges of E'. On the other hand, the edge set E' of a vertex-induced subgraph ( V ) is the maximal subset of Ε such that the end vertices of all of its edges are in V. Later we shall see that a "component" (Section 1.4) of a graph G is a maximal "connected" subgraph of G, and a "spanning tree" (Chapter 2) of a connected graph G is a minimal "connected" spanning subgraph of G. Next we_ define the complement of a graph. Graph G = (V, E') is called the complement of a simple graph G = (V, E) if the edge (v , v ) is in E' if and_only if it is not in E. In other words two vertices u, and v are adjacent in G if and only if they are not adjacent in G. A graph and its complement are shown in Fig. 1.4. As another example, consider the graph G shown in Fig. 1.5e. In this graphjthere is an edge between every pair of vertices. Hence in the complement G of G there will t

t

f

»*0

Figure 1.4. A graph and its complement, (a) Graph G. (b) Graph G, complement of G.

0»5

(61

Figure 1.5. A graph and its complement, (a) Graph G. (b) Graph G, complement of G.

WALKS, TRAILS, PATHS, AND CIRCUITS

7

be no edge between any pair of vertices; that is, G will contain only isolated vertices. This is shown in Fig. 1.5ft. Let G' = (V, Ε') be a subgraph of a graph G = (V, E). The subgraph G" = (V, Ε - Ε') of G is called the complement of G' in G. For example, in Fig. 1.2, subgraph G" is the complement of G' in the graph G. The following example illustrates some of the ideas presented thus far. Suppose we want to prove the following: At any party with six people there are three mutual acquaintances or three mutual nonacquaintances. Representing people by the vertices of a graph and acquaintance relationship among the people by edges connecting the corresponding vertices, we can see that the above assertion can also be stated as follows: In any simple graph G with six vertices there are three mutually adjacent vertices or three mutually nonadjacent vertices. In view of the definition of the complement of a graph, we see that the above statement is equivalent to the following: _ For any simple graph G with six vertices, G or G contains three mutually adjacent vertices. To prove this, we may proceed as follows: Consider any vertex ν of a simple graph G with six vertices. Note that if υ is not adjacent to three vertices in G, then it will be adjacent to three vertices in G. So, without any loss of generality, we may assume that, in G, υ is adjacent to some three vertices v , v , and u . If any two of these vertices, say v and v , are adjacent in G, then the vertices ν, υ , and υ are mutually adjacent in G, and the assertion is proved. If no two of the three vertices v v , and v are adjacent in G, then it means that u,, v , and v are mutually nonadjacent in G. Hence, by the definition of a complement, the vertices v , v , and v are mutually adjacent in G, and the assertion is again proved. t

i

ι

lt

2

2

2

3

3

t

1.3

3

2

2

2

3

WALKS, TRAILS, PATHS, AND CIRCUITS

A walk in a graph G = (V, E) is a finite alternating sequence of vertices and edges v , e v e ,..., v _ , e , v beginning and ending with vertices such that v,_ and v are the end vertices of the edge e l ^ i ^ k. Alternately, a walk can be considered as a finite sequence of vertices v , v,, v ,...,v , such that (y,_i, ι>,), 1 ^ ι k, is an edge in the graph G. This walk is usually called a v -v walk with υ and v referred to as the end or terminal vertices of this walk. All other vertices are internal vertices of this walk. Note that in a walk, edges and vertices can appear more than once. A walk is open if its end vertices are distinct; otherwise it is closed. In the graph G of Fig. 1.6, the sequence i>,, e , v , e , u , e , v , e , v , e ,v ,e ,v is an open walk, whereas the sequence v ,e ,v ,e ,v ,e ,v , e , v , e,, u, is a closed walk. 0

u

u

x

2

k

l

k

k

t

n

0

0

k

0

2

3

u

2

6

k

k

1

7

2

2

2

3

s

6

1

l

9

2

3

s

5

7

3

8

BASIC CONCEPTS

Ίο

»7

Figure 1.6. Graph G. A walk is a trail if all its edges are distinct. A trail is open if its end vertices are distinct; otherwise, it is closed. In Fig. 1.6, u,, e , v , e , v , e , v , e , v is an open trail, whereas υ,, e,, v , e , v , e , v , e , v , e , u , e , y, is a closed trail. An open trail is a path if all its vertices are distinct. A closed trail is a circuit if all its vertices except the end vertices are distinct. For example, in Fig. 1.6 the sequence u,, e,, v , e , v is a path, whereas the sequence v,, e,, v , e , v , e , v , e , υ, is a circuit. An edge of a graph G is said to be a circuit edge of G if there exists a circuit in G containing the edge. Otherwise the edge is called a noncircuit edge. In Fig. 1.6, all edges except e are circuit edges. The number of edges in a path is called the length of the path. Similarly the length of a circuit is defined. A path is even if it is of even length; otherwise it is odd. Similarly even and odd circuits are defined. The distance between two vertices u and ν in G, denoted by d(u, υ), is the length of the shortest u-υ path in G. If no such path exists, then we define d(u, v) to be infinite. The diameter of G, denoted by diam(G), is the maximum distance between any two vertices of G. The following properties of paths and circuits should be noted: 6

u

3

2

2

3

2

2

3

5

6

4

7

2

5

x

2

2

3

2

4

3

s

4

5

3

5

n

1. In a path the degree of each vertex that is not an end vertex is equal to 2; the end vertices have degrees equal to 1. 2. In a circuit every vertex is of degree 2, and so of even degree. The converse of this statement, namely, the edges of a subgraph in which

CONNECTEDNESS AND COMPONENTS OF A GRAPH

9

every vertex is of even degree form a circuit, is not true. A more general question is discussed in Chapter 3. 3. In a path the number of vertices is one more than the number of edges, whereas in a circuit the number of edges is equal to the number of vertices.

1.4

CONNECTEDNESS AND COMPONENTS O F A GRAPH

An important concept in graph theory is that of connectedness. Two vertices v, and v are said to be connected in a graph G if there exists a v-v path in G. A vertex is connected to itself. A graph G is connected if there exists a path between every pair of vertices in G. For example, the graph of Fig. 1.6 is connected. Consider a graph G = (V, E) which is not connected. Then the vertex set V of G can be partitioned into subsets V , , V , . . . , V such that the vertex-induced subgraphs (V,), i = 1 , 2 , . . . , p, are connected and no vertex in subset V, is connected to any vertex in subset V , / # i. The subgraphs (V ), i = 1, 2 , . . . , p, are called the components of G. It may be seen that a component of a graph G is a maximal connected subgraph of G; that is, a component of G is not a proper subgraph of any other connected subgraph of G. For example, the graph G of Fig. 1.7 is not connected. Its four components G , G , G , and G have vertex sets {v , υ , ν }, (ι> , υ }, {v , υ , u } , and {υ }, respectively. Note that an isolated vertex by itself should be treated as a component since, by definition, a vertex is connected to itself. Further, note that if a graph G is connected, it has only one component that is the same as G itself. We next consider some properties of connected graphs. t

J

t

2

f

t

l

g

2

3

4

t

2

4

3

5

6

Ί

9

Figure 1.7. Graph G with components G,, G , G , and G . 2

3

4

Ά set V is said to be partitioned into subsets V,, V ,..., V if V, U V U · · · U V = V and V, Π V, = 0 for all i and /, iV /'. {V,, V , . . . , V,,} is then called a partition of V. 2

2

p

2

p

10

BASIC CONCEPTS

Theorem 1.3. In a connected graph, any two longest paths have a common vertex. Proof. Consider any two longest paths P, and P in a connected graph G. Let P, be denoted by the vertex sequence υ ,υ ,υ ,... ,v and P by the sequence v' , v\, v' ,...,v' . Assume that P, and P have no common vertex. Since the graph G is connected, then for some i,0
0

Q

2

ί

2

k

2

k

2

a

a

l

2

t

2

/, = length of v -v, path P

u

t = length of v-v

P,

0

2

path

k

/[ = length of ι>ό-ι>' path P ;

t' = length of v'-v' 2

k

}

a

path

, l2

21

,

P, 22

t = length of path P . fl

a

The paths P , P , P , and P n

1 2

2 1

2 2

are also shown in Fig. 1.8. Note that

f, + t = t[ + t' = length of a longest path in G 2

2

and t >0. a

Path P, •'it

i" 1

'iJ

* ^

2

Path P

2

Figure 1.8. Paths P,, P and P„. 2

OPERATIONS ON GRAPHS

11

Without any loss of generality, let

and t' > t' so that Ί +'!

2=

i, + t = t\

+t .

2

2

Now it may be verified that the paths P , P , and P together constitute a v -v' path with its length equal to /, + t[ + t > i, + t because /„ > 0. This contradicts that r, + t is the length of a longest path in G. • n

0

a

0

2i

a

2

2

The following theorem is a very useful one; it is used often in the discussions of the next chapter. In this theorem as well as in the rest of the book, we abbreviate {x} to χ whenever it is clear that we are referring to a set rather than an element. Theorem 1.4. If a graph G = (V, E) is connected, then the graph G' = (V, Ε — e) that results after removing a circuit edge e is also connected. • We leave the proof of this theorem as an exercise.

1.5

OPERATIONS ON GRAPHS

In this section we introduce a few operations involving graphs. The first three operations are binary operations involving two graphs, and the last four are unary operations, that is, operations defined with respect to a single graph. Consider two graphs, G, = (V,, Ej) and G = (V , E ). The union of G, and G , denoted as G, U G , is the graph G = (V, U V , E U E )\ that is, the vertex set of G is the union of V and V , and the edge set of G is the union of £ , and E . For example, two graphs G, and G and their union are shown in Fig. 1.9a, b and c. The intersection of G, and G , denoted as G, Π G , is the graph G = (V, Π V , Ε, Π E ). That is, the vertex set of G consists of only those vertices present in both G and G , and the edge set of G consists of only those edges present in both G, and G . The intersection of the graphs G, and G of Fig. 1.9a and 1.96 is shown in Fig. l.9d. 2

2

2

2

2

3

3

2

l

2

l

3

2

2

2

3

2

2

3

2

l

2

3

2

2

2

12

BASIC CONCEPTS

(6)

"1

o-

"to

(c)

Figure 1.9. Union, intersection, and ring sum operations on graphs, (a) Graph G,. (b) Graph G2. (c) G, U G2. (<*) G, Π G2. (e) G, Θ G2.

OPERATIONS ON GRAPHS

13

"5

(e)

Figure 1.9. (Continued)

The ring sum of two graphs G, and G , denoted as G , © G , is the induced graph G on the edge set £ , Θ E . In other words G has no isolated vertices and consists of only those edges present either in G, or in G , but not in both. The ring sum of the graphs of Fig. 1.9a and 1.96 is shown in Fig. 1.9e. It may be easily verified that the three operations defined are commutative, that is, 3

2

2

2

3

2

G, U G = G U G, , 2

G nG = x

2

G DG

2

2

l

,

Gj Φ G = G φ G, . 2

2

Also note that these operations are binary, that is, they are defined with respect to two graphs. Of course, the definitions of these operations can be extended in an obvious way to include more than two graphs. Next we discuss four unary operations on a graph. Vertex Removal If v, is a vertex of a graph G = (V, E), then G — υ, is the induced subgraph of G on the vertex set V—υ,; that is, G - v, is the graph obtained after removing from G the vertex υ, and all the edges incident on v r

Edge Removal

If e, is an edge of a graph G = (V, E), then G - e, is the

' Note that £, θ £ = (£, - E ) U (£ - £,). 2

2

2

14

BASIC CONCEPTS

subgraph of G that results after removing from G the edge e,. Note that the end vertices of e, are not removed from G. The removal of a set of vertices or edges from a graph is defined as the removal of single vertices or edges in succession. If G, = (V, Ε') is a subgraph of the graph G = (V, E), then by G - G,

(c)

Figure 1.10. Vertex and edge removal operations on a graph, (a) G. (b) G - u , . (c) G-{e ,e ). 7

s

OPERATIONS ON GRAPHS

15

we refer to the graph G' = (V, Ε - Ε'). Thus G — G, is the complement in G of the subgraph G,. The vertex removal and edge removal operations are illustrated in Fig. 1.10. The short-circuiting or identifying operation to be considered next is a familiar one for electrical engineers. Short-Circuiting or Identifying A pair of vertices v, and v in a graph G are said to be short-circuited (or identified) if the two vertices are replaced by a new vertex such that all the edges in G incident on v, and i» are now incident on the new vertex. For example, short-circuiting v and v in the graph of Fig. 1.11a results in the graph shown in Fig. 1.11ft. Contraction By contraction of an edge e we refer to the operation of removing e and identifying its end vertices. A graph G is contractible to a graph Η ύ Η can be obtained from G by a sequence of contractions. For example, the graph shown in Fig. 1.11c is obtained by contracting the edges e, and e of the graph G in Fig. 1.11a. ]

;

3

4

5

«61

Figure 1.11. Short-circuiting and contracting in a graph, (a) Graph G. (ft) Graph G after short-circuiting i> and v . (c) Graph G after contracting e, and e . 3

4

5

BASIC CONCEPTS

16

1.6

SPECIAL GRAPHS

Certain special classes of graphs that occur frequently in the theory of graphs are introduced next. A complete graph G is a simple graph in which every pair of vertices is adjacent. If a complete graph G has η vertices, then it will be denoted by K . It may be seen that K„ has n(n - l ) / 2 edges. As an example, the graph K is shown in Fig. 1.12. A graph G is regular if all the vertices of G are of equal degree. If G is regular with d(v,) = r for all vertices υ, in G, then G is called r-regular. A 4-regular graph is shown in Fig. 1.13. It may be noted that K is an (n - l)-regular graph. A graph G = (V, E) is bipartite if its vertex set V can be partitioned into two subsets K, and V such that each edge of Ε has one end vertex in V and another in V ; (V,, V ) is referred to as a bipartition of G. If in a simple bipartite graph G, with bipartition ( V , , ^ ) , there is an edge (υ,-,υ,) for every vertex υ, in V, and every vertex u in V , then G is called a complete bipartite graph and will be denoted by K if V has m vertices and V has η vertices. A bipartite graph and the complete bipartite graph K are shown in Fig. 1.14. A graph G = (V, E) is k-partite if it is possible to partition V into k subsets V,, V ,..., V such that each edge of G has one end vertex in some V and the other in some V , ΪΦ j . A complete k-partite graph G is a simple fc-partite graph with vertex set partition {V,, V ,..., V ) and with the additional property that for every vertex u, in V and every vertex υ in V , r¥=s, l < r , 5 < / c , (v , u ) is an edge in G. A complete 3-partite graph is shown in Fig. 1.15. n 5

n

2

l

2

2

;

2

mn

i

2

3 4

2

k

l

t

2

r

t

k

;

y

Figure 1.12. Graph K . s

Figure 1.13. A 4-regular graph.

s

SPECIAL GRAPHS

17

(/>)

Figure 1.14. (a) A bipartite graph, (b) The complete bipartite graph

K. 34

We conclude this section with a useful characterization of bipartite graphs. Theorem 1.5. A graph with η =: 2 vertices is bipartite if and only if it contains no circuits of odd length. Proof Necessity Let G = (V, E) be a bipartite graph with bipartition (X, Y). Consider any circuit C of G. Suppose that we traverse C starting from a vertex, say v , of G. Without loss of generality, let v be in X. Since the circuit terminates in X, it follows that every time we traverse an edge (u , v ) with υ, in X and v in Y, then there will be an untraversed edge v ) of C with v in Χ Thus at the end of the traversal we shall have traversed an even number of edges and so C is of even length. 0

0

f

f

k

k

t

18

BASIC CONCEPTS

Figure 1.15. A complete 3-partite graph. Sufficiency Let G = (V, E) be a graph with η > 2 vertices and with no circuits of odd length. Without loss of generality, assume that G is connected. Pick any vertex uolG and define a partition (X, Y) of V as follows: X = {x\d(u, x) is even}, y = {y\d(u, y) is odd} . Clearly χ is in X. We now show that (X, Y) is a bipartition of G. Consider any two vertices ν and w in X. We need to show that υ and w are not adjacent in G. Assume the contrary and let e = (υ, w) denote an edge connecting υ and w in G. Let P, be a shortest u-υ path and P be a shortest u-w path. Since both υ and w are in X, these paths are, by definition, of even lengths. As we traverse P, and P starting from u, let Λ: be the last vertex common to P, and P . Since P and P are shortest paths, the u-x sections of these paths are of the same length. Thus the x-v section of P, and the x-w section of P are both even or are both odd. Therefore, if we concatenate them, then we would get an even path connecting υ and w. This 2

2

2

2

x

2

19

CUT-VERTICES AND SEPARABLE GRAPHS

path and the edge e = (v, w) would then produce a circuit of odd length in G. This is contrary to the hypothesis that G has no odd circuits. Therefore no two vertices of X are adjacent in G. In a similar manner we can show that no two vertices of Y are adjacent. Thus G is a bipartite graph with bipartition (X, Y). •

1.7

CUT-VERTICES AND SEPARABLE GRAPHS

A vertex u, of a graph G is a cut-vertex of G if the graph G - v, consists of a greater number of components than G. If G is connected, then G - v will contain at least two components, that is, G - v, will not be connected. According to this definition, an isolated vertex is not a cut-vertex. We call a graph trivial if it has only one vertex. Thus a trivial graph has no cut-vertex. A nonseparable graph is a connected graph with no cut-vertices. All other graphs are separable. (Note that a graph that is not connected is separable.) The graph G shown in Fig. 1.16a is separable. It has three cut-vertices u , v , and u . A block of a separable graph G is a maximal nonseparable subgraph of G. The blocks of the separable graph G of Fig. 1.16a are shown in Fig. 1.166. The following theorem presents an equivalent definition of a cut-vertex. t

t

2

3

Theorem 1.6. A vertex ν is a cut-vertex of a connected graph G if and only if there exist two vertices u and w distinct from υ such that υ is on every u-w path. Proof Necessity Since υ is a cut-vertex of G, G - υ is, by definition, not connected. Let G, be one of the components of G - v. Let be the vertex set of G, and V the complement of V, in V- v. Let u and w be two vertices such that u is in V and w is in V . Consider any u-w path in G. If the cut-vertex υ does not lie on this path, then this path is also in G - u; that is, the vertices u and w are connected in G - v. However, this is a contradiction since u and w are in different components of G - v. Hence the vertex ν lies on every u-w path. 2

x

2

Sufficiency If υ is on every u-w path, then the vertices u and w are not connected in G - v. Thus the graph G - υ is not connected. Hence, by definition, υ is a cut-vertex. • An equivalent definition of a separable graph follows from Theorem 1.6: A connected graph G is separable if and only if there exists a vertex ν in G such that it is the only vertex common to two proper nontrivial subgraphs G, and G whose union is equal to G. 2

20

BASIC CONCEPTS

δ » .

(b)

Figure 1.16. A separable graph and its blocks, (a) A separable graph G. (ϋ>) Blocks of G. Suppose a graph G is separable. Is it possible that all the vertices of G are cut-vertices? The answer is " n o " as we prove in the next theorem. Theorem 1.7. Every nontrivial connected graph contains at least two vertices that are not cut-vertices. Proof. Consider a nontrivial connected graph G. Let u and υ be two vertices of G such that d(u, v) = diam(G). We claim that neither u nor υ is a

CUT-VERTICES AND SEPARABLE GRAPHS

21

cut-vertex of G. Suppose, on the contrary, that one of u and v, say v, is a cut-vertex of G. Then let w be a vertex belonging to a component of G — υ not containing u. Since very u-w path contains v, it follows that d{u, w) > d(u, v) = diam(G). A contradiction, since diam(G) is the maximum distance between any two vertices in G. Hence υ is not a cut-vertex. Similarly, u is also not a cut-vertex. Thus the desired result follows. • A family of u-v paths in a graph G is said to be internally disjoint if no vertex of G is an internal vertex of more than one of these paths. In the following theorem due to Whitney [1.1], we present an important characterization of a block.

Theorem 1.8 (Whitney). A graph G with η > 3 vertices is a block if and only if any two vertices of G are connected by at least two internally disjoint paths. Proof Necessity Let G be a block. Consider any two vertices u and ν with distance d(u, υ). We prove that there are at least two internally disjoint u-v paths. Proof is by induction on d(u, v). Consider, first, the case d(u, v) = l. Clearly in this case u and ν are adjacent. Let e = (u, v). Since G is a block, it has no cut-vertices. In particular, neither u nor υ is a cut-vertex. So the graph G - e is connected and there is a u - υ path in G not containing the edge e. In other words G has two internally disjoint u-v paths whenever d(u, v) = 1. Assume that for any two vertices at distance less than k, there are at least two internally disjoint paths between these two vertices. Let d(u, v) = k 2:2. Consider a u-v path of length k and let w be the last vertex that precedes ν on this path. Then d(u, w) = k - 1, and so by the induction hypothesis there are two internally disjoint u-w paths Ρ and Q in G. If one of these paths contains v, then clearly G has two internally disjoint u-v paths. Suppose that neither Ρ nor Q contains v. Since G is a block, G - w is connected. So, in G there is a u - ν path P* that does not contain w. Let χ be the last vertex of P* that is also in one of the two paths Ρ and Q. Since u is present in all these three paths such a vertex χ exists. Without loss of generality, assume that χ is in P. Then, G has two internally disjoint u-v paths; one of them is a concatenation of the u-x section of Ρ and the x-v section of P* and the other is obtained by adding the edge (w, v) to Q. Sufficiency If any two vertices of G are connected by at least two internally disjoint paths, then clearly G is connected and also, by Theorem 1.6, G has no cut-vertex. Hence G is a block. •

22

BASIC CONCEPTS

Several equivalent characterisations of a block are given in Exercise 1.24. A block G is also called 2-connected or biconnected because at least two vertices have to be removed from G to disconnect it. The concepts of vertex and edge connectivities and generalized versions of Theorem 1.8, called Menger's theorems, will be discussed in Chapter 8.

1.8

ISOMORPHISM AND 2-ISOMORPHISM

The two graphs shown in Fig. 1.17 are "seemingly" different. However, one can be redrawn to look exactly like the other. Thus these two graphs are

Figure 1.17. Isomorphic graphs, (a) G,. (b) G . 2

ISOMORPHISM AND 2-ISOMORPHISM

23

"equivalent" in some sense. This equivalence is stated more precisely as follows: Two graphs G, and G are said to be isomorphic if there exists a one-to-one correspondence between their vertex sets and a one-to-one correspondence between their edge sets so that the corresponding edges of G, and G are incident on the corresponding vertices of G and G . In other words, if vertices v and v in G, correspond respectively to vertices v\ and v' in G , then an edge in G, with end vertices u, and v should correspond to an edge in G with v\ and v as end vertices and vice versa. According to this definition, the two graphs shown in Fig. 1.17 are isomorphic. The correspondences between their vertex sets and edge sets are as follows: 2

2

t

1

2

2

2

2

2

2

2

Vertex correspondence u, «-»i> , 2

υ ** Κ 4

Edge

>

*~*

2

V

'i

V

>

y

3 *"* ' \ V

V «-» v' . s

5

correspondence e *-*e\ , i

e

2

e\,

e +*e , 3

t « 4 * *

^

2

t

«6>

Consider next the two separable graphs G, and G shown in Fig. 1.18a and 1.186. These two graphs are not isomorphic. Suppose we "split" the cut-vertex u, in G, into two vertices so as to get the two edge-disjoint graphs shown in Fig. 1.18c. If we perform a similar "splitting" operation on the cut-vertex v\ in G , we get the two edge-disjoint graphs shown in Fig. 1.18d. It may be seen that the graphs of Fig. 1.18c and 1.18d are isomorphic. Thus the two graphs G, and G become isomorphic after splitting the cut-vertices. Such graphs are said to be l-isomorphic. 2-isomorphism to be defined next is a more general type of isomorphism. Two graphs G, and G are 2-isomorphic if they become isomorphic after one or more applications of either or both of the following operations: 2

+

2

2

2

1. Splitting a cut-vertex in G, and/or G into two vertices to get two edge-disjoint graphs. 2. If one of the graphs, say G,, has two subgraphs G[ and G" that have exactly two common vertices u, and v , the interchange of the names 2

2

* Two graphs are edge-disjoint if they have no edge in common, and vertex-disjoint if they have no vertex in common.

24

BASIC CONCEPTS

(c)

(d)

Figure 1.18. 1-isomorphic graphs, (a) G,. (b) G . (c) Graph after splitting u, in G,. (d") Graph after splitting v\ in G . 2

2

of these vertices in one of the subgraphs. (Geometrically this operation is equivalent to turning around one of the subgraphs G\ and G" at the common vertices v and v .) l

2

Consider the graphs G and G shown in Fig. 1.19a and 1.196. After performing in G a splitting operation on vertex v and a turning around operation at vertices υ and v , we get the graph G[ shown in Fig. 1.19c. A splitting operation on vertex v' in G results in the graph G shown in Fig. 1.19d. These two graphs Gj and G' are isomorphic. Hence G, and G are 2-isomorphic. The next theorem presents an important result on 2-isomorphic graphs. 2

x

x

2

χ

2

2

2

2

2

2

FURTHER READING

(c)

Figure 1.19. Illustration of 2-isomorphism. (a) G

v

25

(d) (b) G . (c) G\. (d) G' . 2

2

Theorem 1.9. Two graphs G, and G are 2-isomorphic if and only if there exists a one-to-one correspondence between their edge sets such that the circuits in one graph correspond to the circuits in the other. • 2

The fact that circuits in Gj will correspond to circuits in G , when G, and G are 2-isomorphic, is fairly obvious. However, the proof of the converse of this result is too lengthy to be discussed here. Whitney's original paper [1.2] on 2-isomorphic graphs discusses this. 2

2

1.9

FURTHER READING

Harary [1.3], Berge [1.4], Bondy and Murty [1.5], Lovasz [1.6], Chartrand and Lesniak [1.7], and Bollobas [1.8] are excellent references for several of

26

BASIC CONCEPTS

the topics covered in this book. Berge [1.4] also discusses hypergraphs and matroids. Bondy and Murty [1.5] include a collection of a number of unsolved problems in graph theory. Lovasz [1.6] discusses graph theory through a number of problems and solutions. Among other things, Bollobas [1.8] gives an introduction to extremal graph theory and random graphs. See Swamy and Thulasiraman [1.9] for a detailed discussion of graph algorithms and electrical network applications of graph theory. For an elegant introduction to topics in graph theory including matroids see Wilson [1.10]. Other textbooks on graph theory include Liu [1.11] and Deo [1.12]. Several books and monographs devoted exclusively to special topics such as extremal graph theory, random graphs, and so forth are also available. These will be referred to in the appropriate chapters of this book.

1.10

EXERCISES

1.1

Let G be a graph with η vertices and m edges such that the vertices have degree k or k + 1. Prove that if G has n vertices of degree k and k+i vertices of degree k + 1, then n = (k + \)n - 2m. k

n

k

1.2

Prove or disprove: (a) The union of any two distinct closed walks joining two vertices contains a circuit. (b) The union of any two distinct paths joining two vertices contains a circuit.

1.3

If in a graph G there is a path between any two vertices a and b, and a path between any two vertices b and c, then prove that there is a path between a and c.

1.4

Let P, and P be two distinct paths between any two vertices of a graph. Prove that Ρ, Θ P is a circuit or the union of some edgedisjoint circuits of the graph. 2

2

1.5

Prove that a closed trail with all its vertices of degree 2 is a circuit.

1.6

Show that if two distinct circuits of a graph G contain an edge e, then in G there exists a circuit that does not contain e.

1.7

Show that in a simple graph G with S(G) ^ k there is a path of length at least k. Also show that G has a circuit of length at least k + 1, if ik>2.

1.8

Prove that a graph G = (V, E) is connected if and only if, for every partition (V , V ) of V with V and V nonempty, there is an edge of G joining a vertex in V to a vertex in V . x

2

x

x

1.9

2

2

Prove that a simple graph G with η vertices and k components can

EXERCISES

27

have at most (n - k)(n - k + l ) / 2 edges. Deduce from this result that G must be connected if it has more than (n - l)(n - 2)12 edges. 1.10 Prove that if a graph G (connected or disconnected) has exactly two vertices of odd degree, then there must be a path joining these two vertices. 1.11 Prove that if a simple graph G is not connected, then its complement G is. 1.12

If G is a graph with η vertices and m edges such that m < η - 1, then prove that G is not connected.

1.13

If, for a graph G with η vertices and m edges, m^n, G contains a circuit edge.

1.14

Prove that a simple η-vertex graph G is connected if 5(G) ^ (η - 1)/ 2.

1.15

Show that a simple graph G with at least two vertices contains two vertices of the same degree.

1.16

Show that if a graph G = (V, E) is simple and connected but not complete, then G has three vertices u, v, and w such that the edges (w, υ) and (υ, w) are in Ε and the edge (u, w) is not in E.

1.17

Show that if a simple graph G has diameter greater than 3, then G has diameter less than 3.

then prove that

1.18 The girth of a graph G is the length of a shortest circuit in G. If G has no circuits, we define the girth of G to be infinite. Show that a ^-regular graph of girth 4 has at least 2k vertices. 1.19

A simple grapji G is self-complementary if it is isomorphic to its complement G. Prove that the number of vertices of a selfcomplementary graph must be of the form 4k or 4k + 1 where k is an integer.

1.20

Prove that an «-vertex simple graph is not bipartite if it has more than n /4 edges. 2

1.21 Let G be a simple graph with maximum degree Δ. Show that there exists a Δ-regular graph containing G as an induced subgraph. 1.22

Construct a simple cubic graph with 2n(n s 3) vertices having no triangles. (Note: A graph is cubic if it is 3-regular. A triangle is a circuit of length 3.)

1.23

Prove that if is a cut-vertex of a simple graph G, then υ is not a cut-vertex of G.

28

BASIC CONCEPTS

1.24

Prove that the following properties of a graph G with η ^ 3 vertices are equivalent: (a) G is nonseparable. (b) Every two vertices of G lie on a common circuit. (c) For any vertex υ and any edge e of G there exists a circuit containing both. (d) Every two edges of G lie on a common circuit. (e) Given two vertices and one edge of G, there is a path joining the vertices that contains the edge. (f) For every three distinct vertices of G there is a path joining any two of them that contains the third. (g) For every three distinct vertices of G there is a path joining any two of them that does not contain the third.

1.25

Show that if a graph G has no circuits of even length, then each block of G is either K or K or a circuit of odd length. t

2

1.26

Show that a connected graph that is not a block has two blocks that have a common vertex. Let b(v) denote the number of blocks of a graph G = (V, E) containing vertex v. Show that the number of blocks of G is equal to ρ + Σ ip(v) - 1), where ρ is the number of components of G.

1.27

Let c(B) denote the number of cut-vertices of a connected graph G that are vertices of the block B. Then the number c(G) of cut-vertices of G is given by c(G)-l=

Σ

[c(B)-l].

all b l o c k s

1.28

A bridge of a graph G is an edge e such that G - e has more components than G. Prove the following:

Figure 1.20

REFERENCES

29

Figure 1.21

(a) An edge e of a connected graph G is a bridge if and only if there exist vertices u and w such that e is on every u-w path of G. (b) An edge of a graph G is a bridge of G if and only if it is on no circuit of G. 1.29

Are the graphs shown in Fig. 1.20 isomorphic? Why?

1.30

Show that the two graphs of Fig. 1.21 are not isomorphic.

1.31 Determine all nonisomorphic simple graphs of order 3 and those of order 4. (Note: There are exactly four nonisomorphic graphs on three vertices, and 11 on four vertices.) 1.32

1.11

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Prove that any two simple connected graphs with η vertices, all of degree 2, are isomorphic.

REFERENCES

H. Whitney, "Nonseparable and Planar Graphs," Trans. Amer. Math. Soc, Vol. 34, 339-362 (1932). H. Whitney, "2-Isomorphic Graphs," Am. J. Math., Vol. 55, 245-254 (1933). F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976. L. Lovasz, Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979. G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Coles, Pacific Grove, Calif. 1986. B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979.

30

1.9

BASIC CONCEPTS

Μ. Ν. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, New York, 1981. 1.10 R. J. Wilson, Introduction to Graph Theory, Oliver and Boyd, Edinburgh, 1972. 1.11 C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. 1.12 N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Englewood Cliffs, N.J., 1974.

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 2

TREES, CUTSETS, AND CIRCUITS

The graphs that are encountered in most of the applications are connected. Among connected graphs trees have the simplest structure and are perhaps the most important ones. If connected graphs are important, then the set of edges disconnecting a connected graph should be of equal importance. This leads us to the concept of a cutset. In this chapter we study trees and cutsets and several results associated with them. We also bring out the relationship between trees, cutsets, and circuits.

2.1

TREES, SPANNING TREES, AND COSPANNING TREES

A graph is said to be acyclic if it has no circuits. A tree is a connected acyclic graph. A tree of a graph G is a connected acyclic subgraph of G. A spanning tree of a graph G is a tree of G having all the vertices of G. A connected subgraph of a tree Τ is called a subtree of T. Consider, for example, the graph G shown in Fig. 2.1a. The graphs G, and G of Fig. 2.16 are two trees of G. The graphs G and G of Fig. 2.1c are two of the spanning trees of G. The cospanning tree T* of a spanning tree Γ of a graph G is the subgraph of G having all the vertices of G and exactly those edges of G that are not in T. Note that a cospanning tree may not be connected. The cospanning trees G* and G* of the spanning trees G and G of Fig. 2.1c are shown in Fig. 2.1d. The edges of a spanning tree Τ are called the branches of T, and those of the corresponding cospanning tree T* are called links or chords. 2

3

3

4

4

31

32

TREES, CUTSETS, AND CIRCUITS

Id

Figure 2.1. Trees, spanning trees, and cospanning trees, (a) Graph G. (b) Trees G, and G2 of G. (c) Spanning trees G3 and G4 of G. (d) Cospanning trees GJ and GJ of G.

TREES, SPANNING TREES, AND COSPANNING TREES

"4

"3

0

C

*3

C

33

0

*4

id)

Figure 2.1. (Continued) A spanning tree Τ uniquely determines its cospanning tree T*. As such, we refer to the edges of Τ * as the chords or links of T. We now proceed to discuss several properties of a tree. While the definition of a tree as a connected acyclic graph is conceptually simple, there exist several other equivalent ways of characterizing a tree. These are discussed in the following theorem. Theorem 2.1. The following statements are equivalent for a graph G with η vertices and m edges: 1. 2. 3. 4. 5.

G is a tree. There exists exactly one path between any two vertices of G. G is connected and m = η - 1. G is acyclic and m = η - 1. G is acyclic, and, if any two nonadjacent vertices of G are connected by an edge, then the resulting graph has exactly one circuit.

Proof 1=P2

See Exercise 1.2b.

2Φ3 We first note that G is connected because there exists a path between any two vertices of G. We prove that m = η - 1 by induction on the number of vertices of G. This is obvious for connected graphs with one or two vertices. Assume that this is true for connected graphs with fewer than η vertices. Consider any edge em G. The edge e constitutes the only path between the end vertices of e. Hence in G — e there is no path between these

34

TREES, CUTSETS, AND CIRCUITS

vertices. Thus G - e is not connected. Further, it should contain exactly two components, for otherwise the graph G will not be connected. Let G, and G be the two components of G - e. Let η and m, be, respectively, the numbers of vertices and edges in G Similarly, let n and m be denned for G . Then we have 2

λ

v

2

2

2

n =m + n l

2

and m = m, + m + 1. 2

Note that G and G satisfy the hypothesis of statement 2, namely, there exists exactly one path between any two vertices in G, and G . Since η, < η and n
{

2

2

m, = «, - 1 and m = n - 1. 2

2

Hence m = /j,-l + n - l + l = n - l. 2

3^4

Let the graph G be renamed as G ; that is, G = G. 0

0

Suppose that G has some circuits. Then let G, be the graph that results after removing from G a circuit edge, say e ; that is, G = G - e . Since G„ is connected, it follows from Theorem 1.4 that G, is also connected and has all the η vertices of G . Further, the number of edges in G, is equal to m-l. If G, is not acyclic, let e be a circuit edge of G,. Again the graph G = G, - e = G - e, - e should be connected and should have all the η vertices of G,. Also, G has m - 2 edges. If G is not acyclic, repeat this process until we get a connected graph G that is acyclic. Note that G has η vertices and m— ρ edges. Since G is connected and acyclic, it must be a tree. Hence it follows from the previous statement of the theorem that 0

0

{

x

0

x

0

2

2

2

0

2

2

2

p

p

p

m — ρ = η — 1. Since by hypothesis m = η - 1, we get ρ = 0. Thus the graph G = G is acyclic. 0

TREES, SPANNING TREES, AND COSPANNING TREES

35

4φ5 Let G,, G , . . . , G be the ρ components of G with n, and m, denoting, respectively, the numbers of vertices and edges in the component G,. Then 2

p

m = m, + m + · • · + m 2

p

and η = η, + « + · · · + n . 2

p

Each component G, is connected, and it is also acyclic because G is. Hence G, is a tree. Then, by statement 3 of the theorem, m, = n, = 1 for all 1 s ι < ρ . Hence we get ρ

m -

Σ

ρ

m,

=Σ

/= 1

ι=

(η, - 1) = η - ρ .

1

However, by hypothesis m= η- 1 . Hence we get p = l. Thus G consists of exactly one component. Hence it is connected. Since it is also acyclic, it is a tree. Then, by statement 2 of the theorem, there exists exactly one path between any two distinct vertices of G. Hence if we add an edge e = (υ,, υ ) to G, then this edge together with the unique path between i>, and υ will form exactly one circuit in the resulting graph. 2

2

5φ1 Suppose that G is not connected. Consider any two vertices v and v that are in different components of G. Then v and v are not connected in G. Addition of an edge (v , v ) to G does not produce a circuit since in G there is no path between v and v . This, however, contradicts the hypothesis. Hence the assumption that G is not connected is false. Thus G is connected. Since G is also acyclic, it must be a tree, by definition. • a

b

a

a

b

b

a

b

It should be clear that each of statements 1 through 5 of Theorem 2.1 represents a set of necessary and sufficient conditions for a graph G to be a tree. An immediate consequence of this theorem is the following.

36

TREES, CUTSETS, AND CIRCUITS

Corollary 2.1.1. Consider a subgraph G' of an w-vertex graph G. Let G' have η vertices and m' edges. Then the following statements are equivalent: 1. 2. 3. 4. 5.

G' is a spanning tree of G. There exists exactly one path between any two vertices of G'. G' is connected and m' = η - 1. G' is acyclic and m' - η - 1. G' is acyclic, and, if any two nonadjacent vertices of G' are connected by an edge, then the resulting graph has exactly one circuit. •

A condition that is not covered by Corollary 2.1.1 but can be proved easily is stated next. Corollary 2.1.2. A subgraph G' of an η-vertex graph G is a spanning tree of G if and only if G' is acyclic, connected, and has η - 1 edges. • It should now be obvious that a subgraph of an π-vertex graph G having any three of the following properties should be a spanning tree of G: 1. 2. 3. 4.

It It It It

has η vertices. is connected. has η - 1 edges. is acyclic.

The question then arises whether any two of these four properties will be sufficient to define a spanning tree. This question is answered next. (See also Exercise 2.3.) Theorem 2.2. A subgraph G' of an η-vertex graph G is a spanning tree of G if and only if G' is acyclic and has η - 1 edges. Proof. Necessity follows from Theorem 2.1, statement 4. To show the sufficiency, we have to prove that G' is connected and has all the η vertices of G. Let G' consist of ρ components G,, G , . . . , G , with «, denoting the number of vertices in component G,. Let n' be the number of vertices in G'. Then 2

p

ρ

n'

=Σ

η,.

ι= 1

Each G, is connected. It is also acyclic because G is. Thus each G, is a tree

TREES, SPANNING TREES, AND COSPANNING TREES

37

and hence has η, - 1 edges. Thus the total number of edges in G' is equal to Ρ

Σ (η,-1) = "'-/>· But by hypothesis n' - ρ = η - 1 . Since n' < n and ρ ^ 1, it is clear that the above equation is true if and only if n' - η and ρ = 1. Thus G' is connected and has η vertices. Since it is also acyclic, it is, by definition, a spanning tree of G. • Suppose a graph G has a spanning tree T. Then G should be connected because the subgraph Τ of G is connected and has all the vertices of G. Next, we would like to prove the converse of this result, namely, that a connected graph has at least one spanning tree. If a connected graph G is acyclic, then it is its own spanning tree. If not, let e, be some circuit edge of G. Then, by Theorem 1.4, the graph G, = G - e, is connected and has all the vertices of G. If G, is not acyclic, repeat the process until we get a connected acyclic graph G that has all the vertices of G. This graph G will then be a spanning tree of G. The results of this discussion are summarized in the next theorem. p

p

Theorem 2.3. A graph G is connected if and only if it has a spanning tree. • Since a spanning tree Γ of a graph G is acyclic, every subgraph of Τ is an acyclic subgraph of G. Is it then true that every acyclic subgraph of G is a subgraph of some spanning tree of G? The answer is "yes" as proved in the next theorem. Theorem 2.4. A subgraph G' of a connected graph G is a subgraph of some spanning tree of G if and only if G' is acyclic. Proof. Necessity is obvious. To prove the sufficiency, let Γ be a spanning tree of a graph G. Consider the graph G, = TU G'. It is obvious that G' is a subgraph of G,. G, is connected and has all the vertices of G because Γ is a subgraph of G,. If G, is acyclic, then it is a spanning tree of which G' is a subgraph, and the theorem is proved. (Note that if G, is acyclic, G, = Γ and G' is then a subgraph of T.) Suppose G, has a circuit C . Since G' is acyclic, it follows that not all the edges of C, are in G'. Thus C, must have at least one edge, say e,, which is x

38

TREES, CUTSETS, AND CIRCUITS

not in G'. Removal of this circuit edge e, from G, results in the graph G = G, - e,, which is also connected and has all the vertices of G,. Note that G' is a subgraph of G . If G is acyclic, then it is a required spanning tree. If not, repeat the process until a spanning tree of which G' is a subgraph is obtained. • 2

2

2

Next we prove an interesting theorem on the minimum number of pendant vertices, that is, vertices of degree 1, in a tree. Theorem 2.5. In a nontrivial tree there are at least two pendant vertices. Proof. Suppose a tree Τ has η vertices. Then, by Theorem 2.1, it has η - 1 edges. We also have, by Theorem 1.1, that η

Σ d{v,) = 2 x number of edges in Τ . Thus + d(v ) + ·•• + d(v„) = 2

2n-2.

This equation will be true only if at least two of the terms on its left-hand side are equal to 1, that is, Τ has at least two pendant vertices. •

2.2

/(-TREES, SPANNING /(-TREES, AND FORESTS

A k-tree* is an acyclic graph consisting of k components. Obviously, each component of a k-tree is a tree by itself. Note that a 1-tree is the same as a tree. If a k-tree is a spanning subgraph of a graph G, then it is called a spanning k-tree of G. The cospanning k-tree T* of a spanning &-tree Τ of G is the spanning subgraph of G containing exactly those edges of G that are not in T. For example, the graph of Fig. 2.2ft is a 2-tree of the graph G shown in Fig. 2.2a. A spanning 3-tree Τ of G and the corresponding cospanning 3-tree T* are shown in Fig. 2.2c and 2.2a". Let the k components of a spanning A>tree of an η-vertex graph G be denoted by T,, T ,..., T . If n, is the number of vertices in Γ,, then 2

k

η = «, + n + · · · + n . 2

k

' This definition of &-tree has been used extensively in the electrical network literature. In current graph theory literature this term has a different meaning. See Rose [2.1].

fc-TREES, SPANNING fc-TREES, AND FORESTS

39

(0 id) Figure 2.2. Illustrations of the definitions of a A:-tree, a spanning &-tree, and a cospanning fc-tree. (a) Graph G. (b) A 2-tree of G. (c) A spanning 3-tree Τ of G. (d) Cospanning 3-tree T*. Since each T is a tree, we have, by Theorem 2.1, i

m, = n, - 1 , where m, is the number of edges in T,. Thus the total number of edges in the spanning Λ-tree Γ is equal to *

Σ 1=1

k

tn ι {

= Σ (w, -

1) = η - k .

1=1

If m is the number of edges in G, then the cospanning A>tree T* will have m - n + k edges.

40

TREES, CUTSETS, AND CIRCUITS

A. forest of a graph G is a spanning fc-tree of G, where k is the number of components in G. If a graph G has ρ components, then for any spanning fc-tree of G, k 2: p . Since a forest Τ of G is a spanning Λ-tree of G with k= p, it is necessary that each component of Τ be a spanning tree of one of the components of G. Thus a forest Γ of a graph G with ρ components G,, G , . . . , G consists of ρ components T ,T ,... ,T such that T, is a spanning tree of G , l s i ' < 2

X

2

p

(

p

PThe co-forest T* of a forest Γ of a graph G is the spanning subgraph of G containing exactly those edges of G that are not in T. "9

"2

"3

"β

"7

»11

(C)

Figure 2.3. Forest and co-forest, (a) Graph G. (6) A forest Τ of G. (c) Co-forest Γ*.

FUNDAMENTAL CIRCUITS

41

Note that forest and spanning tree are synonymous in the case of a connected graph. A forest Τ and the corresponding co-forest T* of a graph are shown in Fig. 2.3.

2.3

RANK AND NULLITY

Consider a graph G with m edges, η vertices, and k components. The rank of G, denoted by p ( G ) , is defined equal to n- k, and the nullity of G, denoted by p-(G), is defined equal to m - η + k. Note that p(G) + ,*(G) = m . It follows from the definition of a forest and a co-forest that the rank p(G) of a graph G is equal to the number of edges in a forest of G, and the nullity p-(G) of G is equal to the number of edges in a co-forest of G. The numbers p ( G ) and /x.(G) are among the most important ones associated with a graph. As we shall see in Chapter 4, they define the dimensions of the cutset and circuit subspaces of a graph.

2.4

FUNDAMENTAL CIRCUITS

Consider a spanning tree 7 of a connected graph G. Let the branches of Τ be denoted by b , b ,..., b _ , and let the chords of Τ be denoted by c,, c , . . . , c _ , where m is the number of edges in G, and η is the number of vertices in G. While Τ is acyclic, by Theorem 2.1 the graph Τ U c, contains exactly one circuit C,. This circuit consists of the chord c, and those branches of Τ that lie in the unique path in Τ between the end vertices of c,. The circuit C, is called the fundamental circuit of G with respect to the chord c, of the spanning tree T. The set of all the m - η + 1 fundamental circuits C,, C , . . . , C _ of G with respect to the chords of the spanning tree Τ of G is known as the fundamental set of circuits of G with respect to T. An important feature of the fundamental circuit C, is that it contains exactly one chord, namely, chord c,. Further, chord c, is not present in any other fundamental circuit with respect to T. Because of these properties, the edge set of no fundamental circuit can be expressed as the ring sum of the edge sets of some or all of the remaining fundamental circuits. We also show in Chapter 4 that every circuit of a graph G can be expressed as the ring sum of some fundamental circuits of G with respect to a spanning tree of G. It is for these reasons that the "fundamental" circuits are called so. A graph G and a set of fundamental circuits of G are shown in Fig. 2.4. x

2

m

2

n

x

n + 1

2

m

n + 1

42

TREES, CUTSETS, AND CIRCUITS

10

Ic)

Figure 2.4. A set of fundamental circuits of a graph G. (a) Graph G. (b) A spanning tree Γ of G. (c) Set of five fundamental circuits of G with respect to T. (Chords are indicated by dashed lines.) 2.5

CUTSETS

A cutset 5 of a connected graph G is a minimal set of edges of G such that its removal from G disconnects G, that is, the graph G - S is disconnected. For example, consider the subset 5, = {e,, e , e , e, } of edges of the graph G in Fig. 2.5a. The removal of S, from G results in the graph G, = G - S, of Fig. 2.5i>. G, is disconnected. Furthermore, the removal of any proper subset of S, cannot disconnect G. Thus S, is a cutset of G. 3

7

0

43

CUTS

Figure 2.5. Illustration of the definition of a cutset, (a) Graph G. (b) G, = G - S,, 5, = {«?,,
2

2

7

5

0

2

2

2

2

5

7

Q

3

2

g

Consider next the set S = {e,, e , e , e , e }. The graph G = G - S shown in Fig. 2.5c is disconnected. However, the set S' = {e , e , e , e }, which is a proper subset of S , also disconnects G. The graph G = G - S' is shown in Fig. 2.5d. Thus 5 is not a cutset of G. Note that, by the definition of a cutset given above, if S is a cutset of a graph G, then the ranks of G and G-S differ by at least 1; that is, p(G)-p(G-5)>l. Seshu and Reed [2.2] define a cutset as follows: A cutset 5 of a connected graph G is a minimal set of edges of G such that removal of 5 connects G into exactly two components; that is, p(G) p ( G - 5 ) = l. The question now arises whether these two definitions of a cutset are equivalent. The answer is "yes" and its proof is left as an exercise (Exercise 2.15). 2

2

s

7

9

2

2

2

1

2

2

3

5

g

2

2

2.6

CUTS

We now define the concept of a cut, which is closely to that of a cutset.

44

TREES, CUTSETS, AND CIRCUITS

Consider a connected graph G with vertex set V. Let V, and V be two mutually disjoint subsets of V such that V=V UV ; that is, V, and V have no common vertices and together contain all the vertices of V. Then the set S of all those edges of G having one end vertex in V, and the other in V is called a cut of G. This is usually denoted by (V V ). Reed [2.3] refers to a cut as a seg (the set of edges of segregating the vertex set V). For example, for the graph G shown in Fig. 2.6, if V, = {u,, v , u , v ) and V - {v , υ , υ }, then the cut (V,, V ) of G is equal to the set {e , e , e } of edges. Note that the cut (V,, V ) of G is the minimal set of edges of G whose removal disconnects G into two graphs G, and G , which are induced subgraphs of G on the vertex sets V, and V . G and G may not be connected. If both these graphs are connected, then is also the minimal set of edges disconnecting G into exactly two components. Then, by definition, (V,, V ) is a cutset of G. Suppose that for a cutset 5 of G, V, and V are, respectively, the vertex sets of the two components G and G of G - 5. Then S is the cut (V ,V ). Thus we have the following theorem. 2

1

2

2

2

U

2

2

2

5

6

Ί

2

3

A

6

7

8

2

2

2

x

(ν^,^) 2

2

2

t

2

l

Figure 2.6. Definition of a cut. (a) Graph G. (b) Cut of G. 2

2

CUTS

45

Theorem 2.6

1. A cut (V,, V ) of a connected graph G is a cutset of G if the induced subgraphs of G on vertex sets V and V are connected. 2. If S is a cutset of a connected graph G, and V, and V are the vertex sets of the two components of G - S, then S = (V,, V ). • 2

t

2

2

2

Any cut (V,, V ) in a connected graph G contains a cutset of G, since the removal of {Vj, V ) from G disconnects G. In fact, we can prove that a cut in a graph G is the union of some edge-disjoint cutsets of G. Formally, we state this in the following theorem. 2

2

Theorem 2.7. A cut in a connected graph G is a cutset or union of edge-disjoint cutsets of G. • The proof of this theorem is not difficult, and it is left as an exercise (Exercise 2.19). Consider next a vertex v in a connected graph. The set of edges incident on u, forms the cut (v V— u , ) . The removal of these edges disconnects G into two subgraphs. One of these subgraphs containing only the vertex υ, is, by definition, connected. The other subgraph is the induced subgraph G' of 1

u

Ο «2

"2 O

».o-

-o»«

Figure 2.7. Illustration of Theorem 2.8. (a) Graph G. (b) Induced subgraph of G on the vertex set {v , v , ι> , i> , v , v , t>„}. 2

3

4

5

6

7

46

TREES, CUTSETS, AND CIRCUITS

G on the vertex set V- v Thus the cut (v ,V—v ) is a cutset if and only if G' is connected. However, G' is connected if and only if v is not a cut-vertex (Section 1.7). Thus we have the following theorem. v

l

i

x

Theorem 2.8. The set of edges incident on a vertex υ in a connected graph G is a cutset of G if and only if υ is not a cut-vertex of G. • For example, consider the separable graph G shown in Fig. 2.7a in which i>, is a cut-vertex. The induced subgraph of G on the vertex set V—v = {v , v , v , v , v , i> , v ) is shown in Fig. 2.7b. This subgraph consists of three components and is not connected. Thus the edges incident on the cut-vertex UJ do not form a cutset of G. l

2

2.7

3

4

s

7

6

g

FUNDAMENTAL CUTSETS

It was shown in Section 2.4 that a spanning tree of a connected graph can be used to obtain a set of fundamental circuits of the graph. We show in this section how a spanning tree can also be used to define a set of fundamental cutsets. Consider a spanning tree Γ of a connected graph G. Let b be a branch of 7. Now, removal of the branch b disconnects Γ into exactly two components Tj and 7 . Note that 7, and T are trees of G. Let V and V , respectively, denote the vertex sets of 7, and T . V and V together contain all the vertices of G. Let G, and G be, respectively, the induced subgraphs of G on the vertex sets V, and V . It can be seen that 7, and T are, respectively, spanning trees of G, and G . Hence, by Theorem 2.3, G, and G are connected. This, in turn, proves (Theorem 2.6) that the cut (V,, V ) is a cutset of G. This cutset is known as the fundamental cutset of G with respect to the branch b of the spanning tree 7 of G. The set of all the η -1 fundamental cutsets with respect to the η - 1 branches of a spanning tree Γ of a connected graph G is known as the fundamental set of cutsets of G with respect to the spanning tree 7. Note that the cutset (Vj, V ) contains exactly one branch, namely, the branch b of 7. All the other edges of (V ,V ) are links of 7. This follows from the fact that (V , V ) does not contain any edge of 7, or 7 . Further, branch b is not present in any other fundamental cutset with respect to 7. Because of these properties, the edge set of no fundamental cutset can be expressed as the ring sum of the edge sets of some or all of the remaining fundamental cutsets. We show in Chapter 4 that every cutset of a graph G can be expressed as the ring sum of the fundamental cutsets of G with respect to a spanning tree 7 of G. A graph G and a set of fundamental cutsets of G are shown in Fig. 2.8. 2

2

i

2

x

2

2

2

2

2

2

2

2

2

X

x

2

2

2

FUNDAMENTAL CUTSETS

47

ie) (0 Figure 2.8. A set of fundamental cutsets of a graph, (a) Graph G. (b) Spanning tree Τ of G. (c) Fundamental cutset with respect to branch e,. (d) Fundamental cutset with respect to branch e . (e) Fundamental cutset with respect to branch e . (/) Fundamental cutset with respect to branch e . 2

6

s

48

2.8

TREES, CUTSETS, AND CIRCUITS

SPANNING TREES, CIRCUITS, AND CUTSETS

In this section we discuss some interesting results that relate cutsets and circuits to spanning trees and cospanning trees, respectively. These results will bring out the "dual" nature of circuits and cutsets. They will also lead to alternate characterizations of cutsets and circuits in terms of spanning trees and cospanning trees, respectively. It is obvious that removal of a cutset S from a connected graph G destroys all the spanning trees of G. A little thought will indicate that a cutset is a minimal set of edges whose removal from G destroys all the spanning trees of G. However, the converse of this results is not so obvious. The first few theorems of this section discuss these questions and similar ones relating to circuits. Theorem 2.9. A cutset of a connected graph G contains at least one branch of every spanning tree of G. Proof. Suppose that a cutset S of G contains no branch of a spanning tree Γ of G. Then the graph G - S will contain the spanning tree Τ and hence, by Theorem 2.3, G - S is connected. This, however, contradicts that 5 is a cutset of G. • Theorem 2.10. A circuit of a connected graph G contains at least one edge of every cospanning tree of G. Proof. Suppose that a circuit C of G contains no edge of the cospanning tree T* of a spanning tree Τ of G. Then the graph G - T* will contain the circuit C. Since G - T* is the same as the spanning tree T, this means that the spanning tree Τ contains a circuit. However, this is contrary to the definition of a spanning tree. • Theorem 2.11. A set 5 of edges of a connected graph G is a cutset of G if and only if 5 is a minimal set of edges containing at least one branch of every spanning tree of G. Proof. Necessity If the set 5 of edges of G is a cutset of G, then, by Theorem 2.9, it contains at least one branch of every spanning tree of G. If it is not a minimal such set, then a proper subset S' of S will contain at least one branch of every spanning tree of G. Then G - S' will contain no spanning tree of G and it will be disconnected. Thus removal of a proper subset S' of the cutset S of G will disconnect G. This, however, would contradict the definition of a cutset. Hence the necessity. Sufficiency If S is a minimal set of edges containing at least one branch of every spanning tree of G, then the graph G - S will contain no spanning

SPANNING TREES, CIRCUITS, AND CUTSETS

49

tree, and hence it will be disconnected. Suppose S is not a cutset. Then a proper subset S' of S will be a cutset. Then, by the necessity part of the theorem, S' will be a minimal set of edges containing at least one branch of every spanning tree of G. This, however, will contradict that 5 is a minimal such set. Hence the sufficiency. • Theorem 2.11 gives a characterization of a cutset in terms of spanning trees. We would like to establish next a similar characterization for a circuit in terms of cospanning trees. Consider a set C of edges constituting a circuit in a graph G. By theorem 2.10, C contains at least one edge of every cospanning tree of G. We now show that no proper subset C of C has this property. It is obvious that C does not contain a circuit. Hence, by Theorem 2.4, we can construct a spanning tree Τ containing C. The cospanning tree T* corresponding to Τ has no common edge with C . Hence for every proper subset C of C, there exists at least one cospanning tree T* that has no common edge with C . In fact, this statement is true for every acyclic subgraph of a graph. Thus we have the following theorem. Theorem 2.12. A circuit of a connected graph G is a minimal set of edges of G containing at least one edge of every cospanning tree of G. • The converse of Theorem 2.12 follows next. Theorem 2.13. The set C of edges of a connected graph G is a circuit of G if it is a minimal set containing at least one edge of every cospanning tree of G. Proof. As shown earlier, the set C cannot be acyclic since there exists, for every acyclic subgraph G' of G, a cospanning tree not having in common any edge with G'. Thus C has at least one circuit C . Suppose that C is a proper subset of C. Then, by Theorem 2.12, C is a minimal set of edges containing at least one edge of every cospanning tree of G. This, however, contradicts the hypothesis that C is a minimal such set. Hence no proper subset of C is a circuit. Since C is not acyclic, C must be a circuit. • Theorems 2.12 and 2.13 establish that a set C of edges of a connected graph G is a circuit if and only if it is a minimal set of edges containing at least one edge of every cospanning tree of G. The new characterizations of a cutset and a circuit as given by Theorems 2.11, 2.12, and 2.13, clearly bring out the dual nature of the concepts of circuits and cutsets. This duality is explored further in Chapter 10, where we discuss the theory of matroids. The next theorem relates circuits and cutsets without involving trees.

TREES, CUTSETS, AND CIRCUITS

50

Theorem 2.14. A circuit and a cutset of a connected graph have an even number of common edges. Proof. Let C be a circuit and S a cutset of a connected graph G. Let V, and V be the vertex sets of the two connected subgraphs G, and G of G - S. If C is a subgraph of G, or of G , then obviously the number of edges common to C and S is equal to zero, an even number. Suppose that C and S have some common edges. Let us traverse the circuit C starting from a vertex, say v , in the set V . Since the traversing should end at υ,, it is necessary that every time we meet with an edge of S leading us from a vertex in V to a vertex in V , there must be an edge of S leading us from a vertex in V back to a vertex in V . This is possible only if C and S have an even number of common edges. • 2

2

2

x

x

x

2

2

x

Theorem 2.14 is a very important one. It forms the basis of the orthogonality relationship between cutsets and circuits. This relationship is discussed in Chapter 4. We would like to point out that the converse of Theorem 2.14 is not quite true. However, we show in Chapter 4 that a set S of edges of a graph G is a cutset (circuit) or the union of some edge-disjoint cutsets (circuits) if and only if 5 has an even number of edges in common with every circuit (cutset). Fundamental circuits and fundamental cutsets of a connected graph have been denned with respect to a spanning tree of a graph. It is, therefore, not surprising that fundamental circuits and cutsets are themselves related as proved next. Theorem 2.15 1. The fundamental circuit with respect to a chord of a spanning tree Γ of a connected graph consists of exactly those branches of Τ whose fundamental cutsets contain the chord. 2. The fundamental cutset with respect to a branch of a spanning tree Τ of a connected graph consists of exactly those chords of Τ whose fundamental circuits contain the branch. Proof 1. Let C be the fundamental circuit of a connected graph G with respect to chord c, of a spanning tree Τ of G. Let C contain, in addition to the chord c,, the branches b ,b ,... ,b of T. Suppose S,, 1 ^ / ^ k, is the fundamental cutset of G with respect to the branch b,,i£i^k, of T. The branch b is the only branch common to both Cand S,. The chord c, is the only chord in C. Since C and 5, must have an even number of common edges, it is necessary x

2

k

t

EXERCISES

51

that the fundamental cutset 5, contain c,. Next we show that no other fundamental cutset of Τ contains c,. Suppose the fundamental cutset S with respect to branch b of Τ contains c,. Then c, will be the only common edge between C and S . This will contradict Theorem 2.14. Thus the chord c, is present only in those cutsets denned by the branches b , b ,..., b. 2. Proof of this part is similar to that of part 1. • k+i

k+l

k+X

x

2.9

2

k

FURTHER READING

The concept of a tree is central in the development of several results relating circuits and cutsets of a graph. It so happens that the spanning trees, circuits, and cutsets of a connected graph are, respectively, the bases, circuits, and cocircuits of a matroid defined on the edge set of a graph. So the results of this chapter will be of help in understanding our development of matroid theory in Chapter 10. Electrical network theory is one of the earliest areas of applications of graph theory. The results of this chapter and those of Chapters 4 and 6 form the foundation of the graph-theoretic study of electrical networks. The pioneering work of Seshu and Reed [2.2] and textbooks by Kim and Chien [2.4], Chen [2.5], Mayeda [2.6] and Swamy and Thulasiraman [2.7] are highly recommended for further reading on this topic. Several questions relating to trees have been extensively studied in the literature. Some of these are discussed in the subsequent chapters of the book.

2.10

EXERCISES

2.1

Show that there are exactly 6 nonisomorphic trees on 6 vertices, and 11 on 7 vertices. Draw all of them.

2.2

Show that a tree is a bipartite graph.

2.3

Consider a subgraph G of a connected graph G having η vertices. Show that except for the pair (b) and (d), no other pair of the following conditions implies that G is a spanning tree of G: (a) G contains η vertices. (b) G contains η — 1 edges. (c) G is connected. (d) G contains no circuits. s

s

s

s

s

s

2.4

Prove that each pendant edge (the edge incident on a pendant vertex) in a connected graph G is contained in every spanning tree of G.

52

TREES, CUTSETS, AND CIRCUITS

2.5

Prove that each edge of a connected graph G is a branch of some spanning tree of G.

2.6

Prove that in a tree every vertex of degree greater than 1 is a cut-vertex.

2.7

Prove that each edge of a nonseparable graph G can be made a chord of some cospanning tree of G.

2.8

Prove or disprove: Any two edges of a nonseparable graph can be contained in some fundamental circuit.

2.9

Let d , d ,... ,d denote the degrees of the vertices of a nontrivial tree T. Prove that the number of pendant vertices of Τ equals x

2

n

2+

Σ

(4-2)

2.10

Prove that a nonseparable graph has nullity equal to 1 if and only if it is a circuit.

2.11

Show that the nullity of a graph is nonnegative. Give an example of a graph with nullity equal to 0.

2.12

Consider the following two operations on a graph G: (a) If only two edges, e = (υ,, v„) and e = {υ , υ ), are incident on vertex v , then replace e and e by a single edge connecting u, and v . (b) Replace any edge (v ,v ) by two edges (υ ,υ ) and (v , v ) where v is a new vertex not in G. Prove that the nullity of G is invariant under these operations. x

2

a

x

2

α

2

2

x

2

ι

β

2

a

a

2.13

A connected graph G is minimally connected if for every edge e of G the graph G - e is not connected. Prove that a connected graph is a tree if and only if it is minimally connected.

2.14

Prove that a subgraph G of a connected graph G is a spanning tree of G if and only if it is a maximal subgraph of G containing no circuits.

2.15

Show that a cutset of a connected graph G is a minimal set S of edges of G such that removal of S disconnects G into exactly two components; that is, p(G) - p(G - 5) = 1.

2.16

Prove that every connected graph contains a cutset.

2.17

Prove that a subgraph of a connected graph G can be included in a cospanning tree of G if and only if it contains no cutsets of G.

2.18

Prove that a subset S of edges of a connected graph G forms a cospanning tree of G if and only if it is a maximal subset of edges containing no cutsets of G.

s

EXERCISES

53

2.19

Prove that a cut in a connected graph G is a cutset or union of edge-disjoint cutsets of G.

2.20

Prove that every cutset of a nonseparable graph with more than two vertices contains at least two edges.

2.21

Prove that a graph G is nonseparable if and only if every two edges of G lie on a common cutset.

2.22

Let C be a circuit in a graph G. Let a and b be any two edges in C. Prove that there exists a cutset 5 such that S Π C = {a, b}.

2.23

Let Γ, and T be spanning trees of a connected graph G. Show that if e is any edge of T , then there exists an e d g e / o f T with the property that (7", - e) Uf (the graph obtained from Γ, by replacing e by / ) is also a spanning tree of G. Show also that 7", can be "transformed" into T by replacing the edges of T one at a time by the edges of T in such a way that at each stage we obtain a spanning tree of G. 2

x

2

2

2.24

x

2

(a) Let C, and C be two circuits in a graph G. Let e be an edge that is in both C, and C , and let e be an edge that is in C, but not in C . Prove that there exists a circuit C such that C C ( C , U C ) - e and e is in C . (b) Repeat part (a) when the term "circuit" is replaced by a "cutset." (Note: This result is one of the postulates used by Whitney [2.8] to define "circuits" of a matroid (Chapter 10).) 2

x

2

2

2

2

3

x

2

3

3

2.25

Let Τ be an arbitrary tree on k + 1 vertices. Show that if G is simple, connected and 5(G) ^ k, then G has a subgraph isomorphic to T.

2.26

Show that a graph G contains A: edge-disjoint spanning trees if and only if for each partition (V , V ,..., V ) of V the number of edges that have their end vertices in different parts of the partition is at least k(l- 1) (Nash-Williams [2.9] and Tutte [2.10]). x

2

t

2.27

A center of a graph G = (V, E) is a vertex u such that max„ , {d(u, v)} is as small as possible, where d(u, v) is the distance between u and v. Show that a tree has exactly one center or two adjacent centers.

2.28

The tree graph of an η-vertex connected graph G is the graph whose vertices are the spanning trees Γ,, T ,..., T of G, with 7", and T adjacent if and only if they have exactly n—2 edges in common. Show that the tree graph of any connected graph is connected (Cummins [2.11]). (Hint: See Exercise 2.23.)

ev

2

r

f

54

2.11

2.1

TREES, CUTSETS, AND CIRCUITS

REFERENCES

D. J. Rose, "On Simple Characterizations of fc-trees," Discrete Maths., Vol. 7, 317-322 (1974). 2.2 S. Seshu and M.B. Reed, Linear Graphs and Electrical Networks, AddisonWesley, Reading, Mass., 1961. 2.3 Μ. B. Reed, "The Seg: A New Class of subgraphs," IRE Trans. Circuit Theory, Vol. CT-8, 17-22 (1961). 2.4 W. H. Kim and R. T. Chien, Topological Analysis and Synthesis of Communication Networks, Columbia University Press, New York, 1962. 2.5 W. K. Chen, Applied Graph Theory: Graphs and Electrical Networks, NorthHolland, Amsterdam, 1971. 2.6 W. Mayeda, Graph Theory, Wiley-Interscience, New York, 1972. 2.7 Μ. N. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, New York, 1981. 2.8 H. Whitney, "On the Abstract Properties of Linear Dependence," Am. J. Math., Vol. 57, 509-533 (1935). 2.9 C. St. J. A. Nash-Williams, "Edge-Disjoint Spanning Trees of Finite Graphs," /. London Math. Soc, Vol. 36, 445-450 (1961). 2.10 W. T. Tutte, "On the Problem of Decomposing a Graph into η Connected Factors," /. London Math. Soc, Vol. 36, 221-230 (1961). 2.11 R. L. Cummins, "Hamilton Circuits in Tree Graphs," IEEE Trans. Circuit Theory, Vol. CT-13, 82-90 (1966).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 3

EULERIAN AND HAMILTONIAN GRAPHS

Many discoveries in graph theory can be traced to attempts to solve "practical" problems—puzzles, games, and so on. One of these problems was the celebrated Konigsberg bridge problem. This problem may be stated as follows: There are two islands in the Pregel river in Konigsberg, Germany. These islands were connected to each other and to the banks of the river by seven bridges as shown in Fig. 3.1a. The problem was to start at any one of the four land areas (marked as A, B, C, and D in Fig. 3.1a), walk through each bridge exactly once, and then return to the starting point, that is, to establish a closed walk across all the seven bridges without recrossing any of them. Many were convinced that there was no solution to this problem. Indeed, Euler, the great Swiss mathematician, proved so in 1736 and laid the foundation of the theory of graphs. Euler first showed that the problem was equivalent to establishing a closed trail along the edges of the graph of Fig. 3.16, where the vertices A, B, C, and D represent the land areas, and the edges represent the bridges connecting the land areas. He then generalized the problem and established a characterization of graphs in which such a closed trail exists. These graphs have come to be known as Eulerian graphs. The discussion of Section 3.1 concern these graphs. In 1859 Sir William Hamilton, another great mathematician, invented a game. The game challenges the player to establish a closed route along the edges of a dodecahedron that passes through each vertex exactly once. Hamilton's game, in graphical terms, is equivalent to determining whether in the graph of the dodecahedron shown in Fig. 3.2 there exists a spanning 55

56

EULERIAN AND HAMILTONIAN GRAPHS

(a)

(b)

Figure 3.1. (a) Königsberg bridge problem, (b) Graph of the Königsberg bridge problem.

18

17

Figure 3.2. Graph of Hamilton's game.

EULERIAN GRAPHS

57

circuit, that is, a circuit containing all the 20 vertices. It may be verified that the sequence of vertices, 1 , 2 , . . . ,20, 1 constitutes one such circuit in the graph. All graphs in which a spanning circuit exists have come to be known as Hamiltonian graphs. The discussions of Section 3.2 concern these graphs.

3.1

EULERIAN GRAPHS

An Euler trail in a graph G is a closed trail containing all the edges of G. An open Euler trail is an open trail containing all the edges of G. A graph possessing an Euler trail is an Eulerian graph. Consider the graph G, shown in Fig. 3.3a. The sequence of edges e,, e , e , e , e , e , e , e , e , e , e , and e constitutes an Euler trail in G,. Hence G, is Eulerian. In the graph G of Fig. 3.36, the sequence of edges e , e , e , e , e , e , e , e , e , e , , e , e , and e constitutes an open Euler trail. However, there is no Euler trail in G . Hence G is not Eulerian. A non-Eulerian graph G with no open Euler trail is shown in Fig. 3.3c. The following theorem gives simple and useful characterizations of Eulerian graphs. 2

3

4

5

7

s

9

6

7

s

9

l0

n

12

2

0

u

x

X2

2

3

4

5

6

l3

2

2

3

Figure 3.3. (a) G,, an Eulerian graph, (b) G , a non-Eulerian graph having an open Euler trail, (c) G , a non-Eulerian graph with no open Euler trail. 2

3

58

EULERIAN AND HAMILTONIAN GRAPHS

Figure 3.3. (Continued)

Theorem 3.1. The following statements are equivalent for a connected graph G: 1. G is Eulerian. 2. The degree of every vertex in G is even. 3. G is the union of edge-disjoint circuits.

EULERIAN GRAPHS

59

Proof ΙΦ2 Let Τ be an Euler trail in G. Suppose we traverse Τ starting from any vertex, say v , in G. Let Τ be x

f I

%2> ^2> * 3 > · · · '

·*1 ) €\>

—

» - ^ Λ ' ^r> - * r + l

—

where, of course, all the edges are distinct; the vertices x ,...,x may not all be distinct and some of these vertices may be v . Then it is clear that the pair of successive edges e, and e , , l s i < r - l , contributes 2 to the degree of the vertex x . In addition, vertex i>, gets a contribution of 2 to its degree from the initial and the final edges e, and e . Thus all the vertices are of even degree. 2

r

x

l +

l+l

r

2φ3 Since G is connected and every vertex in G has even degree, it follows that the degree of each vertex in G is greater than 1. Thus G has no pendant vertices. Hence G is not a tree by Theorem 2.5. This means that G has at last one circuit, say C,. Consider the graph G = G - C . Since every vertex in C, is also of even degree, it follows that every vertex in G must have even degree. However, G, may be disconnected. If G, is totally disconnected, that is, G, contains only isolated vertices, then G = C and statement 3 is proved. Otherwise G, has at least one circuit C. Consider next the graph G = G, - C = G - C, — C . Again every vertex in G has even degree. If G is totally disconnected, then G = C, U C . Otherwise repeat the procedure until we obtain a totally disconnected graph G„ = G - C, - C — · · · - C , where C,, C , . . . , C„ are circuits of G, no two of which have common edges. Then x

x

x

x

2

2

2

2

2

2

2

2

2

2

n

G = C, U C U · · · U C„ 2

and statement 3 is proved. 3φ1 Let G be the union of the edge-disjoint circuits C,, C , . . . , C . Consider any of these circuits, say C,. Since G is connected, there must be at least one circuit, say C , which has a common vertex v with C . Let T be a closed trail beginning at u, and traversing C, and C in succession. This trail obviously contains all the edges of C, and C . Again, since G is connected, T must have a common vertex v with at least one circuit, say C , different from C, and C . The closed trail Γ , beginning at i» and traversing T and C in succession will contain all the edges of C,, C , and C . Repeat this procedure until the closed trail T ... containing all the edges of C,, C , . . . , C„ is obtained. This closed trail is an Euler trail in G. Hence G is Eulerian. • 2

x

2

n

x

X2

2

2

l2

2

2

3

2

2

x2

3

x2i

2

23

3

n

60

EULERIAN AND HAMILTONIAN GRAPHS

By this theorem, the graph G, of Fig. 3.3a is Eulerian because every vertex in G is of even degree, whereas the graphs G and G of Fig. 3.3ft and 3.3c are not Eulerian since they contain some vertices of odd degree. It may also be verified that the Eulerian graph G, is the union of the edge-disjoint circuits whose edge sets are 2

x

3

{e , e , e } , 3

7

g

{e , e , e } , 2

{ i> e

9

\\i

e

10

e

\ i i

•

The following result is a consequence of statement 3 of Theorem 3.1. Corollary 3.1.1. Every vertex of an Eulerian graph is contained in some circuit. • Though an Euler trail does not exist in a graph that contains some vertices of odd degree, it is possible to construct in such a graph a set of edge-disjoint open trails that together contain all the edges of the graph. This is proved in the next theorem. Theorem 3.2. Let G = (V, E) be a connected graph with 2k odd degree vertices, k^ 1. Then Ε can be partitioned into subsets E ,E ,...,E such that each E constitutes an open trail in G. l

2

k

i

Proof. Let r, and s , 1 £ i k, be the 2k odd degree vertices of G. Now to G add k new vertices iv,, w ,... ,w together with 2k edges (r,, w,) and (s,, w,), 1 < ι < &. In the resulting graph G' every vertex is of even degree, and hence G' is Eulerian. It may be noted that in any Euler trail of G \ the edges (r,, iv,) and (s , w,) for any 1 s i < k appear consecutively. Removal of these 2k edges will then result in k edge-disjoint open trails of G such that each edge of G is present in precisely one of these trails. These open trails give the required partition of E. • t

2

k

f

Corollary 3.2.1. Let G be a connected graph with exactly two odd degree vertices. Then G has an open trail (which begins at one of the odd degree vertices and ends at the other) containing all the edges of G. • For example, the graph G of Fig. 3.3ft has exactly two odd degree vertices v and v , and the open trail {e,, e , e , e , e , e , e , c , e , e , e , u > \ i ) contains all the edges of G . This trail begins at v and ends at u . The graph G of Fig. 3.3c has four odd degree vertices. This graph has two edge-disjoint open trails constituted by the following sets of edges: 2

6

e

3

2

3

4

5

6

7

g

9

10

u

e

2

3

6

3

EULERIAN GRAPHS

61

»4

Figure 3.4. A graph randomly Eulerian from two vertices. { i e

2i

e

ι

{ 6'

e

e

T

e

C

ii

E

4> ^5}

8 ' 9i e

W

\\i

e

e

e

n }

·

A graph G is said to be randomly Eulerian from a vertex υ if, whenever we start from υ and traverse along the edges of G in an arbitrary way, we eventually obtain an Euler trail. It should be noted that if a graph G is randomly Eulerian from a vertex v, then it should be possible to extend every closed v-v trail not containing all the edges to an Euler trail of G. In other words, if an Eulerian graph G is not randomly Eulerian from a vertex 1», then there must be a closed v-v trail containing all the edges incident on ν but not containing all the edges of G. For example, consider the Eulerian graph of Fig. 3.4. This graph is randomly Eulerian from vertices v and v . It is not randomly Eulerian from the other vertices. It may be verified that for each vertex u, different from v and v there exists a closed v v, trail containing all the edges incident on v, but not containing all the edges of G. For example, the closed v -v trail consisting of the edges e , e , e , and e has this property. The next theorem gives a characterization of a graph that is randomly Eulerian from a vertex v. x

2

x

2

r

3

4

x

2

3

3

Theorem 3.3. An Eulerian graph G is randomly Eulerian from a vertex ν if and only if every circuit of G contains v. Proof Necessity Suppose graph G is randomly Eulerian from a vertex v. Assume that there exists a circuit C in G that does not contain v. Consider the graph G' = G - C. Every vertex in G' is of even degree. G' may not be connected. However, G", the component of G' containing v, is Eulerian, and it contains all the edges incident on v. Thus in G", there exists an Euler

62

EULERIAN AND HAMILTON IAN GRAPHS

trail Τ starting and ending at vertex v. This trail necessarily contains all the edges incident on v. Therefore it cannot be extended to include the edges of C. This contradicts the fact that G is randomly Eulerian from v. Sufficiency Let vertex υ in an Eulerian graph G be present in every circuit of G. Assume that G is not randomly Eulerian from v. Then there exists a closed v-v trail Τ containing all the edges incident on ν but not containing all the edges of G. Further, there exists a vertex u Φ ν such that it is the end vertex of an edge not in T. On removing from G the edges of T, a graph G' in which υ is an isolated vertex results. In G' every vertex is of even degree. So the component of G' containing u is an Eulerian graph. By Corollary 3.1.1 there is a circuit containing w. This circuit obviously does not contain vertex v. This contradicts the hypothesis that ν is in every circuit of G. • It may be verified that in the graph G of Fig. 3.4, vertices v and v are present in every circuit of G. Thus G is randomly Eulerian from both these vertices. On the other hand, for each one of the other vertices there exists a circuit not containing it. A graph is randomly Eulerian if it is randomly Eulerian from each of its vertices. It then follows from Theorem 3.3 that all the vertices of a randomly Eulerian graph G are in exactly one circuit C of G and there is no other circuit in G. In other words, G is randomly Eulerian if and only if it is a circuit. The characterizations of Eulerian graphs given in Theorem 3.1 are not constructive in nature. An efficient algorithm to construct an Euler trail in an Eulerian graph will be presented in Chapter 11 (Section 11.6) where we shall also discuss, among other things, the well-known Chinese postman problem, an important application of Eulerian graphs. x

3.2

2

HAMILTONIAN GRAPHS

A Hamilton circuit in a graph G is a circuit containing all the vertices of G. A Hamilton path in G is a path containing all the vertices of G. A graph G is defined to be Hamiltonian if it has a Hamilton circuit. The graph G, shown in Fig. 3.5a is Hamiltonian because the sequence of edges e,, e , e , e , e , e constitutes a Hamilton circuit in G,. The graph of Fig. 3.56 has a Hamilton path formed by the edges e,, e , e* , e , but it has no Hamilton circuit. Whereas an Euler trail is a closed walk passing through each edge exactly once, a Hamilton circuit is a closed walk passing through each vertex exactly once. Thus there is a striking similarity between an Eulerian graph and a Hamiltonian graph. This may lead one to expect that there exists a simple, useful, and elegant characterization of a Hamiltonian graph, as in the case 2

3

4

5

6

2

3

4

HAMILTONIAN GRAPHS

63

(«)

Figure 3.5. (a) A Hamiltonian graph. (b) A non-Hamiltonian graph having a Hamilton path.

101

of Eulerian graphs. Such is not the case; in fact, development of such a characterization is a major unsolved problem in graph theory. However, several sufficient conditions have been established for a simple graph to be Hamiltonian. (Recall that a graph is simple if it has neither parallel edges nor self-loops.) We consider some of these conditions in this section. A sequence d, s d s • · · < d„ is said to be graphic if there is a graph G with η vertices v , v v„ such that the degree d(v,) of u, equals d, for each i. (d,, d , . . . , d„) is then called the degree sequence of G. If 2

x

2

2

d, < d, < 1

and S*:

2

···s

d π

d,*
are graphic sequences such that d* > d, for 1 < / < n, then S* is said to majorize S. The following result is due to Chvatal [3.1]. Theorem 3.4. A simple graph G = (V, E) of order n, with degree sequence • · s d„, is Hamiltonian if d,
(3.1) Proof. First note that if d s A:, then the number of vertices with degrees k

64

EULERIAN AND HAMILTONIAN GRAPHS

not exceeding k is at least k. Similarly if d _ s η - k, then the number of vertices whose degrees are not exceeded by η - k is at least k + 1. Further, if a graphic sequence satisfies (3.1), then so does every graphic sequence that majorizes it. We now prove the theorem by contradiction. Let there be a simple non-Hamiltonian graph whose degree sequence satisfies (3.1). Then this graph is a spanning subgraph of a simple maximal non-Hamiltonian graph G = (V, E) whose degree sequence d, < d < · · • < d„ also satisfies (3.1). Let u and υ be two nonadjacent vertices in G such that d(w) + d(v) is as large as possible and d(u) s d(v). Since G is maximal non-Hamiltonian, it follows that addition of an edge joining u and υ will result in a Hamiltonian graph. Thus in G there is a Hamilton path « = u , w , u , . . . , u = υ with u and ν as the end vertices (Fig. 3.6). Let n

k

2

2

x

3

n

5 = {i|(u,, « , + , ) £ £ } , T={i\(u„u )EE}. n

Now there is n o / e S Π Τ. For if jε S Π Τ, then the edges («,, w ) and (u , u„) would be in G, and so the cyclic sequence of vertices Uj, u i,..., u,, u , u , . . · , « „ , w, would form a Hamilton circuit in G. Since the vertex M„ = υ is neither in S nor in 7\ it follows that S U 7 C { 1 , 2 , . . . , η - 1}. Therefore, J+1

!+1

/+2

d(«) + d(u) = | 5 | + | T | < n and <*(")< 3 « .

where |A"| denotes the number of elements in set X. Since S Π Τ = 0 , no w with / £ 5 is adjacent to v. The choice of d(u) and d(i>) then implies that for jES, d ( w ) < d ( « ) . Thus there are at least \S\ = d(u) vertices whose degrees do not exceed d(u). So if we set k = d(«), then we get d ^ Λ < 5 « , and therefore by (3.1) d _ ^nk. This means that there are at least k + 1 vertices whose degrees are not exceeded by η - k. The vertex u can be adjacent to at most k of these k + 1 vertices because d(u) = Thus there is a vertex w with d(w) > π - k, which is not adjacent to u. But then d(u) + d[w)szn>d(u) + d(v) contradicting the choice of d(u) and d(v). • y

;

t

n

Figure 3.6

k

HAMILTONIAN GRAPHS

65

We establish, in the following corollary, the sufficient conditions developed by Dirac [3.2], Ore [3.3], Posa [3.4], and Bondy [3.5] for a graph to be Hamiltonian. Corollary 3.4.1. A simple graph G = (V,E) of order n > 3 with degree sequence d, < d < · · · < d is Hamiltonian if one of the following conditions is satisfied: 2

1. (Dirac)

2. (Ore)

n

l<Jt|n. t

(κ, v)0Εφ

d(u) + d(v)> n.

3. (Posa) 1k. 4. (Bondy) j < k, d < j , d < k - 1 Φ d + d > k

;

k

t

k

n.

Proof. We prove by showing that all these conditions imply (3.1). 1Φ2 Clearly any degree sequence that satisfies condition 1 also satisfies condition 2. If this is not true, then there exists a t such that 1 =£ t< \n and

2Φ3

Now suppose there exists an / with l< t and (v„

v,)0E.

Then

d, + d, < 2 d , < 2 · \n = n, contradicting condition 2. Therefore the induced subgraph of G on the vertices i>,, v ,..., v, is a complete graph. Since d, ^ t, every vertex υ,, 1 ^ ι' ^ f, is adjacent to at most one vertex v , r + l < ; ' t. So there exists a vertex v f + l < / < n , which is not adjacent to any v,, l ^ i ^ t . Thus d < π - t - 1. But then 2

t

r

t

d, + d < r + n - f- l ;

Thus were have (v„

v )0E f

and d, + d . < η contradicting condition 2. y

If this is not true, then there exists a j
f

k

;

;

Therefore we have / < 5/1 and d, s t contradicting condition 3. 4φ(3.1) If this is not true, then there exists a t such that d, and d„_, < η — t - 1. But then

t < \n

66

EULERIAN AND HAMILTONIAN GRAPHS

d, + d _,
n-

n

n - l ,

=

contradicting condition 4.

t-\

•

It is easy to see that if a graphic sequence satisfies any one of the conditions stated in Theorem 3.4 and Corollary 3.4.1, so does every graphic sequence that majorizes it. Chvatal's condition, the weakest of these five conditions, has the interesting property that it is the best condition of this kind. That is, if a graphic sequence fails to satisfy Chvatal's condition, then it is majorized by the degree sequence of a non-Hamiltonian graph [3.1]. Though, in general, it is difficult to establish the non-Hamiltonian nature of a graph, in certain cases it may be possible to do so by the use of some shrewd arguments. This is illustrated in Liu [3.6] through the following example. Consider the graph G shown in Fig. 3.7. We show that there is no Hamilton path in G. Among all the edges incident on any vertex at most two can be included in any Hamilton path. In the graph G, vertex i> is of degree 5, and hence at least three of the edges incident on u cannot be included in any Hamilton path. The same is true of the vertices v and v . Since the degrees of v ,v , v , and u are equal to 3, at least one of the three edges incident on each of 8

8

l0

6

12

16

»4

Figure 3.7. A non-Hamiltonian graph.

2

4

FURTHER READING

67

these vertices cannot be included in a Hamilton path. Thus at least 13 of the 27 edges of G cannot be included in any Hamilton path. Hence there are not enough edges to form a Hamilton path on the 16 vertices of G. Thus G has no Hamilton path. A graph is randomly Hamiltonian from a vertex υ if any path starting from υ can be extended to be a Hamilton v-v circuit. A graph is randomly Hamiltonian if it is randomly Hamiltonian from each of its vertices. The following theorem completely characterizes randomly Hamiltonian graphs. Proof of this theorem may be found in Behzad and Chartrand [3.7]. Theorem 3.5. A simple graph of order η is randomly Hamiltonian if and only if it is a circuit or a complete graph or the complete bipartite graph K , the last one being possible only when η is even. • nl2nl2

We conclude this section with a reference to the traveling salesman problem. The problem is as follows: A salesman is required to visit a number of cities. What is the route he should take, if he has to start at his home city, visit each city exactly once, and then return home traveling the shortest distance? Suppose we represent cities by the vertices of a graph and roads by edges connecting the vertices. The length of a road may be represented as a weight associated with the corresponding edge. If between every pair of vertices there is a road connecting them, then it may be seen that the traveling salesman problem is equivalent to finding a shortest Hamilton circuit in a complete graph in which each edge is associated with a weight/ In a complete graph of order η there exist (n - l ) ! / 2 Hamilton circuits. One "brute-force" approach to solving the traveling salesman problem is to generate all the (n - l)!/2 Hamilton circuits and then pick the shortest one. The labor in this approach is too great (even for a computer), even for values of η as small as 50. For arbitrary values of n, no efficient algorithm for solving this problem exists. More discussions on the traveling salesman problem and related algorithms may be found in Lin and Kernighan [3.8], Belmore and Nemhauser [3.9], Held and Karp [3.10] and [3.11], Rosenkrantz, Stearns, and Lewis [3.12], and Lawler, Lenstra, Kan, and Shmoys [3.13].

3.3

FURTHER READING

Berge [3.14] proves, in a unified manner, several sufficient conditions for a graph to be Hamiltonian. This book also discusses the question of partitioning the edge set of a graph into paths, circuits, and so on. See also Bondy f

The weight of a circuit is the sum of the weights of the edges in the circuit.

68

EULERIAN AND HAMILTONIAN GRAPHS

and Murty [3.15], Chartrand and Lesniak [3.16], Bollobas [3.17], NashWilliams [3.18], and Lesniak-Foster [3.19]. We mentioned in Section 3.2 that if the degree sequence of a graph does not satisfy Chvatal's condition (3.1), then it is majorized by the degree sequence of a non-Hamiltonian graph. Further there are graphic sequences that do not satisfy (3.1), but that are necessarily degree sequences of Hamiltonian graphs. See Nash-Williams [3.20]. Motivated by Ore's sufficient condition (see Corollary 3.4.1), Bondy and Chvatal introduced in [3.21] the concept of the closure of a graph and obtained a sufficient condition for a graph to be Hamiltonian. They also showed how this result can be used to deduce Theorem 3.4 and Corollary 3.4.1. Generalizations of these ideas may be found in [3.17]. Also see [3.13], [3.15], and [3.16] and Exercises 3.14 and 3.16.

3.4

3.1

EXERCISES

Let G = (V, E) be a connected graph with 2k odd degree vertices, k 2: 1. Then Ε can be partitioned into subsets E ,..., E so that for each i, E, is a trail connecting odd degree vertices and such that at most one of these trails has odd length (Chartrand, Polimeni, and Stewart [3.22]). 2

k

3.2

Prove that a nontrivial connected graph G is Eulerian if and only if every edge of G lies on an odd number of circuits (Toida [3.23]).

3.3

Prove that if a graph G is randomly Eulerian from a vertex v, then ν is the only cut-vertex or G has no cut-vertices.

3.4

Let G be an Eulerian graph with η ^ 3 vertices. Prove that G is randomly Eulerian from none, one, two, or all of its vertices.

3.5

If a graph G is randomly Eulerian from a vertex v, then prove that A(G) = d(v), where A(G) is the maximum degree in G.

3.6

Let G be a randomly Eulerian graph from a vertex v. If d(u) = d(v), UJ^V, then prove that G is randomly Eulerian from u.

3.7

Do there exist graphs in which an Euler trail is also a Hamilton circuit? Characterize such graphs.

3.8

Show that the graph of Fig. 3.8 does not have a Hamilton path.

3.9

Let G be a connected simple graph with η > 25(G) vertices. Prove that the length of a longest path of G should be greater than or equal to 25(G), where 5(G) is the minimum degree in G (Dirac [3.2]).

3.10 Let G be a simple connected graph with η & 3 vertices such that for any two nonadjacent vertices u and υ in G

EXERCISES

69

Figure 3.8 d(u) + d(v)

> A:.

Show that if k = n, G is Hamiltonian and if k < n, G contains a path of length k and a circuit of length (k + 2)12. 3.11 Let d, s d s · • · < d be the degrees of a simple graph with η ^ 2 vertices. If 2

n

d^sfc-ls^n-ΐφ d _ n+l

k

^ η- k ,

then prove that G has a Hamilton path (Chvatal [3.1]). 3.12

Let G be a simple graph with η vertices and m edges such that n ^ 3 and m > ( n - 3 n + 6)/2. Prove that G is Hamiltonian (Ore [3.3]). 2

3.13

Let G be a simple graph with η vertices, and u and ν be nonadjacent vertices in G such that d(u) + d(u) > η . Denote by G ' the graph obtained by adding edge («, v) to G. Show that G is Hamiltonian if and only if G' is Hamiltonian.

3.14

The closure of a simple graph G with η vertices is the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least η until no such pair remains. Show that the closure of a graph is well-defined. (Note: An easy consequence of Exercise 3.13 is that a graph is Hamiltonian if and only if its closure is Hamiltonian. So G is Hamiltonian if its closure is complete. For an algorithm to determine

70

EULERIAN AND HAMILTONIAN GRAPHS

a Hamilton circuit of G given a Hamilton circuit of the closure of G, see [3.21] and [3.13].) 3.15

Prove that the maximum number of pairwise edge-disjoint Hamilton circuits in the complete graph K is [(n - l ) / 2 j . (Note: [x\ means the greatest integer less than or equal to x.) n

3.16

A simple graph G is Hamilton connected if for every pair of distinct vertices u and ν of G there exists a Hamilton u-υ path. If a simple graph G is of order n, then the (n + \)-closure of G is the graph obtained from G by recursively joining pairs of nonadjacent vertices whose degree sum is at least η + 1 until no such pair remains. Show that if the (n + l)-closure of a simple graph G is complete, then G is Hamilton connected. (Note: An easy corollary of this result is that a simple graph G is Hamilton connected if d(u) + d(v) s η + 1, for all distinct nonadjacent vertices u and ν of G. In fact, it is this result that suggested the definition of the (n + l)-closure of a graph. See Ore [3.24] and Erdos and Gallai [3.25].)

3.17

Show that the tree graph (defined in Exercise 2.28) of a connected graph is Hamiltonian (Cummins [3.26], Shank [3.27]).

3.5

REFERENCES

3.1

V. Chvatal, "On Hamilton's Ideals," /. Combinatorial Theory B, Vol. 12, 163-168 (1972). 3.2 G. A. Dirac, "Some Theorems on Abstract Graphs," Proc. London Math. Soc, Vol. 2, 69-81 (1952). 3.3 O. Ore, "Arc Coverings of Graphs," Ann. Mat. Pura Appl., Vol. 55, 315-321 (1961). 3.4 L. Posa, "A Theorem Concerning Hamilton Lines," Magyar Tud. Akad. Mat. Kutato Int. Kozl., Vol. 7, 225-226 (1962). 3.5 J. A. Bondy, "Properties of Graphs with Constraints on Degrees," Studia Sci. Math. Hungar., Vol. 4, 473-475 (1969). 3.6 C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. 3.7 M. Behzad and G. Chartrand, Introduction to the Theory of Graphs, Allyn and Bacon, Boston, 1971. 3.8 S. Lin and B. W. Kernighan, "An Effective Heuristic Algorithm for the Traveling Salesman Problem," Oper. Res., Vol. 21, 498-516 (1973). 3.9 M. Belmore and G. L. Nemhauser, "The Traveling Salesman Problem: A Survey," Oper. Res., Vol. 16, 538-558 (1968). 3.10 M. Held and R. M. Karp, "The Traveling Salesman Problem and Minimum Spanning Trees," Oper. Res., Vol. 18, 1138-1162 (1970).

REFERENCES

71

3.11 Μ. Held and R. M. Karp, "The Traveling Salesman Problem and Minimum Spanning Trees: Part II," Math. Programming, Vol. 1, 6-25 (1971). 3.12 D. J. Rosenkrantz, R. E. Stearns, and P. M. Lewis, "An Analysis of Several Heuristics for the Travelling Salesman Problem," SIAM J. Comput., Vol. 6, 563-581 (1977). 3.13 E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, Eds.; The Travelling Salesman Problem, Wiley-Interscience, New York, 1985. 3.14 C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. 3.15 J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976. 3.16 G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Coles, Pacific Grove, Calif., 1986. 3.17 B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979. 3.18 C. St. J. A. Nash-Williams, "Hamiltonian Circuits," in Studies in Graph Theory, Part II, MAA Press, Washington, D.C., 1975, pp. 301-360. 3.19 L. Lesniak-Foster, "Some Recent Results in Hamiltonian Graphs," J. Graph. Theory, Vol. 1, 27-36 (1977). 3.20 C. St. J. A. Nash-Williams, "Hamilton Arcs and Circuits," in Recent Trends in Graph Theory, Springer, Berlin, 1971, pp. 197-210. 3.21 J. A. Bondy and V. Chvatal, "A Method in Graph Theory," Discrete Math., Vol. 15, 111-135 (1976). 3.22 G. Chartrand, A. D. Polimeni, and M. J. Stewart, "The Existence of 1-Factors in Line Graphs, Squares, and Total Graphs," Indag. Math., Vol. 35, 228-232 (1973). 3.23 S. Toida, "Properties of an Euler Graph," J. Franklin Institute, Vol. 295, 343-345 (1973). 3.24 O. Ore, "Hamilton Connected Graphs," J. Math. Pures. Appl., Vol. 42, 21-27 (1963). 3.25 P. Erdos and T. Gallai, "On Maximal Paths and Circuits of Graphs," Acta Math. Acad. Sci. Hung., Vol. 10, 337-356 (1959). 3.26 R. L. Cummins, "Hamilton Circuits in Tree Graphs," IEEE Trans. Circuit Theory, Vol. CT-13, 82-90 (1966). 3.27 H. Shank, "A Note on Hamilton Circuits in Tree Graphs," IEEE Trans. Circuit Theory, Vol. CT-15, 86 (1968).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 4

GRAPHS AND VECTOR SPACES

Modern algebra has proved to be a very valuable tool for engineers and scientists in the study of their problems. Identifying the algebraic structure associated with a set of objects has been found to be very useful since the powerful and elegant results relating to the algebraic structure can then be brought to bear upon the study of such a set. Systems theory, electrical network theory, coding theory, switching circuits (sequential and combinational), and computer science are some examples of areas that have benefited by taking such an approach. In this chapter we show that a vector space can be associated with a graph, and we study in detail the properties of two important subspaces of this vector space, namely, the cutset and circuit spaces. In the first two sections of this chapter, we provide an introduction to some elementary algebraic concepts and results that will be used in the discussions of the subsequent sections. For more detailed discussions of these concepts and related results in linear algebra MacLane and Birkhoff [4.1], Halmos [4.2], and Hohn [4.3] may be consulted.

4.1

GROUPS AND FIELDS

Consider a finite set S = {a, ft, c,...}. Let + denote a binary operation defined on S. This operation assigns to every pair of elements a and b of 5 a unique element denoted by α + ft. The set S is said to be closed under + if the element a + ft belongs to S whenever a and ft are in 5. The operation + is said to be associative if 72

GROUPS AND FIELDS

a + (b + c) = (a + b) + c

73

for all a, b, and c in 5 .

Further, it is said to be commutative if a+b=b +a

for all α and binS

.

We are now in a position to define a group. A set S with a binary operation + , called addition, is a group if the following postulates hold: 1. 5 is closed under +. 2. + is associative. 3. There exists a unique element e in S such that a + e = e + a = a for all a in 5. The element e is called the identity of the group. 4. For each element a in S there exists a unique element b such that b + a = a + b = e. The element 6 is called the inverse of a and vice versa. Clearly the identity element e is its own inverse. A group is said to abelian if the operation + is commutative. A common example of a group is the set S = {..., - 2 , - 1 , 0 , 1 , 2 , . . . } of all integers, with + defined as the usual addition operation. In this group, 0 is the identity element and —a is the inverse of a for all a in S. Note that this group is abelian. Is the set 5 of all integers with the multiplication operation a group? (No. Why?) An important example of a group is the set Z = { 0 , 1 , 2 , . . . , ρ - 1} of integers with modulo ρ addition operation. In this group, 0 is the identity element. Further the integer ρ - a is the inverse of the integer a, for all a not equal to 0. Of course, 0 is its own inverse. This group is also abelian. As an example, the addition table for Z is as follows: p

+

5

-Ι-

0

1

2

3

4

Ο 1 2 3 4

0 1 2 3 4

1 2 3 4 0

2 3 4 0 1

3 4 0 1 2

4 0 1 2 3

Next we define a field. A set F with two operations + and ·, called, respectively, addition and multiplication, is a field if the following postulates are satisfied: If a = mp + q, 0 s q < ρ - 1, then in modulo arithmetic a = ^(modulo p).

f

74

GRAPHS AND VECTOR SPACES

1. F is an abelian group under + , with the identity element denoted as e. 2. The set F-{e) is an abelian group under ·, the multiplication operation. 3. The multiplication operation is distributive with respect to addition; that is, a • (b + c) = (a · b) + (a • c)

for all a, b, and c in F .

As an example, consider again the set Z = { 0 , l , 2 , . . . , p - 1 } with addition (modulo p) and multiplication (modulo p) as the two operations. As shown earlier, Z is an abelian group under modulo ρ addition, with 0 as the identity element. It can be shown that the set Z - {0} = { 1 , 2 , . . . , ρ - 1} is a group under modulo ρ multiplication if and only if ρ is prime. This group is also abelian. The fact that modulo ρ multiplication is distributive with respect to modulo ρ addition may be easily verified. Thus the set Z is a field if and only if ρ is prime. The field Z is usually denoted as G¥(p) and is called a Galois field. The following multiplication table for GF(5) is given as an illustration: p

p

p

p

0 1 2 3 4

0

1

2

3

4

0 0 0 0 η

0 1 2 3 4

0 2 4 1 3

0 3 1 4 2

0 4 3 2 1

A field that is of special interest to us is GF(2), the set of integers modulo 2. In this field

4.2

0+ 0= 0

0-0 = 0

1+ 0= 0+1 = 1

1-0 = 0 - 1 = 0

1 + 1= 0

1-1 = 1 .

VECTOR SPACES

Consider a set S with a binary operation B . Let F be a field with + and · denoting, respectively, the addition and multiplication operations. A multiplication operation, denoted by *, is also defined between elements in F and those in S. This operation assigns to each ordered pair (a, s), where a is in F and s in S, a unique element denoted by a * s of 5. The set 5 is a vector space over F if the following postulates hold:

VECTOR SPACES

75

1. 5 is an abelian group under Ξ . 2. For any elements a and β in F, and any elements s, and s in S, 2

a * ( s , E ) s ) = (a * j , ) E ( a * s ) 2

2

and (α+β)*ί =(«*ϊ )Ξ(/8*ϊ ). 1

1

Ι

3. For any elements α and /3 in F and any element s in 5, (a · /3)*s = a *(/3 *s). 4. For any element s in S, 1 * s = s, where 1 is the multiplicative identity in F. Next we give an important example of a vector space. Consider the set W of all n-vectors over a field F. (Note that the elements of the η-vectors are chosen from F.) The symbols + and • will, respectively, denote the addition and multiplication operations in F, and the symbols 0 and 1 will denote the additive and multiplicative identities in F. Let Ξ , an addition operation on W, and *, a multiplication operation between the elements of F and those of W, be defined as follows: +

1. If ω, = (α,, α , . . . , α ) and ω = (>3 , β ,..., then 2

η

2

1

2

β ) are elements of W, η

ω, • ω = (α, + β,, α + / 3 , . . . , α„ + β„). 2

2

2

2. If α is in F, then a * ω, = (α · α,, α · α , . . . , α · α ) . 2

π

With Ξ defined as above, we can easily establish that W is an abelian group under • , with the /i-vector ( 0 , 0 , 0 , . . . ,0) as the identity element. Thus W satisfies the first postulate in the definition of a vector space. We can easily show that the elements of IV and F also satisfy the other three requirements of a vector space. For example, the set W of the following eight 3-vectors is a vector space over GF(2): 'An η-vector over F is a row vector with η elements from the field F.

76

GRAPHS AND VECTOR SPACES

ω = (000),

ω, = ( 0 0 1 ) ,

ω = (0 1 0 ) ,

ω = (011),

ω = (100),

ω = (101),

ω = (110),

ω = (111) .

0

4

2

5

3

6

7

This vector space is used in all illustrations in this section. A few important definitions and results (without proof) relating to a vector space are stated next. Consider a vector space S over a field F. Vectors and Scalars Elements of 5 are called vectors and those of F are called scalars. Linear Combination If an element s in 5 is expressible as s = (a, * s,) Ξ ( a * s ) Ξ · • · Ξ (a, * s ) , 2

2

;

where s,'s are vectors and a,'s are scalars, then s is said to be a linear combination of s,, s ,...,

Sj.

2

Linear Independence Vectors s ,s ,... ,s are said to be linearly independent if no vector in this set is expressible as a linear combination of the remaining vectors in the set. Basis Vectors Vectors s , s ,... ,s form a basis in the vector space 5 if they are linearly independent and every vector in 5 is expressible as a linear combination of these vectors. The vectors s,, s ,... ,s are called basis vectors. It can be shown that the representation of a vector as a linear combination of basis vectors is unique for a given basis. A vector space may have more than one basis. However, it can be proved that all the bases have the same number of vectors. Dimension The dimension of the vector space 5, denoted as dim(S), is the number of vectors in a basis of 5. Subspace If 5 ' is a subset of the vector space S over F, then 5 ' is a subspace of 5 if 5 ' is also a vector space over F. Direct Sum The direct sum 5, • S of two subspaces 5, and S of S is the set of all vectors of the form s , S i , , where s, is in 5, and s is in S . l

x

2

2

J

n

2

n

2

2

;

2

It can be shown that S, ED 5 is also a subspace, and that its dimension is given by 2

dim(5, • 5 ) = dim(5,) + dim(S ) - d i m ^ Π 5 ) . 2

2

2

Note that S Π S is also a subspace whenever Sj and 5 are subspaces. Let us illustrate these definitions using the vector space W of 3-vectors over GF(2). The vectors ω , ω ,..., ω of W are as defined earlier. t

2

2

0

χ

7

VECTOR SPACES

77

1. ω, is a linear combination of ω and ω since 6

7

ω, = ( 1 * ω ) Ξ ( 1 * ω ) . 6

7

2. Vectors ω and ω are linearly independent since neither of these two can be expressed in terms of the other. Note that ω and ω, are not linearly independent, for any i. 3. Vectors ω,, ω , and ω form a basis in W since they are linearly independent, and the remaining vectors can be expressed as linear combinations of these: 3

4

0

2

4

ω = (0*ω,)Ξ(0*ω )Ξ(0*ω ), 0

2

4

ω = (1*ω,)Ξ(1*ω ), 3

2

ω = (1*ω,)Ξ(1*ω ), 5

4

ω = (1*ω )Ξ(1*ω ), 6

2

4

ω = (1 * ω,) Β (1 * ω ) Ξ (1 * ω ) . 7

2

4

It may be verified that the vectors ω,, ω and ω also form a basis. 4. Dimension of W is equal to 3 since there are three vectors in a basis of W. 3

7

5. The sets W

= {ω , ω,, ω , ω } 0

2

3

and W"= {ω , ω,, ω , ω } 6

0

7

are subspaces of W. It may be verified that {ω,, ω } is a basis for and {ω,, ω } is a basis for W". Thus 2

6

dim(W) = 2 and dim(W") = 2 . 6. If W and W" are as defined above, then W Ξ W" = {ω , ω,, ω , ω , ω , ω , ω , ω } . 0

2

3

7. The dimension of W El W is given by

4

5

6

7

W,

78

GRAPHS AND VECTOR SPACES

dim( W Ξ W") = dim(lV') + dim(W") - dim( W η W")

= 4-dim(W'n W"). Since w'nw"

= {u> ,a> }, 0

x

we have dim( W Π W") = 1. Thus dim(W'0W") = 4 - l = 3. This can also be obtained by noting that in this case W ' 0 W " = W. Next we define isomorphism between two vector spaces defined over the same field. Let S and 5 ' be two η-dimensional vector spaces over a field F. Then S and 5 ' are said to be isomorphic if there exists a one-to-one correspondence between S and 5 ' such that the following hold true: 1. If vectors s, and s of 5 correspond to the vectors s[ and s' of S, respectively, then the vector s, Ξ s corresponds to the vector s[ Δ s' , where Ξ and Δ are corresponding operations in S and S'. 2. For any a in F, the vector a*s corresponds to the vector a As' if s corresponds to s', where * and Δ are corresponding operations in S and S'. 2

2

2

2

Consider an /i-dimensional vector space S over a field F and the ndimensional vector space W of Λ-vectors over F. Let the vectors s ,s ,. • • ,s„ form a basis in 5. Suppose we define that a vector s in S corresponds to the vector ω = (α,, a ,..., α γ of W if and only if s = (a, * s,) • (a * s ) EE] · · · Ξ (a„ * s ). Then it is not difficult to show that this one-to-one correspondence defines an isomorphism between 5 and W. Thus we have the following important result. l

2

2

2

2

η

n

Theorem 4.1. Every η-dimensional vector space over a field F is isomorphic to the vector space W of η-vectors over F. •

*a,

a„ are called t h e

coordinates

of

s relative

to t h e basis { j , ,

s

2

s„}.

VECTOR SPACES

79

Theorem 4.1 provides the main link connecting vector spaces and matrices. It implies that an η-dimensional vector space over a field F can be studied in terms of the η-dimensional vector space W of all η-vectors over F. We conclude this section with the definition of two more important concepts—dot product (or inner product) and orthogonality. Let ω, = (α, α · · · α„) 2

and ω = (/3,

β --β„)

2

2

be two vectors in the vector space W of η-vectors over F. The dot product of ω, and ω , denoted by ( ω , , ω ) , is a scalar defined as 2

2

<ω,, ω ) = α, · 0, + α · β + • • • + α • β„ . 2

2

2

η

For example, if ω, = (0

1 0

0

1)

and ω = (1 2

0

1 1

1),

then ( ω, ω ) = 0 · 1 + 1 · 0 + 0 · 1 + 0 · 1 + 1 · 1 2

=0+0+0+0+1 = 1 . Vectors ω, and ω, are orthogonal to each other if (ω,, ω ) = 0, where 0 is the additive identity in F. For example, the vectors ω, = (1

1 0

1

1)

ω = (1

1 1 0

0)

and 2

are orthogonal over GF(2) since

80

GRAPHS AND VECTOR SPACES

<«,, ω > = 1-1 + 1-1 + 0-1 + 1-0+ 1-0 2

=1+1+0+0+0 = 0. Two subspaces W and W" of W are orthogonal subspaces of W if each vector in one subspace is orthogonal to every vector in the other subspace. Two subspaces W and W" of W are orthogonal complements if they are orthogonal to each other and their direct sum W Ξ W" is equal to the vector space W. For example, consider again the three-dimensional vector space W of 3-vectors over GF(2). In this vector space the subspaces W = {ω , ω,, ω , ω ) and W" = {ω , ω } are orthogonal to each other. It may be verified that the direct sum of W and W" is equal to W. Hence they are orthogonal complements. 0

4.3

2

3

0

4

VECTOR SPACE OF A GRAPH

In this section we show how we can associate a vector space with a graph, and we also identify two important subspaces of this vector space. Consider a graph G = (V, E). Let W denote the collection of all subsets of E, including the empty set 0 . First we show that W is an abelian group under Θ, the ring sum operation between sets. After suitably defining multiplication between the elements of the field GF(2) and those of W , we show that W is a vector space over GF(2). The following are easy to verify: G

c

G

o

1. W is closed under Θ. 2. θ is associative. 3. φ is commutative. c

Further, for any element E, in W , G

Ε θ0=Ε ί

ι

and £,©£, = 0 .

Thus for the operation Θ, 0 is the identity element, and each E, is its own inverse. Hence W is an abelian group under Φ, thereby satisfying the first requirement in the definition of a vector space. G

VECTOR SPACE OF A GRAPH

81

Let *, a multiplication operation between the elements of GF(2) and those of W , be defined as follows: For any £,· in W , G

G

l*E, = E, and 0*E, = 0 . With this definition of * we can easily verify that the elements of W satisfy the following other requirements for a vector space: For any elements α and β in GF(2) = {0,1}, and any elements £, and £ in W ,

c

;

c

1. 2. 3. 4.

(a + β)*Ε = α*(Ε,ΦΕ ) = (α·β)·Ε, = 1 *£, = £,. ι

ι

(α*Ε,)®(β*Ε ). (α*Ε,)Φ(α*Ε ). α*(β*Ε,). ί

)

Note that 1 is the multiplicative identity in GF(2). Thus W is a vector space over GF(2). If Ε - {e,, e ,..., e }, then it is easy to see that the subsets {«,}, {e }, • • •, {e } will constitute a basis for W . Hence the dimension of W is equal to m, the number of edges in G. Since each edge-induced subgraph of G corresponds to a unique subset of E, and by definition (see Chapter 1), the ring sum of any two edge-induced subgraphs corresponds to the ring sum of their corresponding edge sets, it is clear that the set of all edge-induced subgraphs of G is also a vector space over GF(2) if we define the multiplication operation * as follows: For any edge-induced subgraph G, of G, G

2

2

m

m

G

G

1 * G, = G, and O*G, = 0 , the null graph having no vertices and no edges. This vector space will also be referred to by the symbol W . Note that W will include 0 , the null graph. We summarize the results of this discussion in the following theorem. G

G

Theorem 4.2. For a graph G having m edges W is an m-dimensional vector space over GF(2). • G

82

GRAPHS AND VECTOR SPACES

Since, in this chapter, we are concerned only with edge-induced subgraphs, we refer to them simply as subgraphs without the adjective "edgeinduced." However, we may still use this adjective in some places to emphasize the edge-induced nature of the concerned subgraph. Next we show that the following subsets of W are subspaces: G

1. W , the set of all circuits (including the null graph 0 ) and unions of edge-disjoint circuits of G. 2. W , the set of all cutsets (including the null graph 0 ) and unions of edge-disjoint cutsets of G. c

s

This result will follow once we show that W and W are closed under Θ, the ring sum operation. c

s

Theorem 4.3. W , the set of all circuits and unions of edge-disjoint circuits of a graph G, is a subspace of the vector space W of G. c

G

Proof. By Theorem 3.1, a graph can be expressed as the union of edgedisjoint circuits if and only if every vertex in the graph is of even degree. Hence we may regard W as the set of all edge-induced subgraphs of G, in which all vertices are of even degree. Consider any two distinct members C, and C of W : C and C are edge-induced subgraphs with the degrees of all their vertices even. Let C denote the ring sum of C and C . To prove the theorem, we need only to show that C belongs to W . In other words we should show that in C every vertex is of even degree. Consider any vertex υ in C . Obviously, this vertex should be present in at least one of the subgraphs C, and C . Let AT,, i = 1 , 2 , 3 , denote the set of edges incident on ν in C,. Let \X \ denote the number of edges in X . Thus \X \ is the degree of the vertex υ in C,. Note that \X \ and |A" | are even and one of them may be zero. Further \X \ is nonzero. Since C = C, Θ C , we get c

2

c

2

x

3

t

3

2

3

c

3

2

t

t

t

2

X

3

3

2

X

3

'Χ φ X · χ

2

Hence 1^1 = 1^1 +

1 ^ 1 - 2 1 ^ 0 ^ 1 .

It is now clear from this equation that \X \ is even, because |ΑΊ| and \X \ are both even. In other words the degree of vertex υ in C is even. Since this should be true for all vertices in C , it follows that C belongs to W , and the theorem is proved. • 3

2

3

3

3

W will be referred to as the circuit subspace of the graph G. c

c

VECTOR SPACE OF A GRAPH

83

As an example to illustrate the result of Theorem 4.3, consider the graph G shown in Fig. 4.1a. The subgraphs C, and C shown in Fig. 4.1b and 4.1c are unions of edge-disjoint circuits of G, since the degree of each vertex in these subgraphs is even. Thus they belong to the subspace W of W . The ring sum C of C, and C is shown in Fig. 4.Id. It may be seen that all the vertices in C are of even degree. Hence C also belongs to W . 2

c

3

c

2

3

3

c

Figure 4.1. (a) Graph G. (b) Subgraph C, of G. (c) Subgraph C of G. (d) Subgraph C, θ C of G. 2

2

84

GRAPHS AND VECTOR SPACES

Next we show that W , the set of all cutsets and unions of edge-disjoint cutsets of G, is a subspace of W . By Theorem 2.7, a cut is a cutset or union of some edge-disjoint cutsets. Thus every cut of G belongs to W . We now prove that every element of W is a cut. While doing so we also prove that W is a subspace of W . s

G

s

s

s

G

Theorem 4.4. The ring sum of any two cuts in a graph G is also a cut in G. Proof. Consider any two cuts S, = (V,, V ) and S = (V , V ) in a graph G = (V, E). Note that 2

2

3

4

ν^υν = ν υν = ν 2

3

4

and

ν,ην = ν ην = 0 . 2

3

4

Let A = V, Π V , 3

c= v nv , d = v η v. 2

3

2

4

It is easy to see that the sets A, B, C, and D are mutually disjoint. Then S = (AUB,

CUD)

t

= (A, C) U (A, D) U ( B , C) U ( B , D ) and 5 =
= U(i4,D>U(C,B>U . Hence, we get S

i

e S

2

= (A,C)U(B,

D)U(A,B)L)(C,D)

.

Since (/4UD,flUC) = ( / l , C ) U ( f l D ) U ( A , 5 ) U ( C D ) , (

we can write

1

VECTOR SPACE OF A GRAPH

S1®S2 = (AUD,BUC)

85

.

Because ADD and B U C are mutually disjoint and together include all the vertices in V, S, Θ S2 is a cut in G. Hence the theorem. ■ Since the ring sum of two disjoint sets is the same as their union, we get the following corollary of Theorem 4.4. Corollary 4.4.1. The union of any two edge-disjoint cuts in a graph G is also a cut in G. ■ Since a cutset is also a cut, it is now clear from Corollary 4.4.1 that Ws is the set of all cuts in G. Further, by Theorem 4.4, Ws is closed under the ring sum operation. Thus we get the following theorem. Theorem 4.5. Ws, the set of all cutsets and unions of edge-disjoint cutsets in a graph G, is a subspace of the vector space WG of G. ■ Ws will be referred to as the cutset subspace of the graph G. As an example illustrating the result of Theorem 4.5, consider the following cuts 5, and S2 in the graph G of Fig. 4.2: •jj

\^ι> e y, e 4, e$, e^, e-j) =

02~

e2'

e

4> e5>

e

SI

= (v3,vA), where

»e

Figure 4.2

86

GRAPHS AND VECTOR SPACES

Vl =

{Vi,

V,

V)

2

,

4

V = {v ,v ,v )

,

V = {v ,v ,v )

,

2

3

3

5

l

4

6

5

V ={v ,v ,v }. 4

2

3

6

Then S, φ S — {e , e , e , e , e ) . 2

2

3

6

7

s

If A

V,nV =

Β

v nv = {υ }, v nv = { v ) , v nv = {v ,v ),

C D

{ i ;

3

1

l

4

2

2

3

5

2

4

, i ;

3

4

} ,

6

then it is verified that the set 5, Φ 5 = {e , e , e , e , e } can be written (as in the proof of Theorem 4.4) as 2

5, Φ 5 = (A U D, Β U C) = ({v

2

2

4.4

u

3

7

6

s

v , v , v ), {v , v )) . 4

3

6

2

5

DIMENSIONS OF CIRCUIT AND CUTSET SUBSPACES

In this section we show that the dimensions of the circuit and cutset subspaces of a graph are equal to the nullity and the rank of the graph, respectively. We do this by proving that the set of fundamental circuits and the set of fundamental cutsets with respect to some spanning tree of a connected graph are bases for the circuit and cutset subspaces of the graph, respectively. Let Γ be a spanning tree of a connected graph G with η vertices and m edges. The branches of Τ will be denoted by ft,, b ,..., b _, and the chords by c,, c , . . . , c _ . Let C, and S, refer to the fundamental circuit and the fundamental cutset with respect to c, and b„ respectively. By definition, each fundamental circuit contains exactly one chord, and this chord is not present in any other fundamental circuit. Thus no fundamental circuit can be expressed as the ring sum of the other fundamental circuits. Hence the fundamental circuits C,, C , . . . , C _ are independent. Similarly, the fundamental cutsets S , S ,...,S„_ are also independent since each of these contains exactly one branch that is not present in the others. 2

2

m

n

n+l

2

t

m

2

i

n + 1

DIMENSIONS OF CIRCUIT AND CUTSET SUBSPACES

87

To prove that C,, C , . . . , C _ (S,, S , . . . , S„_,) constitute a basis for the circuit (cutset) subspace of G, we need only to prove that every subgraph in the circuit (cutset) subspace of G can be expressed as a ring sum of C,'s (S,'s). Consider any subgraph C in the circuit subspace of G. Let C contain the chords c , c , . . . , c . Let C denote the ring sum of the fundamental circuits C -, C , , . . . , C . Obviously, the chords c , c , , . . . , c, are present in C , and C contains no other chords of T. Since C also contains these chords and no others, C ' 0 C contains no chords. We now claim that C ' 9 C is empty. If this is not true, then by the preceding arguments, C ' ® C contains only branches and hence has no circuits. On the other hand, being a ring sum of circuits, C θ C is, by Theorem 4.3, a circuit or the union of some edge-disjoint circuits. Thus the assumption that C φ C is not empty leads to a contradiction. Hence C φ C is empty. This implies that C = C = C ® C, © · · · φ C,. In other words, every subgraph in the circuit subspace of G can be expressed as a ring sum of C,'s. In an exactly similar manner we can prove that every subgraph in the cutset subspace of G can be expressed as a ring sum of 5,'s. Thus we have the following theorem. 2

M

N

2

+ I

(

(

2

2

(]

2

Theorem 4.6. Let a connected graph G have m edges and η vertices. Then: 1. The fundamental circuits with respect to a spanning tute a basis for the circuit subspace of G, and hence the circuit subspace of G is equal to m - η + 1, the 2. The fundamental cutsets with respect to a spanning tute a basis for the cutset subspace of G, and hence the cutset subspace of G is equal to η - 1, the rank

tree of G constithe dimension of nullity of G. tree of G constithe dimension of of G. •

It is now easy to see that in the case of a graph G that is not connected, the set of all the fundamental circuits with respect to the chords of a forest of G, and the set of all the fundamental cutsets with respect to the branches of a forest of G are, respectively, bases for the circuit and cutset subspaces of G. Thus we get the following corollary of the previous theorem. Corollary 4.6.1. If a graph G has m edges, η vertices, and ρ components, then: 1. The dimension of the circuit subspace of G is equal to m - η + ρ, the nullity of G. 2. The dimension of the cutset subspace of G is equal to η — ρ, the rank of

G.

m

88

GRAPHS AND VECTOR SPACES

Figure 4.3

As an example, consider the graph G shown in Fig. 4.3. The edges marked as bx, b2, b3, b4 constitute a spanning tree of G. The chords of T are marked as cx, c2, c3, c4, and c5. The fundamental circuits C,, C2, C3, C4, and C5 with respect to the chords cx, c2, c3,c4, and c5, and the fundamental cutsets 5,, S2, S3, and S4 with respect to the branches bx, b2, b3, and b4 are obtained as •S, = {bx,cx,c2} C2 = (c2,bx,b2,b3} C3 = {c 3 ,ft 3 ,6 4 }, ft

Q = {c4,ft2> 4}> C5 = {c5, ft2, 63} .

,

,

o2 = {b2, cx, c2, c4, c 5 ), Λ3 = {o3, c2, c3, c5) , 5

4 = {ft4.C3»C4} ·

Consider first the subgraph C consisting of the edges bx, b2, ò 3 , c2, c3, c 4 , and c5. It may be verified that in C every vertex is of even degree, and hence C belongs to the circuit subspace of G. Since C contains the chords c2,c3,c4, and c5, it follows from the arguments preceding Theorem 4.6 that C should be equal to the ring sum of the fundamental circuits C2, C3,C4, and C5. This can be verified to be true as follows: C2S>C3G>C4®CS = {c2, bx, b2, b3)Θ{c3, b3, b4}Φ{c4, b2, b4) ®{cs,b2,b3) — \bx, o2, o 3 , c2, c3, c4, c5) = C.

RELATIONSHIP BETWEEN CIRCUIT AND CUTSET SUBSPACES

89

Consider next the cut S consisting of the edges b , b , c , c , c . Again, since S contains the branches ft, and b , it should be equal to the ring sum of the cutsets 5, and S . This also can be verified to be true as follows: x

3

t

3

5

3

3

S φ Sj t

=

{ftp

C j , c2}

φ {ft , c , 3

2

Cj,

c} 5

= 5. In the preceding discussions we have shown that, starting from a spanning tree, we can construct bases for the circuit and cutset subspaces of a graph. The bases so constructed are commonly used in the study of electrical networks.

4.5

RELATIONSHIP BETWEEN CIRCUIT AND CUTSET SUBSPACES

We establish in this section a characterization for the subgraphs in the circuit subspace of a graph G in terms of those in the cutset subspace of G. We proved in Section 2.8 (Theorem 2.14) that a circuit and a cutset have an even number of common edges. Since every subgraph in the circuit subspace of a graph is a circuit or the union of edge-disjoint circuits, and every subgraph in the cutset subspace is a cutset or the union of edgedisjoint cutsets, we get the following as an immediate consequence of Theorem 2.14. Theorem 4.7. Every subgraph in the circuit subspace of a graph G has an even number of common edges with every subgraph in the cutset subspace of G. m In the next theorem we prove the converse of Theorem 4.7. Theorem 4.8 1. A subgraph of a an even number subspace of G. 2. A subgraph of a an even number subspace of G.

graph G belongs to the circuit subspace of G if it has of common edges with every subgraph in the cutset graph G belongs to the cutset subspace of G if it has of common edges with every subgraph in the circuit

Proof 1. We may assume, without any loss of generality, that G is connected. The proof when G is not connected will follow in an exactly similar manner.

90

GRAPHS AND VECTOR SPACES

Let Γ be a spanning tree of G. Let b , b ,... denote the branches of Γ and c , c , . . . denote its chords. Consider any subgraph C of G that has an even number of common edges with every subgraph in the cutset subspace of G. Without any loss of generality, assume that C contains the chords c,, c , . . . , c . Let C denote the ring sum of the fundamental circuits C,, C , . . . , C with respect to the chords c,, c , . . . , c. Obviously, C" consists of the chords c,, c ,...,c and no other chords. Hence C © C consists of no chords. C, being the ring sum of some circuits of G, has an even number of common edges with every subgraph in the cutset subspace of G. Since C also has this property, so does C φ C. We now claim that C ' ® C is empty. If not, C ' ® C contains only branches. Let b be any branch in C Φ C. Then b is the only edge common between C Θ C and the fundamental cutset with respect to b . This is not possible since C'®C must have an even number of common edges with every cutset. Thus C Φ C should be empty. In other words, C = C = C, Φ C Φ · · · Θ C , and hence C belongs to the circuit subspace of G. 2. The proof of this part follows in an exactly similar manner. • x

t

2

r

2

2

2

2

r

r

2

r

t

t

t

2

4.6

r

ORTHOGONALITY OF CIRCUIT AND CUTSET SUBSPACES

By Theorem 4.1, every η-dimensional vector space over a field F is isomorphic to the vector space of all η-vectors over the same field. Hence W , the vector space of a graph G, is isomorphic to the vector space of all m-vectors over GF(2), where m is the number of edges in G. Let e , e ,...,e denote the m edges of G. Suppose we associate each edge-induced subgraph G, of G with an m-vector w, such that the ;'th entry of w, is equal to 1 if and only if the edge e is in G,. Then the ring sum G, Φ G of two subgraphs G, and G will correspond to the m-vector w, + w , the modulo 2 sum of w and w . It can now be seen that the association just described indeed defines an isomorphism between W and the vector space of all m-vectors over GF(2). In fact, if we choose { e j , {e },..., {e } as the basis vectors for W , then the entries of w, are the coordinates of G, relative to this basis. In view of this isomorphism we again use the symbol W to denote the vector space of all the m-vectors associated with the subgraphs of the graph G. Also, W will denote the subspace of m-vectors representing the subgraphs in the circuit subspace of G and similarly W will denote the subspace of those representing the subgraphs in the cutset subspace of G. Consider any two vectors w, and n>j such that w, is in W and w is in W . Because every subgraph in W has an even number of common edges with G

x

2

m

t

;

;

t

y

t

c

2

m

G

a

c

s

c

c

f

s

ORTHOGONALITY OF CIRCUIT AND CUTSET SUBSPACES

91

those in W , it follows that the dot product (w w ) of w and w is equal to the modulo 2 sum of an even number of l's. This means (w , Wj) = 0. In other words, the m-vectors in W are orthogonal to those in W . Thus we have the following theorem. s

;

lt

t

t

c

s

Theorem 4.9. The cutset and circuit subspaces of a graph are orthogonal to each other. • Consider next the direct sum W \BW . C

We know that

S

dim(W Θ W ) = dim(W ) + dim(W ) - dim(W Π c

c

s

Since dim(lV ) + dim(W ) c

W ).

c

s

s

= m, we get

s

dim(W 0ff ) = (n-dim(W Π c

s

c

W ). s

Now the orthogonal subspaces W and W will also be orthogonal complements of W if and only if dim(W • W ) = m. In other words W and W will be orthogonal complements if and only if dimiWcD W ) = 0, that is, W Π W is the zero vector whose elements are all equal to zero. Thus we get the following theorem. c

s

c

G

s

c

s

s

c

s

Theorem 4.10. W and W , the circuit and cutset subspaces of a graph are orthogonal complements if and only if W Π W is the zero vector. • c

s

c

s

Suppose W and W are orthogonal complements. Then it means that every vector in W can be expressed as w + w where tv, is in W and w is in W . In other words every subgraph of G can be expressed as the ring sum of two subgraphs, one belonging to the circuit subspace and the other belonging to the cutset subspace. In particular, the graph G itself can be expressed as above. Suppose W and W are not orthogonal complements. Then, clearly, there exists a subgraph that cannot be expressed as the ring sum of subgraphs in W and W . The question then arises whether, in this case too, it is possible to express G as the ring sum of subgraphs from W and W . The answer is in the affirmative as stated in the next theorem. c

s

G

t

jy

c

)

s

c

s

c

s

c

s

Theorem 4.11. Every graph G can be expressed as the ring sum of two subgraphs one of which is in the circuit subspace and the other is in the cutset subspace of G. • See Chen [4.4] and Williams and Maxwell [4.5] for a proof of this theorem. We conclude this section with an example.

92

GRAPHS AND VECTOR SPACES

<*>

(6)

Figure 4.4. (a) Graph G„. (b) Graph G„. Consider the graph G shown in Fig. 4.4a. It may be verified that no nonempty subgraph of this graph belongs to the intersection of the cutset and circuit subspaces. Hence these subspaces of G are orthogonal complements. Then the set of fundamental circuits and fundamental cutsets with respect to some spanning tree of G constitutes a basis for the vector space of G . One such set with respect to the spanning tree formed by the edges «?,, e , e , and e follows: a

a

a

a

2

3

4

Fundamental cutset vectors S, = ( l

0 0 0

1 1 0)

5 = (0

1 0 0

1 1 0)

2

5 = (0 0 3

1 0

5 = (0 0 0 4

1 0

1)

1 0 0

1)

Fundamental circuit vectors C, = (l

1 1 0

C = (l

1 0 0 0

C = (0

0

2

3

1 0

0)

1 0)

1 1 0 0

1)

We can now verify that every vector can be expressed as the ring sum of a circuit vector and a cutset vector. In particular, the vector (1 1 1 1 1 1 1) representing G itself can be expressed as: a

FURTHER READING

93

i) = 5,e5 2 ©s 4 ec 1 ec 2

( 1 1 1 1 1 1

=

( 1 1 0 1 0 0 1 ) Θ(0

0

1 0

1 1 0 )

where ( 1 1 0 1 0 0 1) represents a cut and (0 0 1 0 1 1 0) represents a circuit in G . Consider next the graph G in Fig. 4.4b. In this graph the edges e , e , e , and e constitute a cut as well as a circuit. Hence the circuit and cutset subspaces of this graph are not orthogonal complements. This means that there exists some subgraphs in G that cannot be expressed as the ring sum of subgraphs from the cutset and circuit subspaces of G . However, according to Theorem 4.11, such a decomposition should be possible for G . This is true since a

b

3

x

2

4

b

b

b

G =

{e ,e ,e }®{e ,e ,e },

b

x

2

s

i

4

6

where {e , e , e } is a cut and {e , e , e } is a circuit in G . x

4.7

2

3

5

4

6

b

FURTHER READING

An early paper on vector spaces associated with a graph is by Gould [4.6], where the question of constructing a graph having a specified set of circuits is also discussed. Chen [4.4] and Williams and Maxwell [4.5] are also recommended for further reading on this topic. We conclude with a brief introduction to the concept of principal partition of a graph. Though motivated by an application in electrical network analysis, this concept is quite profound as we shall see below. Consider a connected graph G = (V, E). The distance d(T , T ) between any two spanning trees T and T of G is defined as x

x

ά(τ ,τ ) χ

2

2

= \τ -τ \

2

χ

=

2

\τ -τ \. 2

χ

In other words d(T , T ) is equal to the number of edges of T (T ), that are not present in T (T ). Two trees T and T are maximally distant if d(T , T )> d(T T ) for every pair of spanning trees T, and of G. Let d denote the maximum distance between any two spanning trees of G. Given a pair of maximally distant spanning trees T and T . Suppose c is a common chord of T and T . The A:-subgraph G of G with respect to c is constructed as follows: x

2

2

x

2

n

X

X

x

2

2

f

m

x

x

2

2

c

1. Let L, be the set of all the edges in the fundamental circuit with respect to Γ, defined by c.

94

GRAPHS AND VECTOR SPACES

2. Let L be the union of all the fundamental circuits with respect to T denned by every edge in L,. 3. Repeating the above, we can obtain a sequence of sets of edges L j , L , . - - until we arrive at a set L = L . Then the induced subgraph on the edge set L is called the fc-subgraph G with respect to c. 2

2

2

k+X

k

k

c

Replacing circuit by cutset, in the above construction, we can define in an exactly dual manner the ^-subgraph G with respect to a common branch b. The principal subgraph G, with respect to common chords is the union of the ^-subgraphs with respect to all the common chords. The principal subgraph G with respect to the common branches is the union of the fc-subgraphs with respect to all the common branches. Kishi and Kajitani [4.7] have shown that G, and G have no common edges. Let E, and E denote the edge sets of G, and G , respectively, and let £ = £ - ( £ , U E ). Then the partition ( £ , £ , , E ) is called the principal partition of G. This partition is unique and is independent of the maximally distant spanning trees used to construct the partition. Let E and E be a partition of the edge set £ of a graph. Let G be the subgraph on the edge set E and let G be the graph obtained by contracting all the edges in G . Ohtsuki, Ishizaki, and Watanabe [4.8] have shown that b

2

2

2

2

0

0

2

a

2

b

a

a

b

a

It can be shown that the quantity p ( G ) + p-(G£) achieves the value d when E = £ , and E = £ U £ . This result is the basis of an optimum choice of variables in an approach for electrical network analysis. See [4.8] and [4.9]. For this reason in electrical network theory d is referred to as the topological degrees of freedom. Lin [4.10] discusses an algorithm for computing the principal partition of a graph. Bruno and Weinberg [4.11] extend the concept of principal partition to matroids. a

a

0

b

m

2

m

4.8

EXERCISES

4.1 Show that the circuits formed by the following sets of edges constitute a basis for the circuit subspace of the graph shown in Fig. 4.5: {e,, e , e , e } ; 3

4.2

4

6

{e , e , e , e } ; 2

3

5

7

{e e ,e }. Jt

2

g

An incidence set at a vertex is the set of edges incident on the vertex. Show that any η — 1 incidence sets in an η-vertex connected graph form a basis of the cutset subspace of the graph.

EXERCISES

95

«ι

Figure 4.5

4.3

If ρ is the rank and μ is the nullity of a graph G, show that (a) The number of distinct bases possible for the cutset subspace of G is equal to ^ ( 2 - 2°)(2 - 2 ' ) ( 2 - 2 ) · · · ( 2 - 2 P

P

P

2

P

P _ 1

).

(b) The number of distinct bases possible for the circuit subspace of G is equal to (2" - 2°)(2 - 2')(2" - 2 ) · · • ( 2 - 2 " ) . μ

2

μ

- 1

4.4

For the graph G shown in Fig. 4.6, obtain the following: (a) A set of basis vectors for the circuit subspace, which are not fundamental circuit vectors with respect to any spanning tree of G. (b) A set of basis vectors for the cutset subspace, which are neither incidence sets nor fundamental cutsets with respect to any spanning tree of G.

4.5

(a) Test whether the cutset and circuit subspaces of the graph G of Fig. 4.6 are orthogonal complements of the vector space of G. (b) Express G as the ring sum of two subgraphs, one from the circuit subspace and the other from the cutset subspace.

4.6

Show that every subset of the edge set of a tree is a cut of the tree.

4.7

Show that a subgraph of a graph has an even number of edges if it belongs to both the cutset and the circuits subspaces of the graph.

4.8

A subset E' of edges of a graph is said to be independent if E' contains no circuits. Prove the following: (a) Every subset of an independent set is independent.

9β

GRAPHS AND VECTOR SPACES

Figure 4.6 (b) If / and / are independent sets containing k and k + 1 edges, respectively, then there exists an edge e that is in J but not in I, such that / U (e) is an independent set. 4.9

Repeat Exercise 4.8 replacing "circuit" by "cutset."

4.9

REFERENCES

4.1 4.2

S. MacLane and G. Birkhoff, Algebra, Macmillan, New York, 1967. P. R. Halmos, Finite-Dimensional Vector Spaces, Van Nostrand Reinhold, New York, 1958. 4.3 F. E. Hohn, Elementary Matrix Algebra, MacMillan, New York, 1958. 4.4 W. K. Chen, "On Vector Spaces Associated with a Graph," SIAM J. Appl. Math., Vol. 20, 526-529 (1971). 4.5 T. W. Williams and L. M. Maxwell, "The Decomposition of a Graph and the Introduction of a New Class of Subgraphs," SIAM J. Appl. Math., Vol. 20, 385-389 (1971). 4.6 R. Gould, "Graphs and Vectors Spaces," J. Math. Phys., Vol. 37, 193-214 (1958). 4.7 G. Kishi and Y. Kajitani, "Maximally Distant Trees and Principal Partition of a Linear Graph," IEEE Trans. Circuit Theory, Vol. CT-16, 323-330 (1969). 4.8 T. Ohtsuki, Y. Ishizaki, and H. Watanabe, "Topological Degrees of Freedom and Mixed Analysis of Electrical Networks," IEEE Trans. Circuit Theory, Vol. CT-17, 491-499 (1970). 4.9 Μ. N. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, New York, 1981. 4.10 P. M. Lin, "An Improved Algorithm for Principal Partition of Graphs," Proc. IEEE Intl. Symp. Circuits and Systems, Munich, Germany, 145-148 (1976). 4.11 J. Bruno and L. Weinberg, "The Principal Minors of a Matroid," Linear Algebra Its Appl., Vol. 4, 17-54, (1971).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 5

DIRECTED GRAPHS

In the last four chapters we developed several basic results in the theory of undirected graphs. Undirected graphs are not adequate for representing several situations. For example, in the graph representation of a traffic network where an edge may represent a street, we have to assign directions to the edges to indicate the permissible direction of traffic flow. As another example, a computer program can be modeled by a graph in which an edge represents the flow of control from one set of instructions to another. In such a representation of a program, directions have to be assigned to the edges to indicate the directions of the flow of control. An electrical network is yet another example of a physical system whose representation requires a directed graph. In this chapter we develop several basic results in the theory of directed graphs. We discuss questions relating to the existence of directed Euler circuits and Hamilton trails. We also discuss directed trees and their relationship with directed Euler trails.

5.1

BASIC DEFINITIONS AND CONCEPTS

We begin with the introduction of a few basic definitions and concepts relevant to directed graphs. A directed graph G = (V, E) consists of two sets: a finite set V of elements called vertices and a finite set £ of elements called edges. Each edge is associated with an ordered pair of vertices. We use the symbols v ,v ,... to represent the vertices and the symbols e , e ,... to represent the edges of a directed graph. If e, = (v„ v ), then υ, x

x

2

2

f

97

98

DIRECTED GRAPHS

and Vj are called the end vertices of the edge e,, v, being called the initial vertex and y the terminal vertex of e,. All edges having the same pair of initial and terminal vertices are called parallel edges. An edge is called a self-loop at vertex υ,, if v, is the initial as well as the terminal vertex of the edge. In the pictorial representation of a directed graph, a vertex is represented by a dot or a circle and an edge is represented by a line segment connecting the dots or the circles that represent the end vertices of the edge. In addition, each edge is assigned an orientation indicated by an arrow that is drawn from the initial to the terminal vertex. For example, if ;

V=

{Όι,υ ,υ ,υ ,υ ,υ ,υ } 2

3

4

5

6

Ί

and Ε — {β,, e , e , e , e , e , e , e } 2

3

A

5

6

7

R

such that ei = e

(v ,v ), x

2

=

(Vi,v ),

«3 =

("1.03)»

2

2

^ = (^3.^1). «5 = ( " 2 . " 4 ) . ^ e

= 7

(«3.^4),

= (v , 4

*e = (*>5.

v)

,

4

v ), 6

then the directed graph G = (V, E) will be represented as in Fig. 5.1. In this graph e, and e are parallel edges, and e is a self-loop. An edge is said to be incident on its end vertices. Two vertices are adjacent if they are the end vertices of some edge. If two edges have a common end vertex, then these edges are said to be adjacent. An edge is said to be incident out of its initial vertex and incident into its terminal vertex. A vertex is called an isolated vertex if no edge is incident on it. The degree d(Vj) of a vertex Vj is the number of edges incident on v The in-degree d~{v ) of Vj is the number of edges incident into and the out-degree ά (υ ) is the number of edges incident out of v And δ and δ ~ will denote the minimum out-degree and the minimum in-degree in a 2

7

r

t

+

+

;

r

99

BASIC DEFINITIONS AND CONCEPTS

Figure 5.1. A directed graph. directed graph. Similarly Δ and Δ" will denote the maximum out-degree and the maximum in-degree, respectively. For any vertex υ the sets Γ (ι>) and Γ~(υ) are defined as follows: +

+

r (v)

=

Γ »

= {νν|(Η>,

+

{w\(v,w)(EE}, v)EE).

For example, in the graph of Fig. 5.1 Γ ( υ , ) = {υ , i> } and Γ~(υ ) = {v , v , v }. Note that a self-loop at a vertex contributes to the in-degree as well as to the out-degree of the vertex. The following result is a consequence of the fact that every edge contributes 1 to the sum of the in-degrees as well as to the sum of the out-degrees of a directed graph. +

3

2

2

3

4

4

Theorem 5 . 1 . In a directed graph, sum of the in-degrees = sum of the out-degrees = m, where m is the number of edges of the graph. • Subgraphs and induced subgraphs of a directed graph are defined as in the case of undirected graphs (Section 1.2). The graph that results after ignoring the orientations of a directed graph G is called the underlying undirected graph of G. This undirected graph will be denoted by G . A directed walk in a directed graph G = (V, E) is a finite sequence of vertices v , i > , , . . . , v , such that (u,_,, υ,), 1 s i < k, is an edge in G. This directed walk is usually called a v -v directed walk. v is called the initial vertex, v is called the terminal vertex of this walk, and all other vertices are called internal vertices. The initial and terminal vertices of a directed walk are called its end vertices. Note that in a directed walk, edges and hence vertices can appear more than once. A directed walk is open if its end vertices are distinct; otherwise it is closed. u

0

k

0

k

k

0

100

DIRECTED GRAPHS

A directed walk is a directed trail if all its edges are distinct. A directed trail is open if its end vertices are distinct; otherwise it is closed. An open directed trail is a directed path if all its vertices are distinct. A closed directed trail is a directed circuit if all its vertices except the end vertices are distinct. A directed graph is said to be acyclic if it has no directed circuits. For example, the directed graph in Fig. 5.2 is acyclic. A sequence of vertices in a directed graph G is a walk in G if it is a walk in the underlying undirected graph Gu. For example, the sequence υ,, υ2, υ3, υ4, υ2, ν3 in the graph of Fig. 5.2 is a walk; but it is not a directed walk. Trail path and circuit in a directed graph are defined in a similar manner. A directed graph is connected if the underlying undirected graph is connected. A subgraph of a directed graph G is a component of G if it is a component of G„. In a directed graph G a vertex v, is strongly connected to a vertex vt if in G there exists a directed path from v, to v, and a directed path from v to v,. If v, is strongly connected to vr then, obviously, vJ is strongly connected to υ,. Every vertex is strongly connected to itself. If v, is strongly connected to vr and vt is strongly connected to vk, then it is easy to see that v, is strongly connected to vk. Therefore, in such a case, we simply state that the vertices υ,, vr and vk are strongly connected. A directed graph is strongly connected if all its vertices are strongly connected. For example, the graph in Fig. 5.3 is strongly connected. A maximal strongly connected subgraph of a directed graph G is called a strongly connected component of G. If a directed graph is strongly connected, then it has only one strongly connected component, namely, itself. Consider a directed graph G = (V, E). It is easy to see that each vertex of G is present in exactly one strongly connected component of G. Thus the vertex sets of the strongly connected components constitute a partition of the vertex set V of G.

Figure 5.2. An acyclic directed graph.

Figure 5.3. A strongly connected directed graph.

BASIC DEFINITIONS AND CONCEPTS

101

For example, the directed graph of Fig. 5.4a has three strongly connected components with {v , v , u , v }, {v }, and {v } as their vertex sets, which form a partition of the vertex set {«,, v , v , v , v , v } of the directed graph. It is interesting to note that there may be some edges in a directed graph that do not belong to any strongly connected component of the graph. For example, in the graph of Fig. 5.4a, the edges e,, e , e , e , and e are not present in any strongly connected component. Thus while the "strongly connected" property induces a partition of the vertex set of a graph, it may not induce a partition of the edge set. Union, intersection, ring sum, and other operations involving directed graphs are defined in exactly the same way as in the case of undirected graphs (see Section 1.5). The graph that results after contracting all the edges in each strongly connected component of a directed graph G is called the condensed graph G of G. For example, the condensed graph of the graph of Fig. 5.4a is shown in Fig. 5.4b. The vertex in G , which corresponds to a strongly connected component, is called the condensed image of the component. 2

3

4

5

t

6

2

3

4

5

6

6

7

g

c

c

ib) Figure 5.4. A graph and its condensed graph.

xo

102

DIRECTED GRAPHS

The rank and nullity of a directed graph are the same as those of the corresponding undirected graph. Thus if a directed graph G has m edges, η vertices, and ρ components, then the rank ρ and the nullity μ of G are given by ρ =

η-ρ

and μ =m — η +ρ . Next we define a minimally connected directed graph and study some properties of such graphs. A directed graph G is minimally connected if G is strongly connected, and the removal of any edge from G destroys its "strongly connected" property. For example, the graph of Fig. 5.5 is minimally connected. Obviously, a minimally connected directed graph can have neither parallel edges nor self-loops. We know that an undirected graph is minimally connected if and only if it is a tree (Exercise 2.13). By Theorem 2.5, a tree has at least two vertices of degree 1. Thus in a minimally connected undirected graph there are at least two vertices of degree 1. We now establish an analogous result for the case of directed graphs. In a strongly connected directed graph the degree of every vertex should be at least 2, since at each vertex there should be one edge incident out of the vertex and one edge incident into the vertex. In the following theorem we prove that a minimally connected directed graph has at least two vertices of degree 2. Theorem 5.2. If a minimally connected directed graph G has more than one vertex, then G has at least two vertices of degree 2. Proof. Since G is strongly connected and has more than one vertex, it must have at least one directed circuit. Thus for G, the nullity μ > 1. We shall prove the theorem by induction on μ.

Figure 5.5. A minimally connected directed graph.

BASIC DEFINITIONS AND CONCEPTS

103

If μ = 1, then G must be a directed circuit, and hence the theorem is true for this case. Let the theorem be true for all minimally connected directed graphs for which the nullity μ < k, with k > 2. Now consider a minimally connected graph G with μ = k. We shall show that the theorem is true for G. Two cases now arise. Case 1

Every directed circuit in G is of length 2.

In this case any two adjacent vertices of G are joined by exactly two edges oriented in different directions. Let G' be a simple undirected graph that has the same vertex set as G and in which two vertices are adjacent if and only if they are adjacent in G. Since G is connected, G' is also connected. Further, G' has no circuits because G has no directed circuit of length greater than 2. Thus G' is a tree. Hence, by Theorem 2.5, it has at least two pendant vertices. These two vertices have degree 2 in G, and the theorem is therefore true for this case. Case 2

G has a directed circuit C of length / > 3.

Since G is minimally connected, there is only one edge in G between any two adjacent vertices of C, and further, there is no edge in G between any two vertices that are not adjacent in C. Suppose G' is the graph that results after contracting the edges of C. Then G' has m -I edges and η -I +1 vertices, where m and η are, respectively, the number of edges and the number of vertices in G. Therefore the nullity of G' is equal to (m - / ) - ( « - / + 1 ) + 1 = k-

1.

Since G' is also minimally connected, it follows from the induction hypothesis that G' has at least two vertices v and v of degree 2. If one of these vertices, say v , is the condensed image of C, then C has at least one vertex, say v , which is of degree 2. In such a case v and v are two vertices of degree 2 in G. On the other hand, if neither v nor v is the condensed image of C, then v and v are two vertices of degree 2 in G. Hence the theorem. • x

2

x

3

2

x

x

3

2

2

We conclude this section with the definition of the "quasi-strongly connected" property. A graph is said to be quasi-strongly connected if for every pair of vertices v and v there is a vertex v from which there is a directed path to v and a directed path to v . Note that v need not be distinct from i>, or v . For example, the graph of Fig. 5.6 is quasi-strongly connected. x

2

3

2

x

3

2

DIRECTED GRAPHS

104

"β

Figure 5.6. A quasi-strongly connected graph. 5.2

GRAPHS AND RELATIONS

A binary relation on a set X = {*,, x ,...} is a collection of ordered pairs of elements of X. For example, if the set X is composed of men and R is the relation "is son of," then the ordered pair (*,, x ) means that x, is the son of Xj. This is also denoted as x, R x . A most convenient way of representing a binary relation Λ on a set X is by a directed graph, the vertices of which stand for the elements of X and the edges stand for the ordered pairs of elements of X defining the relation R. For example, the relation "is a factor o f on the set X= {2, 3, 4, 6, 9} is shown in Fig. 5.7. Consider a set X = {*,, x ,...} and a relation R on X. 2

t

f

2

1. Λ is reflexive if every element x, is in relation R to itself; that is, for every x, R x 2. R is symmetric if R x implies x R x,. 3. R is transitive if x, R Xj and x R x imply x R x . 4. R is an equivalence relation if it is reflexive, symmetric, and transitive. If R is an equivalence relation denned on a set 5, then we can uniquely partition S into subsets 5,, S ,...,S such that two elements χ and y of S belong to 5, if and only if χ R y. The subsets 5 , , S ,..., S are all called the equivalence classes induced by the relation R on the set S. r

t

s

)

2

k

t

k

k

2

Suppose the set X is composed of positive integers. Then: 1. The relation "is a factor o f is reflexive and transitive.

k

DIRECTED TREES OR ARBORESCENCES

105

Figure 5.7. A directed graph representing the relation "is a factor of." 2. The relation "is equal t o " is reflexive, symmetric, and transitive and is therefore an equivalence relation. The directed graph representing a reflexive relation is called a reflexive directed graph. In a similar way symmetric and transitive directed graphs are defined. We can now make the following observations about these graphs: 1. In a reflexive directed graph, there is a self-loop at each vertex. 2. In a symmetric directed graph, there are two oppositely oriented edges between any two adjacent vertices. Therefore an undirected graph can be considered as representing a symmetric relation if we associate with each edge two oppositely oriented edges. 3. The edge (v ,v ) is present in a transitive graph G if there is a directed path in G from u, to v . l

2

2

5.3

DIRECTED TREES OR ARBORESCENCES

A vertex υ in a directed graph G is a root of G if there are directed paths from ν to all the remaining vertices of G. For example, in the graph G of Fig. 5.6, the vertex i>, is a root. It is clear that if a graph has a root, it is quasi-strongly connected. In the following theorem we prove that the quasi-strongly connected property implies the existence of a root. Theorem 5.3. A directed graph G has a root if and only if it is quasi-strongly connected.

106

DIRECTED GRAPHS

Proof Necessity

Obvious.

Sufficiency Consider the vertices x , x ,...,x of G. Since G is quasistrongly connected, there exists a vertex y from which there is a directed path to x and a directed path to x . For the same reason there is a vertex y from which there is a directed path to y and a directed path to x . Clearly, y is also connected to x and x by directed paths through y . Proceeding in this manner, we see that there is a vertex y from which there is a directed path to y _ and x . Obviously y is a root since it is also connected to x ,... ,x„_ through y _ , . • x

2

n

2

x

2

3

2

3

x

3

2

2

n

n

x

x

x

n

n

n

A directed graph G is a tree if the underlying undirected graph is a tree. A directed graph G is a directed tree or arborescence if G is a tree and has a root. A vertex ν in G is called a leaf if d (i>) = 0. For example, the graph of Fig. 5.8 is a directed tree. In this graph the vertex υ, is a root and it is the only root. We present in the next theorem a number of equivalent characterizations of a directed tree. +

Theorem 5.4. Let G be a directed graph with η > 1 vertices. Then the following statements are equivalent: 1. G is a directed tree. 2. There exists a vertex r in G such that there is exactly one directed path from r to every other vertex of G. 3. G is quasi-strongly connected and loses this property if any edge is removed from it. 4. G is quasi-strongly connected and has a vertex r such that
»7

P

S

Figure 5.8. A directed tree.

»

4

DIRECTED TREES OR ARBORESCENCES

107

and ίΓ(υ) = 1,

υΦτ.

5. G has no circuits (not necessarily directed circuits) and has a vertex r such that rf»

=0

and d~(p) = \ ,

υΦτ.

6. G is quasi-strongly connected without circuits. Proof 1Φ2 The root r in G has the desired property. 2φ3 Obviously G is quasi-strongly connected. Suppose this property is not destroyed when an edge (u, v) is removed from G. Then there is a vertex ζ such that there are two directed paths from z, one to u and the other to v, which do not use the edge (u, v). Thus in G, there are two directed paths from ζ to υ and hence two directed paths from r to v. This contradicts statement 2. 3φ> 4 Since G is quasi-strongly connected, it has a root r. So, for every vertex υ Φ r, d~[v)>\

.

Suppose, for some vertex v, that d~(v)>l. Then there are two edges, say (*, v) and (y, v), incident into v. Thus there are two distinct paths from r to v, one through (x, v) and the other through (y, v). So even if we remove one of these two edges, the resulting graph still has r as a root and hence remains quasi-strongly connected. This contradicts statement 3 and so, d~(v) = l ,

υΦτ.

Finally, no edge can be incident into r, for otherwise the graph that results after removing this edge from G will still have r as its root and hence will be quasi-strongly connected. This will again contradict statement 3. Therefore d'(r) = 0.

108

DIRECTED GRAPHS

4φ5 The sum of in-degrees in G is equal to n — l. Therefore by Theorem 5.1, G has η - 1 edges. Since G is also connected, by Theorem 2.1, statement 3, it is a tree and hence is circuitless. 5=£> 6 G has η - 1 edges because the sum of in-degrees in G is equal to n — l. Since it is also circuitless, it follows from Theorem 2.2 that it is a tree. Therefore, there exists a unique path in G from vertex r to every other vertex. Such a path should be a directed path, for otherwise at least one of the vertices in this path will have its in-degree greater than 1, contradicting statement 5. Thus vertex r is a root of G. Therefore, by Theorem 5.3, G is quasi-strongly connected. 6^>1 Since G is quasi-strongly connected, it has a root, by Theorem 5.3. Further, it is a tree because it is connected and has no circuits. • A subgraph of a directed graph G is a directed spanning tree of G if the subgraph is a directed tree and contains all the vertices of G. For example, the subgraph consisting of the edges e , e , e , e , and e is a directed spanning tree of the graph of Fig. 5.6. We know that a graph G has a spanning tree if and only if G is connected. The corresponding theorem in the case of directed graphs follows. x

2

}

4

s

Theorem 5.5. A directed graph G has a directed spanning tree if and only if G is quasi-strongly connected. Proof Necessity If G has a directed spanning tree, then obviously the root of the directed tree is also a root of G. Therefore, by Theorem 5.3, G is quasi-strongly connected. Sufficiency Suppose G is quasi-strongly connected and it is not a directed tree. Then, by Theorem 5.4, statement 3, there are edges whose removal from G will not destroy the quasi-strongly connected property of G. Therefore, if we remove successively all these edges from G, then the resulting graph is a directed spanning tree of G. • A binary tree is a directed tree in which the out-degree of every vertex is at most 2. For example, the graph of Fig. 5.9 is a binary tree. In the study of certain aspects of computer science, for example, analysis of algorithms, search techniques, and so on, binary trees are found to be very useful. One problem in this context is the following: Given η weights w , w ,..., w and η lengths binary tree with η leaves v , v ,...,v„ such that: x

2

n

x

2

1. w , 1 s j < η, is the weight of vertex υ,. t

/ , . . . , /„, construct a 2

DIRECTED TREES OR ARBORESCENCES

000

001 010

011

110

109

111

Figure 5.9. A binary tree. 2. / , 1 < ϊ < «, is the length of the path from the root to υ,. 3. Σ " w,l,, called the weighted path length, is minimum. (

=1

A situation where a more general problem arises is now described. The set of words such that no word is the beginning of another is called a prefix code. Clearly, if a sequence of letters is formed by concatenating the words of a prefix code, the sequence can be decomposed into individual words of the prefix code by reading the sequence from left to right. For example {000, 001, 01, 10, 11} is a prefix code and the sequence 1011001000 formed by concatenating the words of this code is easily decomposed into the words 10, 11, 001, and 000. Suppose S = {0, 1, 2 , . . . , m - 1} is an alphabet of m letters. Let Γ be a directed tree such that: 1. The out-degree of each vertex is at most m. 2. Each out-going edge at a vertex is associated with a letter of the alphabet 5 such that no two such edges are associated with the same letter. Then each leaf ν can be associated with a word that is formed by concatenating all the letters in the order in which they appear on the edges along the path from the root to v. If may be verified that the words so associated with the leaves form a prefix code. For example, the prefix code corresponding to the directed tree of Fig. 5.9 is the following: {000,001,010,011,10, 110,111} . Now the problem is: Given m and the lengths /,, l ,... , / „ , construct a prefix code using the letters of the alphabet 5 = {0, 1, 2 , . . . ,m - 1} with /,, / , . . . , /„ as the lengths of the code words. A discussion of this and certain related problems may be found in Swamy and Thulasiraman [5.1]. 2

2

110

5.4

DIRECTED GRAPHS

DIRECTED EULERIAN GRAPHS

A directed Euler trail in a directed graph G is a closed directed trail that contains all the edges of G. An open directed Euler trail is an open directed trail containing all the edges of G. A directed graph possessing a directed Euler trail is called a directed Eulerian graph. The graph G of Fig. 5.10 is a directed Eulerian graph since the sequence of edges e , e , e , e , e , e constitutes a directed Euler trail in G. The following theorem gives simple and useful characterizations of directed Eulerian graphs. The proof of this theorem follows along the same lines as that for Theorem 3.1. x

2

3

4

5

6

Theorem 5.6. The following statements are equivalent for a connected directed graph G: 1. G is a directed Eulerian graph. 2. For every vertex υ of G, d~(v) = d (v). 3. G is the union of some edge-disjoint directed circuits. +

Consider, for example, the directed easy to verify that it satisfies property union of the edge-disjoint circuits {e , Another theorem that can be easily 2

•

Eulerian graph G of Fig. 5.10. It is 2 of Theorem 5.6. Further it is the e } and {e , e , e , e }. proved is the following one. 3

x

4

s

6

Theorem 5.7. A directed connected graph possesses an open directed Euler trail if and only if the following conditions are satisfied: 1. In G there are two vertices v and v such that x

2

and d~(v ) = d (v ) +

2

Figure 5.10. A directed Eulerian graph.

2

+ l. e

3

Figure 5.11. A graph with an open directed Euler trail.

DIRECTED EULERIAN GRAPHS

111

2. For every vertex v, different from v and v , x

d »

2

= d » .

•

For example, the graph G of Fig. 5.11 satisfies the conditions of Theorem 5.7. The sequence of edges e , e , e , e , e , e is an open directed Euler trail in G. An interesting application of Theorem 5.6 is discussed next. Let S = { 0 , 1 , . . . , s - 1} be an alphabet of s letters. Obviously we can construct s" different words of length η using the letters of the alphabet S. A de Bruijn sequence is a circular sequence a , a,, a ,...,a _ of length L = s" such that for every word ω of length n, there is a unique ι such that x

2

3

4

5

6

0

i i

**/

1

+

+n - 1

2

L

x

'

where the computation of the indices is modulo L. de Bruijn sequences find application in the study of coding theory and communication circuits. Reference [5.2] may be consulted for more information on these sequences. We now wish to consider the following problem: For every s a 2 and every integer n, does there exist a de Bruijn sequence over the alphabet S = {0, 1, 2 , . . . , s - 1}? The answer is in the affirmative, as we shall see now. We show that for every s a 2 and every η there exists a directed Eulerian graph G „ such that each directed Euler trail in this graph corresponds to a de Bruijn sequence of length s" over the alphabet S = { 0 , 1 , 2 , . . . , s - 1}. The graph G = (V, E) is constructed as follows: s

sn

1. V consists of all the s"' words of length η -1 over the alphabet S. 2. £ is the set of all the s" words of length η over S. 3. The edge b b --b has b b - · · b _ as its initial vertex and ^ 2 ^ 3 · · · ft„ as its terminal vertex. Note that at each vertex there are s incoming edges and s outgoing edges. Thus for each vertex v, d~(v) = d » . 1

l

2

n

l

2

n

x

For example, the graphs G and G are as shown in Fig. 5.12a and 5.12ft. Suppose there exists a directed Euler trail C in G . If we concatenate the first letters of the words, represented by the edges of G , in the order in which they appear in C, Then we get a sequence of length s" such that every subsequence of η consecutive letters in this sequence will correspond to a unique word and no two different subsequences will correspond to the same word. Thus the sequence constructed will be a de Bruijn sequence. For example, the sequence of edges 000, 001, 011, 111, 110, 101, 010, 100 is a directed Euler trail in G . Concatenating the first letters of these words we get the de Bruijn sequence 00011101. 2 4

2

3

sn

sn

2

3

0000

1111 (a) 000

111 (b)

Figure 5.12. (e) Graph G2A. (b) Graph G2 3. 112

DIRECTED SPANNING TREES AND DIRECTED EULER TRAILS

113

Thus to show that there exists a de Bruijn sequence for every $ s 2 and every n, we have to prove that G „ is a directed Eulerian graph. This we do in the following theorem. s

Theorem 5.8. G

is a directed Eulerian graph.

sn

Proof. G „ is connected since for any two vertices i>, = b b · · · fr„_, and 2 ~ i z " ' n - i > there is a directed path from i>, to v consisting of the following edges: s

C

υ

x

c

2

c

2

i

b

b

2

•

··

b

„-i i> c

bb 2

3

•• •b _ c c ,6 _,c,c n

x

x

n

2

2

· · · c„_, .

Since, as we have seen before, the in-degree of every vertex in G „ is equal to its out-degree, it follows from Theorem 5.6 that G is a directed Eulerian graph. • s

s n

Corollary 5.8.1. For every i > 2 and every η there exists a de Bruijn sequence. •

5.5

DIRECTED SPANNING TREES AND DIRECTED EULER TRAILS

Let G be a directed Eulerian graph without self-loops. In this section we relate the number of directed Euler trails in G to the number of directed spanning trees of G. Let v ,v ,... ,v„ denote the vertices of G. Consider a directed Euler trail C in G. Let e be any edge of G incident into u,. For every ρ = 2, 3 , . . . , η, let e denote the first edge on C to enter vertex υ after traversing ev . For example, in the directed Eulerian graph shown in Fig. 5.10, the sequence of edges e,, e , e , e , e , e is a directed Euler trail. If we choose e = e , then e = e , e = e , and e = e . Let Η denote the subgraph of G on the edge set [e , e^,..., e^. t

2

H

;

2

;)

5

J2

x

4

3

;j

5

6

6

u

4

J2

Lemma 5.1. Let C be a directed Euler trail of a directed Eulerian graph G. The subgraph H, as defined earlier, is a directed spanning tree of G with root υ j . Proof. Clearly in H, d~(v ) = 0, and d'{v ) = \ for all ρ = 2,3,...,η. Suppose Η has a circuit C. Then v is not in C , for otherwise either d ( i ) 0 or )> 1 for some other vertex υ on C . For the same reason C is a directed circuit. Since d~(v) = 1 for every vertex ν on C , no edge not in C enters any vertex in C. This means that the edge e, which is the first edge of C to enter a vertex of C after traversing , does not belong to H, contradicting the definition of H. Thus Η has no circuits. Now it follows from Theorem 5.4, statement 5, that Η is a directed spanning tree of G. • l

p

1

u

>

114

DIRECTED GRAPHS

Given a directed spanning tree Η of an η-vertex directed Eulerian graph G without self-loops, let υ, be the root of Η and e an edge incident into u, in G. Let e for ρ = 2, 3 , . . . , η be the edge entering v in H. We now describe a method for constructing a directed Euler trail in G. h

y

p

1. Start from vertex υ, and traverse backward on any edge entering υ, other than e if such an edge exists, or on e if there is no other alternative. 2. In general on arrival at a vertex v , leave it by traversing backward on an edge entering v that has not yet been traversed and, if possible, other than e^. Stop if no untraversed edges entering v exist. h

h

p

p

p

In this procedure every time we reach vertex v ¥=Vi, there will be an untraversed edge entering v because the in-degree of every vertex in G is equal to its out-degree. Thus this procedure terminates only at the vertex υ after traversing all the edges incident into i>,. Suppose there exists in G an untraversed edge («, υ) when the procedure terminates at υ,. Since the in-degree of u is equal to its out-degree, there exists at least one untraversed edge incident into u. If there is more than one such untraversed edge, then one of these will be the edge y entering u in H. This follows from step 2 of the procedure. This untraversed edge y will lead to another untraversed edge that is also in H. Finally we shall arrive at v and shall find an untraversed edge incident into υ,. This is not possible since all the edges incident into u, will have been traversed when the procedure terminates at i>,. Thus all the edges of G will be traversed during the procedure we described above, and indeed a directed Euler trail is constructed. Since at each vertex v there are (d~(v ) - 1)! different orders for picking the incoming edges (with e at the end), it follows that the number of distinct directed Euler trails that we can construct from a given directed spanning tree Η and e is p

p

λ

t

p

p

t

P

h

Π (£/>,)-!)!.

p = l

Further, each different choice of Η will yield a different e for some p = 2 , 3 , . . . , « , which will in turn result in a different entry to v after traversing e in the resulting directed Euler trail. Finally, since the procedure of constructing a directed Euler trail is the reversal of the procedure for constructing a directed spanning tree, it follows that every directed Euler trail can be constructed from some directed spanning tree. Thus we have proved the following theorem due to Van AardenneEhrenfest and de Bruijn [5.3]. p

t

DIRECTED HAMILTONIAN GRAPHS

115

Theorem 5.9. The number of directed Euler trails of a directed Eulerian graph G without self-loops is

r (G) Π („)-1)! d

p = l

where r (G) root. • d

is the number of directed spanning trees of G with v

t

as

Since the number of directed Euler trails is independent of the choice of the root, we get the following. Corollary 5.9.1. The number of directed spanning trees of a directed Eulerian graph is the same for every choice of root. • We establish in Section 6.9 a formula for evaluating

5.6

T (G). d

DIRECTED HAMILTONIAN GRAPHS

A directed circuit in a directed graph G is a directed Hamilton circuit of G if it contains all the vertices of G. A directed path in G is a directed Hamilton path of G if it contains all the vertices of G. A graph is a directed Hamiltonian graph if it has a directed Hamilton circuit. For example, the sequence of edges e , e , e , e , e , e is a directed Hamilton circuit in the graph of Fig. 5.13a. In Fig. 5.136, the sequence of edges e,, e , e , e , e is a directed Hamilton path. Note that this latter graph has no directed Hamilton circuit. Characterizing a directed Hamiltonian graph is as difficult as characterizing an undirected Hamiltonian graph. However, there are several sufficient conditions that guarantee the existence of directed Hamilton circuits and paths. We now discuss a few of these conditions. A directed graph G is complete if its underlying undirected graph is complete. The following theorem is due to Moon [5.4]. x

2

3

4

2

3

4

5

6

5

Theorem 5.10. Let u be any vertex of a strongly connected complete directed graph with « s 3 vertices. For each k, 3 s f c < « , there is a directed circuit of length k containing u. Proof. Let G - (V, E) be a strongly connected complete directed graph with n > 3 vertices. Consider any vertex u of G. Let 5 = T ( u ) and T = F~(u). Since G is strongly connected, neither S nor Γ is empty. For the +

116

DIRECTED GRAPHS

e

V Figure 5.13. (a) A directed Hamiltonian graph, (b) A directed graph with a Hamilton path but with no Hamilton circuits.

i

'if

'7

e

i'·

\f3

(6)

same reason there is a directed edge from a vertex v E S to a vertex wET. Thus « is in a directed circuit of length 3 (see Fig. 5.14). We prove the theorem by induction on k. Assume that u is in directed circuits of all lengths between 3 and p, where p
Figure 5.14

DIRECTED HAMILTONIAN GRAPHS

117

Figure 5.15 on C such that the directed edges (v„ v) and (v, v ) are in G. Thus in this case u is in the directed circuit u = v , i>,, v ,..., υ,, v, ..., υ = u of length ρ + 1. Otherwise, let S be the set of all those vertices not in C, which are the terminal vertices of edges directed away from the vertices of C, and let Τ be the set of all those vertices not in C, which are the initial vertices of edges directed toward vertices in C. Again, since G is strongly connected, neither S nor Τ is empty. Further, there exists an edge directed from a vertex υ £ S to a vertex wET. Thus u is in the directed circuit u = υ , υ, w, v ,..., v = u of length ρ + 1 (see Fig. 5.15). • l+i

0

2

0

Corollary 5.10.1. A Hamiltonian. •

strongly

connected

complete

2

directed

p

graph

is

Next we state without proof a very powerful theorem due to GhouilaHouri. The proof of this theorem may be found in [5.5] (pp. 196-199). Theorem 5.11. Let G be a strongly connected η-vertex graph without parallel edges and self-loops. If for every vertex ν in G

then G has a directed Hamilton circuit.

•

The following result generalizes Dirac's result (Corollary 3.4.1) for undirected graphs, and it can be proved using Theorem 5.11 and Exercise 5.6. Corollary 5.11.1. Let G be a directed η-vertex graph without parallel edges or self-loops. If min(S~, δ ) ^ 5 Π > 1 , then G contains a directed Hamilton circuit. • +

118

DIRECTED GRAPHS

We conclude this section with a result on the existence of a directed Hamilton path. Theorem 5.12. If a directed graph G = (V, E) is complete, then it has a directed Hamilton path. Proof. Consider a directed path P: a , a ,..., a of maximum length in G. Suppose b is a vertex not lying on P. Then the edge (b,at)0E, for otherwise the path b, a,, a ,...a will be longer than P. Therefore, (a b)EE. Now (b,a2)0E, for otherwise a,, b, a ,a ,... ,a will be a path longer than P. Thus (a , b)E.E. Repeating this argument we find that (a , b) G E. But this is a contradiction, for α ,α ,.... ,a , b will be a path longer than P. Thus there is no vertex outside P, and hence Ρ is a directed Hamilton path in G. • x

2

2

p

p

u

2

3

p

2

p

λ

2

p

For more general conditions for the existence of Hamilton circuits and paths, see Exercises 5.14 to 5.16.

5.7

ACYCLIC DIRECTED GRAPHS

In this section we study some properties of an important class of directed graphs, namely, the acyclic directed graphs. As we know, a directed graph is acyclic if it has no directed circuits. Obviously the simplest example of an acyclic directed graph is a directed tree. The main result of this section is that we can label the vertices of an η-vertex acyclic directed graph G with integers from the set { 1 , 2 , . . . , n) such that the presence of the edge (/, j) in G implies that i* < j . Note that the edge (t, / ) is directed from vertex / to vertex j . Ordering the vertices in this manner is called topological sorting. For example, the vertices of the acyclic directed graph of Fig. 5.16 are topologically sorted. The proof of the main result of this section depends on the following theorem. Theorem 5.13. In an acyclic directed graph G there exists at least one vertex with zero in-degree and at least one vertex with zero out-degree. Proof. Let P: v , v ,..., v be a maximal directed path in G. We claim that the in-degree of rj, and the out-degree of v are both equal to zero. If the in-degree of υ, is not equal to zero, then there exists a vertex w such that the edge (w, υ,) is in G. We now examine two cases. x

2

p

p

Case 1 Suppose w Φ υ,, for any i, 1 < / < p . Then there is a directed path P': w, D, , D , . . . , v that contains all the edges of P. This contradicts that Ρ is a maximal directed path. 2

p

TOURNAMENTS

119

2

1

Figure 5.16. An acyclic directed graph. Case 2 Suppose w = v, for some i. Then in G there is a directed circuit C: v , v ,..., y,, o,. This is again a contradiction since G has no directed circuits. x

2

Thus there is no vertex w such that the edge (w, is in G. In other words the in-degree of υ, is zero. Following a similar reasoning we can show that the out-degree of v is zero. • p

To topologically sort the vertices of an η-vertex acyclic directed graph G, we proceed as follows. Select any vertex with zero out-degree. Since G is acyclic, by Theorem 5.13, there is at least one such vertex in G. Label this vertex with the integer n. Now remove from G this vertex and the edges incident on it. Let G' be the resulting graph. Since G' is also acyclic, we can now select a vertex whose out-degree in G' is zero. Label this with the integer η - 1. Repeat this procedure until all the vertices are labeled. It is now easy to verify that this procedure results in a topological sorting of the vertices of G. For example, the labeling of the vertices of the graph of Fig. 5.16 has been done according to this procedure.

5.8

TOURNAMENTS

A tournament is a complete directed graph. It derives its name from its application in the representation of structures of round-robin tournaments. In a round-robin tournament several teams play a game that cannot end in a tie, and each team plays every other team exactly once. In the directed graph representation of the round-robin tournament, vertices represent teams and an edge u ) is present in the graph if the team represented by the vertex v defeats the team represented by the vertex v . Clearly, such a 2

t

2

120

DIRECTED GRAPHS

Figure 5.17. A tournament.

directed graph has no parallel edges and self-loops, and there is exactly one edge between any two vertices. Thus it is a complete directed graph and hence a tournament. A tournament is shown in Fig. 5.17. The teams participating in a tournament can be ranked according to their scores. The score of a team is the number of teams it has defeated. This motivates the definition of the score sequence of a tournament. The score sequence of an π-vertex tournament is the sequence (s,, s ,... ,s ) such that each s, is the out-degree of a vertex of the tournament. An interesting characterization of a tournament in terms of the score sequence is given in the following theorem. 2

n

Theorem 5.14. A sequence of nonnegative integers s,, s , . . . , s„ is the score sequence of a tournament G if and only if 2

ι. Σ * , - ^ . Σ

2.

s, > ^ ~ k

^

k

for all k < n.

^

1=1

Proof Necessity Note that the sum Σ" , s, is equal to the number of edges in the tournament G (Theorem 5.1). Since a tournament is a complete graph, it has n(n -1)12 edges, where η is the number of vertices. Thus =

1=

1

To prove condition 2, consider the subtournament on any k vertices i>,, v ,..., v . This subtournament has k(k - l ) / 2 edges. Therefore in the entire tournament 2

k

i-l

k(k-\) 2

EXERCISES

121

since there may be edges directed from the vertices in the subtournament to those outside the subtournament. Sufficiency

See [5.5] (pp. 107-109).

•

Suppose we can order the teams in a round-robin tournament such that each team precedes the one it has defeated. Then we can assign the integers 1 , 2 , . . . , η to the teams to indicate their ranks in this order. Such a ranking is always possible since in a tournament there exists a directed Hamilton path (Theorem 5.12), and it is called ranking by a Hamilton path. Note that ranking by a Hamilton path may not be the same as ranking by the score. Further, a tournament may have more than one directed Hamilton path. In such a case there will be more than one Hamilton path ranking. However, there exists exactly one directed Hamilton path in a transitive tournament. This is stated in the following theorem, which is easy to prove. Theorem 5.15. In a transitive tournament there exists exactly one directed Hamilton path. • For other ranking procedures see Bondy and Murty [5.6], Kendall [5.7], and Moon and Pullman [5.8].

5.9

FURTHER READING

Berge [5.5], Harary [5.9], and Chartrand and Lesniak [5.10] are good references for a number of results on directed graphs. Moon [5.11] is a monograph exclusively devoted to the study of tournaments. See also Nash-Williams [5.12]. Directed trees are used in the computer representation of combinatorial objects. For an extensive discussion of this topic Knuth [5.13] and [5.14], Alio, Hopcroft, and Ullman [5.15], and Reingold, Nievergelt, and Deo [5.16] may be referred. Directed trees also arise in several other applications such as the topological study of electrical networks. In this study one is interested in computing the number of directed trees of a directed graph. This question is discussed in Chapter 6. Harary, Norman, and Cartwright [5.17] and Robert [5.18] discuss several applications of directed graphs. 5.10

EXERCISES

5.1

Let G be a directed graph without self-loops and parallel edges. Let max{5 ~,δ } = k. Prove the following: +

122

DIRECTED GRAPHS

(a) G has a directed path of length at least k, (b) G has a directed circuit of length at least k + 1, if k > 0 . 5.2

Show that the edges of an undirected graph G = (V, E) can be oriented so that in the resulting directed graph \d (v) - d'(v)\ < 1 for all υ E V . +

5.3

A directed cut in a directed graph is a cut (S, S) whose edges are all oriented away from 5 or all oriented toward S. Show that an edge of a directed graph belongs either to a directed circuit or to a directed cutset, but no edge belongs to both. (This is a special case of a more general result known as the "arc coloring lemma," which we prove in Chapter 10. See Theorem 10.31.)

5.4

For a directed graph G with at least one edge, prove that the following are equivalent: (a) G has no directed circuits. (b) Each edge of G is in a directed cutset.

5.5

For the (a) (b) (c)

5.6

Show that an η-vertex directed graph without parallel edges or self-loops is strongly connected if

a connected directed graph G with at least one edge, prove that following are equivalent: G is strongly connected. Every edge of G lies on a directed circuit. G has no directed cutsets.

min{S~, δ } > +

.

5.7

Show that a strongly connected graph that contains a circuit of odd length also contains a directed circuit of odd length.

5.8

Let G = (V, E) be a directed graph such that (a) d\x) - d~(x) = I = d~(y) - d\y) and (b) d " » = d'(v) for υ Ε V- {x, y). Show that there are / edge-disjoint directed x-y paths in G.

5.9

Show that a strongly connected graph has a spanning directed walk.

5.10 What is the longest circular sequence formed out of three symbols x, y, and ζ such that no subsequence of four symbols is repeated? Give one such sequence. 5.11 Find a circular sequence of seven 0's and seven l's such that all four-digit binary sequences except 0000 and 1111 appear as subsequences of the sequence.

EXERCISES

5.12

123

Prove that a directed Hamilton circuit of G corresponds to a directed Euler trail of G _ . Is it true that G „ always has a directed Hamilton circuit? sn

sn

l

s

5.13

Show that the number of directed Euler trails in a directed graph, having η vertices and m>2n edges, is even.

5.14

(a) Let G be a strongly connected nontrivial η-vertex directed graph with no parallel edges or self-loops. Prove that G has a directed Hamilton circuit if, for every pair u, ν of distinct nonadjacent vertices, d(u) + d(v) > In - 1

(Meyniel [5.19]).

(b) Let G be a nontrivial π-vertex directed graph G = (V, E) with no parallel edges or self-loops. Prove that G has a directed Hamilton circuit if, whenever u and i; are distinct vertices with (M, v)f£E, d (u) + d~(v)>n +

5.15

(Woodall [5.20]).

Let G = (V, E) be an η-vertex directed graph without parallel edges or self-loops such that for every pair of distinct nonadjacent vertices u and ν d(u) + d{v) > In - 3 . Prove that G contains a directed Hamilton path. Note: This result can be proved using Exercise 5.14. One corollary of this that can be proved independently is: An Λ-vertex directed graph G = (V, E) without parallel edges or self-loops contains a directed Hamilton path if, for every vertex ν G V, d {v) + +

5.16

d-(v)^n-\.

A directed graph G = (V, E) is Hamilton connected if G contains a directed Hamilton u-υ path for every distinct vertices u and ν of G. Let G be a nontrivial η-vertex directed graph without parallel edges or self-loops such that for every pair u, ν of distinct vertices with (u,v)0E,

d (u) + d~(v)>n +

+l .

Prove that G is Hamilton connected (Overbeck-Larisch [5.21]). 5.17

Show that every tournament is strongly connected or can be transformed into a strongly connected tournament by reversing the orientation of just one edge.

124

DIRECTED GRAPHS

5.18

A subset 5 C V is an independent set of a graph G = (V, E) if no two of the vertices in S are adjacent. Prove that a directed graph G without self-loops has an independent set 5 such that each vertex υ of G not in S can be reached from a vertex in S by a directed path of length at most 2 (Chvatal and Lovasz [5.22]). An interesting corollary of this result is the following: A tournament contains a vertex from which every vertex can be reached by a directed path of length at most 2.

5.19

Show that the scores s, of an η-vertex tournament satisfy Σι, 1=1

5.11

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15

2

= Σ(π-ι-ο . 2

1=1

REFERENCES

Μ. N. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, New York, 1981. S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, 1967. T. Van Aardenne-Ehrenfest and N. G. de Bruijn, "Circuits and Trees in Oriented Linear Graphs," Simon Stevin, Vol. 28, 203-217 (1951). J. W. Moon, "On Subtournaments of a Tournament," Canad. Math. Bull., Vol. 9, 297-301 (1966). C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976. M. G. Kendall, "Further Contributions to the Theory of Paired Comparisons," Biometrics, Vol. 11, 43-62 (1955). J. W. Moon and N. J. Pullman, "On Generalized Tournament Matrices," SIAM Rev., Vol. 12, 384-399 (1970). F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969. G. Chartrand and L. Lesniak, Graphs and Directed Graphs, Wadsworth and Brooks/Cole, Pacific Grove, Calif., 1986. J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York, 1968. C. St. J. A. Nash-Williams, "Hamiltonian Circuits," in Studies in Graph Theory, Part II, MAA Press, 1975, Washington, D.C., pp. 301-360. D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, Addison-Wesley, Reading, Mass., 1968. D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching, Addison-Wesley, Reading, Mass.,1973. Α. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974.

REFERENCES

125

5.16 Ε. Μ. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, N.J., 1977. 5.17 F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs, Wiley, New York, 1965. 5.18 F. S. Robert, Discrete Mathematical Models with Applications to Social, Biological and Environmental Problems, Prentice-Hall, Englewood Cliffs, N.J., 1976. 5.19 M. Meyniel, "Une Condition Suffisante d'Existence d'un Circuit Hamiltonian dans un Graph Oriente," J. Combinatorial Theory, Vol. 14B, 137-147, (1973). 5.20 D. R. Woodall, "Sufficient Conditions for Circuits in Graphs," Proc. London, Math. Society, Vol. 24, 739-755, (1972). 5.21 M. Overbeck-Larisch, "Hamiltonian Paths in Oriented Graphs," J. Combinatorial Theory, Vol. 21B, 76-80 (1976). 5.22 V. Chvatal and L. Lovasz, "Every Directed Graph Has a Semi-Kernel," in Hypergraph Seminar, C. Berge and D. K. Ray-Chaudhuri, Eds., Springer, New York, 1974, p. 175.

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 6

MATRICES OF A GRAPH

In this chapter we introduce the incidence, circuit, cut, and adjacency matrices of a graph and establish several properties of these matrices that help to reveal the structure of a graph. The incidence, circuit, and cut matrices arise in the study of electrical networks because these matrices are the coefficient matrices of Kirchhoff s equations that describe a network. Thus the properties of these matrices and other related results to be established in this chapter form the basis of graph-theoretic study of electrical networks and systems. The properties of the adjacency matrix to be discussed here form the basis of the signal flow graph approach, which is a very powerful tool in the study of linear systems. Signal flow graph theory is developed in Section 6.11. Our discussions of incidence, circuit, and cut matrices are mainly with respect to directed graphs. However, these discussions become valid for undirected graphs too if addition and multiplication are in GF(2), the field of integers modulo 2.

6.1

INCIDENCE MATRIX

Consider a graph G with η vertices and m edges and having no self-loops. The all-vertex incidence matrix A = [a, ] of G has η rows, one for each vertex, and m columns, one for each edge. The element a of A is defined as follows: c

7

it

126

c

INCIDENCE MATRIX

127

G is directed 1,

if the y'th edge is incident on the i'th vertex and oriented away from it; if the y'th edge is incident on the i'th vertex and oriented toward it; if the y'th edge is not incident on the i'th vertex .

-1, 0,

G is undirected _ri,

'/-\0,

fl

if the y'th edge is incident on the i'th vertex; ot otherwise.

A row of A will be referred to as an incidence vector of G. Two graphs and their all-vertex incidence matrices are shown in Fig. 6.1a and 6.1b. c

1

1

-1

0

ο

0

-

0

-

0

1

0

0

-

1

0

o

ι

1

1

ο

0

1

-

0

o

1

o

0

o - i

1

-1

1

is)

Figure 6.1 (a) A directed graph G and its all-vertex incidence matrix, (b) An undirected graph G and its all-vertex incidence matrix.

128

MATRICES OF A GRAPH

"4

V

5

"i 1

1

e

Ί

0

0

0

0

0

0

1

1

1

0

1

1

0

1

0

1

1

0

0 1

1 0

0

0

0

0 0

1

0

e

(i>>

Figure 6.1. (Continued) It should be clear from the preceding definition that each column of A contains exactly two nonzero entries, one +1 and one - 1 . Therefore we can obtain any row of A from the remaining η - 1 rows. Thus any η - 1 rows of A contain all the information about A . In other words the rows of A are linearly dependent. An (n - l)-rowed submatrix A of A will be referred to as an incidence matrix of G. The vertex that corresponds to the row of A that is not in A will be called the reference vertex of A. Note that c

c

c

c

c

c

c

rank(.4) = r a n k ( / l ) < n - l .

(6.1)

c

Now we show that in the case of a connected graph, rank of A is in fact equal to η - 1. This result is based on the following theorem. c

Theorem 6.1. The determinant of any incidence matrix of a tree is equal to ±1.

INCIDENCE MATRIX

129

Proof. Proof is by induction on the number η of vertices in a tree. Any incidence matrix of a tree on two vertices is just a 1 x 1 matrix with its only entry being equal to ± 1 . Thus the theorem is true for η = 2. Note that the theorem does not arise for η = 1. Let the theorem be true for 2 s « < t . Consider a tree Τ with k + 1 vertices. Let A denote an incidence matrix of T. By Theorem 2.5, Γ has at least two pendant vertices. Let the i'th vertex of Γ be a pendant vertex, and let this not be the reference vertex of A. If the only edge incident on this vertex is the /th one, then in A a,, = ± l

and

a, = 0, ;

]ΦΙ.

If we now expand the determinant of A by the /"th row, then det(A) = ± ( - l ) ' ' d e t ( A ' ) +

(6.2)

where A' is obtained by removing the i'th row and the /th column from A. Suppose Τ is the graph that results after removing the i'th vertex and the /th edge from T. Clearly 7" is a tree because the /'th vertex is a pendant vertex and the /th edge is a pendant edge in T. Further it is easy to verify that A' is an incidence matrix of T'. Since T is a tree on η - 1 vertices, we have by the induction hypothesis that detA' = ± l .

(6.3)

This result in conjunction with (6.2) proves the theorem for η = k + 1.

•

Since a connected graph has at least one spanning tree, it follows from Theorem 6.1 that in any incidence matrix A of a connected graph with η vertices there exists a nonsingular submatrix of order η - 1 . Thus for a connected graph, rank(A) = η - 1.

(6.4)

Since rank(A ) = rank(A), we get the following theorem. c

Theorem 6.2. The rank of the all-vertex incidence matrix of an n-vertex connected graph G is equal to η - 1, the rank of G. • An immediate consequence of Theorem 6.2 is the following. Corollary 6.2.1. If an π-vertex graph has ρ components, then the rank of its all-vertex incidence matrix is equal to η - ρ, the rank of G. •

130

MATRICES OF A GRAPH

6.2

CUT MATRIX

To define the cut matrix of a directed graph we need to assign an orientation to each cut of the graph. Consider a directed graph G = (V, E). If V is a nonempty subset of V, then we may recall ^Chapter 2) that the set of edges connecting the vertices in V to those in V is a cut, and this cut is denoted as (ν,,,ν,,). The orientation of {V , V ) may be assumed to be either from V to_ V or from V to V . Suppose we assume that the orientation is from V to V . Then the orientation of an edge in (V ,V ) is said to agree with the orientation of the cut (V ,V ) if the edge is oriented from a vertex in V to a vertex in V . The cut matrix Q = [q ] of a graph G with m edges has m columns and as many rows as the number of cuts in G. The entry q is defined as follows: a

a

a

a

a

a

a

a

a

a

a

a

a

a

c

a

a

a

tj

tj

G is directed 1,

if the /th edge is in the ith cut and its orientation agrees with the cut orientation; I - 1 , if the /th edge is in the ith cut and its orientation does not agree with the cut orientation; 0 , if the /th edge is not in the ith cut. G is undirected _ ί 1, Q'i ~ \ 0 ,

if the /th edge is in the /th cut; otherwise .

A row of Q will be referred to as a cut vector. Three cuts of the directed graph of Fig. 6.1a are shown in Fig. 6.2. In each case the cut orientation is shown in dashed lines. The submatrix of Q corresponding to these cuts is c

c

e

i

Cutl ' 1 Cut 2 1 Cut 3 0

e

e

2

3

0 1 1 0 -1 1

0 0 0

1 0 1

«6

«7

1 0" -1 1 1 0

The corresponding submatrix in the undirected case can be obtained by replacing the - l ' s in the matrix by + l ' s . Consider next any vertex v. The nonzero entries in the corresponding incidence vector represent the edges incident on v. These edges form the cut (ν, V- v). If we assume that the orientation of this cut is from υ to V- v, then we can see from the definitions of cut and incidence matrices that the row in Q corresponding to the cut (ν, V- v) is the same as the row in A corresponding to the vertex v. Thus A is a submatrix of Q . c

c

c

c

CUT MATRIX

Cut

3

131

Figure 6.2. Some cuts of the graph of Fig. 6.1a.

Now we proceed to show that the rank of Q is equal to that of A . To do so we need the following theorem. c

c

Theorem 6.3. Each row in the cut matrix Q can be expressed, in two ways, as a linear combination of the rows of the matrix A . In each case the nonzero coefficients in the linear combination are all + 1 or all - 1 . c

c

Proof. Let (V , V ) be the i'th cut in a graph G with η vertices and m edges, and let be the corresponding cut vector. Let V = {y,, v .. •, v ) and V = {v , v ,... ,v„}. For 1 

r

132

MATRICES OF A GRAPH

We_ assume, without_ any loss of generality, that the orientation of {V , V ) is from V to V , and prove the theorem by establishing that a

a

a

a

q, = «i + a + · · · + a = - ( a 2

r

r + 1

+a

+ · · • + a„).

r+2

(6.5)

Let v and v be the end vertices of the kth edge, 1 < A; ^ m. Let this edge be oriented from v to v so that p

q

p

q

°pk = 1

a

(6.6)

=-l,

qk

Now four cases arise. v EV and v Ε V , that is, ρ s r and q 2: r + 1, so that = 1. v EV and v Ε V , that is, ρ > r + 1 and q < r, so that r + 1, so that = 0.

Case 1 Case 2 Caie 3 Case 4

p

0

p

a

p

p

q

a

q

q

λ

a

a

k

q

a

It is easy to verify, using (6.6), that the following is true in each of these four cases.
= -(«,+!.* + <>r 2.k + ·•· + a ). +

nk

Now (6.5) follows since (6.7) is valid for all 1 < A: < m. Hence the theorem. • To illustrate Theorem 6.3, consider the cut 1 j n Fig. 6.2. This cut separates the vertices in V = {υ,, i> } from those in V . The cut orientation is from V to V . So the corresponding cut vector can be expressed as follows: 3

a

a

a

a

[1 0

1 0

1 1 0] = α, + α = -a

3

-a -a

2

4

5

where a,, a ,..., a are the rows of the matrix A in Fig. 6.1a. An important consequence of Theorem 6.3 is that rank (Q ) s rank(A ). However, rank(Q ) ^ rank(y4 ) because A is a submatrix of Q . Therefore, we get 2

5

c

c

c

c

c

c

rank((2 ) = r a n k ( / l ) . c

c

Thus Theorem 6.2 and Corollary 6.2.1, respectively, lead to the following.

CIRCUIT MATRIX

133

Theorem 6.4. The rank of the cut matrix Q of an η-vertex connected graph G is equal to « - 1, the rank of G. • c

Corollary 6.4.1. The rank of the cut matrix Q of an η-vertex graph G with ρ components is equal to η - ρ, the rank of G. • c

As the above discussions show, the all-vertex incidence matrix A is an important submatrix of the cut matrix Q . Next we identify another important submatrix of Q . We know that a spanning tree Τ of an η-vertex connected graph G defines a set of η - 1 fundamental cutsets—one fundamental cutset for each branch of T. The submatrix of Q corresponding to these η - 1 fundamental cutsets is known as the fundamental cutset matrix Q oiG with respect to T. Let b , b ,..., b„_ denote the branches of T. Suppose we arrange the columns and the rows of Q so that c

c

c

c

f

t

2

i

f

1. For 1 < i < η - 1, the ith column corresponds to the branch b . 2. The ith row corresponds to the fundamental cutset defined by b t

r

If, in addition, we assume that the orientation of a fundamental cutset is so chosen as to agree with that of the defining branch, then the matrix Q can be displayed in a convenient form as follows: f

(6-8)

Qf = [U\Q } fc

where U is the unit matrix of order η - 1 and its columns correspond to the branches of T. For example, the fundamental cutset matrix Q ot the connected graph of Fig. 6.1a with respect to the spanning tree T= {e,, e , e , e } is f

2

e e

x

Qf

2

e

7

1 0 0 0

e 0 1 0 0

2

e 0 0 1 0

6

e 0 0 0 1

e

7

3

1 -1 0 0

«4

-1 1 1 1

6

0 0 1 1

7

(6.9)

It is clear from (6.8) that the rank of Q is equal to η - 1, the rank of Q . Thus every cut vector (which may be a cutset vector) can be expressed as a linear combination of the fundamental cutset vectors. f

6.3

c

CIRCUIT MATRIX

A circuit can be traversed in one of two directions, clockwise or anticlockwise. The direction we choose for traversing a circuit defines its orientation.

134

MATRICES OF A GRAPH

Figure 6.3 The orientation of a circuit can be pictorially shown by an arrow as in Fig. 6.3. Consider an edge e that has υ, and υ as its end vertices. Suppose that this edge is oriented from u, to υ and that it is present in circuit C. Then we say that the orientation of e agrees with the orientation of the circuit if u, appears before u when we traverse C in the direction specified by its orientation. For example, in Fig. 6.3, the orientation of e, agrees with the circuit orientation whereas that of e does not. The circuit matrix B = [b \ of a graph G with m edges has m columns and as many rows as the number of circuits in G. The entry 6 is defined as follows: ;

]

;

A

c

tj

(/

G is directed if the /th edge is in the ith circuit and its orientation agrees with the circuit orientation; if the /'th edge is in the ith circuit and its orientation does not agree with the circuit orientation; if the /th edge is not in the ith circuit. G is undirected if the /th edge is in the ith circuit; otherwise . A row of B will be referred to as a circuit vector of G. As an example, consider again the graph of Fig. 6.1a. Three circuits of this graph with their orientations are shown in Fig. 6.4. The submatrix of B corresponding to these three circuits is c

c

Circuit 1 Circuit 2 Circuit 3

1 -1 1

-1 0 1 1 -1 0

e e e e 1 - 1 0 0 0 0 0 0 1 0 - 1 - 1 4

5

6

7

CIRCUIT MATRIX

135

Circuit 3

Figure 6.4. Some circuits of the graph of Fig. 6.1a. The corresponding submatrix for the undirected graph of Fig. 6.1ft can be obtained by replacing the - l ' s by + l ' s . Next we identify an important submatrix of B . Consider any spanning tree Γ of a connected graph G having η vertices and m edges. Let c,, c , . . . , c _ „ be the chords of T. We know that these m - η + 1 chords define a set of m - η + 1 fundamental circuits. The submatrix of B corresponding to these fundamental circuits is known as the fundamental circuit matrix B of G with respect to the spanning tree T. Suppose we arrange the columns and rows of B so that c

2

m

+ 1

c

f

f

1. For l < i < f f i - B + l , the ith column corresponds to the chord c,. 2. The ith row corresponds to the fundamental circuit defined by c, If, in addition, we choose the orientation of a fundamental circuit to agree with that of the defining chord, then the matrix B can be written as f

B = [U\B ] F

FT

(6.10)

where U is the unit matrix of order m - η + 1 and its columns correspond to the chords of T.

136

MATRICES OF A GRAPH

For example, the fundamental circuit matrix of the graph of Fig. 6.1a with respect to the spanning tree T= {e e , e , e } is u

e 1 0 0

3

e e

4 5

0 1 0

e 0 0 1

5

2

6

7

e «1 -1 1 1 -1 0 0

e

2

6

0 -1 -1

0 -1 -1

It is obvious from (6.10) that the rank of B is equal t o m - n + 1. Since B is a submatrix of B , we get f

f

c

rank(B )>m-n + l.

(6.12)

c

We show in the next section that the rank of B in the case of a connected graph is equal to m - η + 1. Now note that the approach used in the previous section to establish the rank of Q cannot be used in the case of B . (Why?) Does this mean that the "duality" we observed (Chapter 4) between circuits and cutsets is merely accidental? No, we shall see that the arguments of the next section would further confirm this "duality." c

c

6.4

c

ORTHOGONALITY RELATION

We showed in Section 4.6 that in the case of an undirected graph every circuit vector is orthogonal to every cut vector. Now we prove that this result is true in the case of directed graphs too. Our proof is based on the following theorem. Theorem 6.5. If a cut and a circuit in a directed graph have 2k edges in common, then k of these edges have the same relative orientations in the cut and in the circuit, and the remaining k edges have one orientation in the cut and the opposite orientation in the circuit. Proof. Consider a cut (V ,V ) and a circuit C in a directed graph. Suppose we traverse C starting from a vertex j n V . Then, for every edge e that leads us from a vertex in V to a vertex in V , there is an edge e that leads us from a vertex in V„ to a vertex in V . The proof of the theorem will follow if we note that if e,(e ) has the same relative orientation in the cut and in the circuit, then e (e ) has one orientation in the cut and the opposite orientation in the circuit (see Fig. 6.5). • a

a

a

a

a

a

2

2

x

Now we prove the main result of this section.

x

2

ORTHOGONALITY RELATION

137

Figure 6.5 Theorem 6.6 (Orthogonality Relation). If the columns of the circuit matrix B and the cut matrix Q are arranged in the same edge order, then c

c

B Q' = o. c

c

Proof. Consider a circuit and a cut that have 2k edges in common. The inner product of the corresponding circuit and cut vectors is equal to zero, since, by Theorem 6.5, it is the sum of k l's and k - l ' s . The proof of the theorem now follows because each entry of the matrix B Q' is the inner product of a circuit vector and a cut vector. • C

C

The orthogonality relation is a very profound result with interesting applications in graph theory and in network theory. Now we use this relation to establish the rank of the circuit matrix B . Consider a connected graph G with η vertices and m edges. Let B and Q be the fundamental circuit and cutset matrices of G with respect to a spanning tree T. If the columns of B and Q are arranged in the same edge order, then we can write B and Q, as c

f

f

f

f

f

B = [B f

fl

U]

and Q = [u f

By the orthogonality relation 0, that is,

that is, B = fl

-Q' . fc

(6.13)

138

MATRICES OF A GRAPH

Let β = [/3,, ft,..., β„\β ,. . ., ft,], where ρ is the rank of G, be a circuit vector with its columns arranged in the same edge order as B and Q . Then, again by the orthogonality relation, ρ+ί

f

f

fiQ'r

= [A,

ft,...,

· . . , A J [ = ο.

ftlft+„

Therefore

[ft, ft,..., ft]

= -[ft + 1 ,

ft ,..., ft,]G' = [β +2

/e

,

ftjft,.

β ,...,

ρ+ι

ρ+2

Using the above equation we can write [ft, ft,..., ft,] as [ft,

ft,...,

ft,]

= [ f t

+

1

,

ft ,...,

ft,][ft,

+2

U]

= [ f t

+

1

,

ft ,..., +2

ftjft

.

(6.14) Thus any circuit vector can be expressed as a linear combination of the fundamental circuit vectors. So rank(ft) s rank(B ) = m - η + 1 . r

Combining the above inequality with (6.12) establishes the following theorem and its corollary. Theorem 6.7. The rank of the circuit matrix B of a connected graph G with η vertices and m edges is equal to m - η + 1, the nullity of G. • c

Corollary 6.7.1. The rank of the circuit matrix B of a graph G with η vertices, m edges, and ρ components is equal to m - η + ρ, the nullity of c

G.

m

Suppose a = [a,, a , . . . , a , a , . . . , a ] is a cut vector such that its columns are arranged in the same edge order as B and Q , then we can start from the relation 2

p

p + 1

m

f

f

aB' = 0 f

and prove that ο = [α,, α . . . ,a ]Q 2 )

p

f

,

(6.15)

by following a procedure similar to that used in establishing (6.14). Thus every cut vector can be expressed as a linear combination of the fundamental cutset vectors. Since T&nk.(Q ) = η - 1, we get f

SUBMATRICES OF CUT, INCIDENCE, AND CIRCUIT MATRICES

rank(G ) = rank(G ) = c

139

,i-l.

/

This is thus an alternative proof of Theorem 6.4. Note that the ring sum of two subgraphs corresponds to modulo 2 addition of the corresponding vectors. Therefore in the case of undirected graphs, (6.14) and (6.15) are merely restatements of what we have already proved in the course of our arguments leading to Theorem 4.6, namely, a circuit (cut) can be expressed as a ring sum of fundamental circuits (fundamental cutsets).

6.5

SUBMATRICES OF CUT, INCIDENCE, AND CIRCUIT

MATRICES

In this section we characterize those submatrices of Q , A , and B that correspond to circuits, cutsets, spanning trees, and cospanning trees and discuss some properties of these submatrices. c

c

c

Theorem 6.8 1. There exists a linear relationship among the columns of the cut matrix Q that correspond to the edges of a circuit. 2. There exists a linear relationship among the columns of the circuit matrix B that correspond to the edges of a cutset.

c

c

Proof 1. Let us partition Q into columns so that c

Q =

[Q \Q' \...,Q ]. {l

c

2

im)

Let β = [ β , , β ,..., β„] be a circuit vector. Then by the orthogonality relation, we have 2

<2J3' = 0 or A(2

(l>

+ /3 <2 + · · · + /3 <2 (2)

2

(m)

m

=0.

(6.16)

If we assume, without loss of generality, that the first r elements of β are nonzero and the remaining ones are zero, then we have from (6.16) ^β

( 1 >

+ /8 ρ 2

( 2 )

+ ··· + /3 β Γ

( Γ )

=0.

MATRICES OF A GRAPH

140

Thus there exists a linear relationship among the columns Q\ β . · · · . Q of Q correspond to the edges of a circuit. 2. The proof in this case follows along the same lines as that for part 1. • (l

<Ζ)

t n a t

(r)

c

Corollary 6.8.1. There exists a linear relationship among the columns of the incidence matrix that correspond to the edges of a circuit. Proof. The result follows from Theorem 6.8, part 1, because the incidence matrix is a submatrix of Q . • c

Theorem 6.9. A square submatrix of order η - 1 of any incidence matrix A of an η-vertex connected graph G is nonsingular if and only if the edges that correspond to the columns of the submatrix form a spanning tree of G. Proof Necessity Consider the η - 1 columns of a nonsingular submatrix of A. Since these columns are linearly independent, by Corollary 6.8.1, there is no circuit in the corresponding subgraph of G. Since this circuitless subgraph has η - 1 edges, it follows from Theorem 2.2 that it is a spanning tree of G. Sufficiency

This follows from Theorem 6.1.

•

Thus the spanning trees of a connected graph are in one-to-one correspondence with the nonsingular submatrices of the matrix A. This result is in fact the basis of a formula for counting spanning trees that we discuss in Section 6.7. Theorem 6.10. Consider a connected graph G with η vertices and m edges. Let Q be a submatrix of Q with η - 1 rows and of rank η - 1. A square submatrix of Q of order η - 1 is nonsingular if and only if the edges corresponding to the columns of this submatrix form a spanning tree of G. c

Proof Necessity

Let the columns of the matrix Q be rearranged so that Q = [Qn

Qn]

with Q nonsingular. Since the columns of Q are linearly independent, by Theorem 6.8, part 1, there is no circuit in the corresponding subgraph of G. This circuitless subgraph has η — 1 edges and is therefore, by Theorem 2.2, a spanning tree of G. n

Sufficiency

u

Suppose we rearrange the columns of Q so that Q = [Qu

Qn),

SUBMATRICES OF CUT, INCIDENCE, AND CIRCUIT MATRICES

141

and the columns of β π correspond to the edges of a spanning tree T. Then the fundamental cutset matrix Q with respect to Τ is f

Q = [u

Q \.

f

fc

Since the rows of Q can be expressed as linear combinations of the rows of Q , we can write Q as f

Q = [Qu

Qn] = t>Q

f

= D[U

Q ]. fc

Thus Q

n

= DU = D .

Now D is nonsingular because both Q and Q are of maximum rank n — l. So Q is nonsingular, and the sufficiency of the theorem follows. • f

n

A dual theorem is presented next. Theorem 6.11. Consider a connected graph G with η vertices and m edges. Let β be a submatrix of the circuit matrix B of G with m - η + 1 rows and of rank m - η + 1. A square submatrix of Β of order m - η + 1 is nonsingular if and only if the columns of this submatrix correspond to the edges of a cospanning tree. c

Proof Necessity

Let the columns of the matrix Β be rearranged so that B = [B

B]

U

l2

with B nonsingular. Since the columns of B are linearly independent, by Theorem 6.8, part 2, the corresponding subgraph contains no cutsets of G. Further this subgraph has m - η +1 edges. Thus it is a maximal (why?) subgraph of G containing no cutsets and is therefore a cospanning tree of G (see Exercise 2.18). u

Sufficiency

u

Suppose we rearrange the columns of Β so that B = [B

B]

U

l2

and the columns of B correspond to the edges of a cospanning tree of G. Then the fundamental circuit matrix B with respect to this cospanning tree is u

f

B, = [U

Bf,).

142

MATRICES OF A GRAPH

Since the rows of Β can be expressed as linear combinations of the rows of B , we can write Β as f

Β = [B

B ] = DB

u

n

f

= D[U

B] fl

so that B = DU = D. Now D is nonsingular because both Β and B are of maximum rank m — n + 1. Thus B is nonsingular and the sufficiency of the theorem follows. • U

f

u

To illustrate Theorems 6.10 and 6.11, consider the graph G of Fig. 6.1a and the matrices Q and B in (6.9) and (6.11). The fourth-order square submatrix of Q corresponding to the spanning tree {e,, e , e , e } and the third-order submatrix of B corresponding to the cospanning tree {e , e , e ) are, respectively, f

f

3

f

f

5

7

2

4

6

1 0 0 -1 0 0 0 1 0 0 1 1

"1 0 0 .0 and

" 1 0 -1 1 . 0 0

0' 1 1.

It may be verified that the determinants of these matrices are nonzero, and thus these matrices are nonsingular. We conclude this section with the study of an interesting property of the inverse of a nonsingular submatrix of the incidence matrix. Theorem 6.12. Let A be a nonsingular submatrix of order η-I of an incidence matrix A of an η-vertex connected graph G. Then the nonzero elements in each row of A are either all 1 or all - 1 . n

l

xl

Proof. Let A be the incidence matrix with v as the reference vertex. Assume that r

A = [A

u

A ], l2

where A is nonsingular. We know from Theorem 6.9 that the edges corresponding to the columns of A constitute a spanning tree ToiG. Then Q , the fundamental cutset matrix with respect to T, will be u

u

f

Q = [U f

Q] fc

143

SUBMATRICES OF CUT, INCIDENCE, AND CIRCUIT MATRICES

By Theorem 6.3, each cut vector can be expressed as a linear combination of the rows of the incidence matrix. So we can write Q as f

Q = [V

Q ) = D[A

f

fc

A \.

u

u

Thus D=

A- . l

l l

Consider now the /th row q of Q . t

(v ,v„).

Let the corresponding cutset be

f

a

Let {O ,V ,... V }

V„ =

i

2

f

k

and

K = {v ,

v ,...,v }

k+l

k+2

.

n

Suppose that the orientation of the cutset (V , V ) is from V to V . Then we get from (6.5) that a

a

a

a

(6.17)

q, = a + a + --- + a l

2

k

= -(«*+.+ «*

+2

+ · · · + «„),

(6-18)

where a, is the ith row of A . Note that row a corresponding to v will not be present in A. So if v E. V , then to represent q, as a linear combination of the rows of A we have to write q, as in (6.18). If v G V , then we have to write q as in (6.17). In both cases the nonzero coefficients in the linear combination are either all 1 or all - 1 . Thus the nonzero elements in each row of D = A„ are either all 1 or all -1. • c

r

r

r

a

r

a

t

The proof used in Theorem 6.12 suggests the following simple procedure for evaluating ΑΪ*. 1. From A construct the tree Τ for which A is the incidence matrix with vertex v„ as the reference vertex. 2. Label the edges of Τ as e,, e ,..., e„_, so that e, corresponds to the ith column of A . Similarly label the vertices of Τ as υ , , v ,..., i>„_, so that v corresponds to the ith row of A . 3. Let the ith column of correspond to υ,. For each i, l s / < n - l , do the following: Remove c, from Τ and let 7\ and T be the two u

n

2

u

(

2

u

2

144

MATRICES OF A GRAPH

components of_the resulting disconnected graph. Let V be the vertex set of 7\ and V the vertex_set of T , and let e, be oriented from a vertex in V to a vertex in V . If v EV , then the ithjow of Ajl has - 1 in all columns that correspond to the vertices in V„; all the other elements of the ith row will be zero. If v EV , then the ith row has 1 in all the columns that correspond to the vertices in V ; all the other elements in the ith row will be zero. a

a

2

a

a

n

a

n

a

a

As an illustration, let «1

0 1 -1 0

e -1 1 0 0 2

e 0 1 0 0

3

0 0 1 -1

The tree Τ for which A is an incidence matrix is shown in Fig. 6.6a. The reference vertex is i> . To obtain the first row of Λ ' , we first remove e, from Τ and get the two components 7\ and T shown in Fig. 6.6b. Thus u

5

η

2

Figure 6.6. (β) Tree T. (b) Subtrees Γ, and T . 2

UNIMODULAR MATRICES

V = {v a

V,

lt

K =

145

V)

2

5

{v ,v }. 3

4

Since the reference vertex is in V , the first row of A, , will have - 1 in the columns corresponding to the vertices v and v of V , and it is - 1

a

3

[0 0

-1

4

a

-1].

In a similar way we get all the other rows of A\~{, and A,", is 1

0 - 1 1 0 6.6

0 - 1 -Γ 0 0 0 1 1 1 · 0 0 -1.

UNIMODULAR MATRICES

A matrix is unimodular if the determinant of each of its square submatrices is 1, - 1 , or 0. We show in this section that the matrices A , Q , and B are all unimodular. c

f

f

Theorem 6.13. The incidence matrix A of a directed graph is unimodular. c

Proof. We prove the theorem by induction on the order of a square submatrix of A . Obviously the determinant of every square submatrix of A of order 1 is 1, - 1 , or 0. Assume, as the induction hypothesis, that the determinant of every square submatrix of order less than k is equal to 1, - 1 , or 0. Consider any nonsingular square submatrix of A of order k. Every column in this matrix contains at most two nonzero entries, one + 1 and /or one - 1 . Since the submatrix is nonsingular, not every column can have both + 1 and - 1 . For the same reason in this submatrix there can be no column consisting of only zeros. Thus there is at least one column that contains exactly one nonzero entry. Expanding the determinant of the submatrix by this column and using the induction hypothesis, we find that the desired determinant is ± 1 . • c

c

c

Let Q be the fundamental cutset matrix of an η-vertex connected graph G with respect to some spanning tree T. Let the branches of Γ be ft,, b ,.. ·,&„_,. Let G' be the graph that is obtained from G by identifying or shortcircuiting the end vertices of one of the branches, say, branch ft,. Then T- {ft,} is a spanning tree of G'. Let us now delete from Q the row f

2

f

146

MATRICES OF A GRAPH

corresponding to branch />, and denote the resulting matrix as Q' . Then it is not difficult to show that Q' is the fundamental cutset matrix of G' with respect to the spanning tree T- {ft,}. Thus the matrix that results after deleting any row from Q is a fundamental cutset matrix of some connected graph. Generalizing this, we can state that each matrix formed by some rows of Q is a fundamental cutset matrix of some connected graph. For example, consider the graph G of Fig. 6.1a and the fundamental cutset matrix Q given in (6.9). The submatrix consisting of the two rows of Qf corresponding to the branches e, and e is f

f

f

f

f

6

It can be verified that this matrix is the fundamental cutset matrix of the graph shown in Fig. 6.7 with respect to the spanning tree {e ,e }. This graph is obtained from the graph of Fig. 6.1a by identifying the end vertices of e as well as the end vertices of e . l

2

6

7

Theorem 6.14. Any fundamental cutset matrix Q of unimodular. f

a connected graph G is

Proof. Let Q be the fundamental cutset matrix of G with respect to a spanning tree T. Then f

Q = [v f

Figure 6.7

Q Af

THE NUMBER OF SPANNING TREES

147

Let an incidence matrix A of G be partitioned as A = [A A ] where the columns of A correspond to the branches of T. We know from Theorem 6.9 that A is nonsingular. Now we can write Q as u

l2

u

u

f

Q = [U

Q ] = An [A

A ).

l

f

fc

u

l2

If C is any square submatrix of Q of order n — l, where η is the number of vertices of G, and D is the corresponding submatrix of A, then C = A\~{D. Since det D = ± 1 or 0 and det A^ = ± 1 , we get f

detC=±lorO.

(6.19)

Consider next any square submatrix Η of Q of order less than n — l. From the arguments preceding this theorem we know that Η is a submatrix of a fundamental cutset matrix of some connected graph. Therefore, det Η = ± 1 or 0, the proof of which follows along the same lines as that used to prove (6.19). Thus the determinant of every square submatrix of Q is ±1 or 0 and hence Q is unimodular. • f

f

f

Next we show that B is unimodular. f

Theorem 6.15. Any fundamental circuit matrix Z^of a connected graph G is unimodular. Proof. Let B and Q be the fundamental circuit and cutset matrices of G with respect to a spanning tree T. If Q = [U Q ], then we know from (6.13) that f

f

f

B = [-Q'fc f

fc

U]-

Since Q is unimodular, Q' is also unimodular. It is now a simple exercise to show that [-Q' U] is also unimodular. • fc

fc

fc

6.7

THE NUMBER OF SPANNING TREES

In this section we derive a formula for counting the number of spanning trees in a connected graph. This formula is based on Theorem 6.9 and a result in matrix theory, known as the Binet-Cauchy theorem. A major determinant or briefly a major of a matrix is a determinant of maximum order in the matrix. Let Ρ be a matrix of order ρ x q and Q be a matrix of order q x ρ with p^q. The majors of Ρ and Q are of order p. If a major of Ρ consists of the columns i , i , . . . , i of P, then the corresponding major of Q is formed by the rows i , i ,...,i of Q. For example, if i

2

x

p

2

p

148

MATRICES OF A GRAPH

Μί

λ

~\

\]

-

β "

2" 1 2 -1 1 -3 . 1 2.

then for the major -1 2 of P, 2 -3 is the corresponding major of Q. Theorem 6.16 (Binet-Cauchy). If Ρ is a ρ χ ο matrix and β is a ο x ρ matrix, with p^q, then d e t ( P g ) = Σ (product of the corresponding majors of Ρ and Q).

•

Proof of this theorem may be found in Hohn [6.1]. As an illustration, if the matrices Ρ and Q are given as earlier, then applying the Binet-Cauchy theorem we get det(PG) =

1 2 +

-1 2

1 2

2 -1

-1 2

3 -1

2 -3

3 3 -1 2

I Τ

1 2

1 2 -3 1

3 -1 +

-1 2

I Τ

1 3 2 2

1 1

2 2

3 2

-3 1 1 2

167. Theorem 6.17. Let G be a connected undirected graph and A an incidence matrix of a directed graph that is obtained by assigning arbitrary orientations to the edges of G. Then t ( G ) = det(AA') , where t ( G ) is the number of spanning trees of G . Proof. By the Binet-Cauchy theorem det(AA') = E(

Product of the \ ν corresponding majors of A and A I

(6.20)

THE NUMBER OF SPANNING TREES

149

Note that the corresponding majors of A and A' both have the same value equal to 1, - 1 , or 0 (Theorem 6.13). Therefore each nonzero term in the sum on the right-hand side of (6.20) has the value 1. Furthermore, a major of A is nonzero if and only if the edges corresponding to the columns of the major form a spanning tree. Thus there is a one-to-one correspondence between the nonzero terms in the sum of the right-hand side of (6.20) and the spanning trees of G. Hence the theorem. • As an example, consider the graph G shown in Fig. 6.8a. A directed graph obtained by assigning arbitrary orientations to the edges of G is shown in Fig. 6.86. The incidence matrix of this directed graph with vertex i> as the reference is given by 4

»4

(a)

Figure 6.8

150

MATRICES OF A GRAPH

So ' AA'=

1 3 -1

2 -1 0

0 01 -1 2

and det(AA') = 8 . The eight spanning trees of G are {e,,e ,e },

{e,,
{e,, e ,
{e e ,e },

{e ,e ,e },

{e ,e ,e },

4

u

2

5

5

3

3

4

4

3

5

2

3

{e e ,e },

5

u

2

3

{e ,e ,e }.

4

2

A

5

Let υ,, υ ,...,υ„ denote the vertices of an undirected graph G without self-loops. The degree matrix Κ = [k ] of G is an η x η matrix defined as follows: 2

tj

if / Φ j and there are ρ parallel edges connecting v, and v ; t

We can easily see that K= A A' and is independent of our choice of edge orientations for arriving at the all-vertex incidence matrix A . If v is the reference vertex of A, then A A' is obtained by removing the rth row and the rth column of K. Thus the matrix A A' used in Theorem 6.17 can be obtained by an inspection of the graph G. It is clear from the definition of the degree matrix that the sum of all the elements in each row of Κ equals zero. Similarly the sum of all the elements in each column of Κ equals zero. A square matrix with these properties is called an equi-cofactor matrix. As its name implies, all the cofactors of an equi-cofactor matrix are equal [6.2]. Thus from Theorem 6.17 we get the following result, originally due to Kirchhoff [6.3]. C

C

c

r

Theorem 6.18. All the cofactors of the degree matrix of a connected undirected graph has the same value equal to the number of spanning trees of G. m Next we derive a formula for counting the number of distinct spanning trees that can be constructed on η labeled vertices. Clearly this number is the same as the number of spanning trees of K , the complete graph on η labeled vertices. n

Theorem 6.19 (Cayley). There are n"~ labeled trees on η &2 vertices. 2

151

THE NUMBER OF SPANNING 2-TREES

Proof. In the case of K , the matrix A A' is of the form n

Γη-1 -1

-1 n-1

··• ···

L -1

-1

···

-1] -1

n

-

l

j

By Theorem 6.17, the determinant of this matrix gives the number of spanning trees of K„, which is the same as the number of labeled trees on η vertices. To compute det(AA'), subtract the first column of AA' from all the other columns of AA'. Then we get η — 1 —η -1 η - 1 0

L -1

-η • •· -η 0 ••· 0 η ··· 0

0

0

···

nj

Now adding to the first row of the above matrix every one of the other rows, we get "

1 0 -1 η -1 0

L-l

0

0 0 η

·•· ·•· •··

0" 0 0 .

0

···

nj

The determinant of this matrix is n"~ . The theorem now follows since addition of any two rows or any two columns of a matrix does not change the value of the determinant of the matrix. • 2

Several proofs of Cayley's theorem [6.4] are available in the literature. See Moon [6.5] and Priifer [6.6]. 6.8

THE NUMBER OF SPANNING 2-TREES

In this section we relate the cofactors of the matrix AA' oi a graph G to number of spanning 2-trees of the appropriate type. For this purpose need symbols to denote 2-trees in which certain specified vertices required to be in different components. We use the symbol T ljk

rsl

the we are to

152

MATRICES OF A GRAPH

denote spanning 2-trees in which the vertices v„ v v ,... are required to be in one component and the vertices v ,v ,v,... are required to be in the other component of the 2-tree. The number of these spanning 2-trees in the graph G will be denoted by r „, . For example, in the graph of Fig. 6.8a, the edges e, and e form a spanning 2-tree of the type T , and jt

r

k

s

l]k

3

T

14,23

U23

1·

=

In the following we denote by A an incidence matrix of the directed graph that is obtained by assigning arbitrary orientations to the edges of the graph G. However, we shall refer to A as an incidence matrix of G. We shall assume, without any loss of generality, that v„ is the reference vertex for A, and the ith row of A corresponds to vertex ν,. Δ, will denote the (i, j) cofactor of AA'. Let A denote the matrix obtained by removing from A its ith row. If G' is the graph obtained by short-circuiting the vertices υ, and v in G, then we can easily verify the following: ;

n

1. A_, is the incidence matrix of G' with v„ as the reference vertex. 2. A set of edges forms a spanning tree of G' if and only if these edges form a spanning 2-tree T of G. in

Thus there exists a one-to-one correspondence between the nonzero majors of A_, and the spanning 2-trees of the type T, „. Theorem 6.20. For a connected graph G, Δ . = τ- . Proof. Clearly Δ„ = det(A_,A'_,). Proof is immediate since the nonzero majors of A_, correspond to the spanning 2-trees T of G and vice versa. • in

Consider next the (i, /) cofactor Δ of AA', which is given by (;

A = (-l)' 'det(A_,A'_ ).

(6.21)

+

i;

;

By the Binet-Cauchy theorem, det(A_,A' ) = Σ( Productofthe \ ' \ corresponding majors of A _,· and A_ l t

Each nonzero major of A _, corresponds to a spanning 2-tree of the type T, „, and each nonzero major of A corresponds to a spanning 2-tree of the type T . Therefore the nonzero terms in the sum on the right-hand side of (6.22) correspond to the spanning 2-trees of the type T „. Each one of these nonzero terms is equal to a determinant of the type det(F_,F'_ ), where F is the incidence matrix of a 2-tree of the type Γ . in

tl

;

THE NUMBER OF SPANNING 2-TREES

153

Theorem 6.21. Let F denote the incidence matrix of a 2-tree Τ with v„ as the reference vertex. If the i'th row of F corresponds to vertex v , then ηη

t

det(F_,F'_,) = ( - l ) ' \ +

Proof. Let Γ, and T denote the two components of Γ . Assume that v„ is in T . By interchanging some of its rows and the corresponding columns, we can write the matrix FF' as 2

2

5

=

[O4D]'

where 1. C is the degree matrix of Γ,, and 2. D can be obtained by removing from the degree matrix of T the row and the column corresponding to v . 2

n

Let row k' of S correspond to vertex v . Interchanging some rows and the corresponding columns of a matrix does not alter the values of the cofactors of the matrix. So k

(/', ; ' ) cofactor of S = (/, ;') cofactor of (FF') = (-l)' 'det(F_,F'_ ). +

/

(6.23)

By Theorem 6.18 all the cofactors of C have the same value equal to the number of spanning trees of 7 , . So we have (Γ, / ' ) cofactor of c = 1 . Furthermore, det D = 1 . So (f, j') cofactor of S = [(Γ, / ' ) cofactor of C][det D] = 1.

(6.24)

Now, from (6.23) and (6.24) we get det(F_,F'_,) = ( - l ) ' . , +

•

Proof of Theorem 6.21 is by Sankara Rao, Bapeswara Rao, and Murti [6.7].

The following result is due to Mayeda.

154

MATRICES OF A GRAPH

Theorem 6.22. For a connected graph G, Δ =τ·

Ι."

ι/

Proof. Since each nonzero term in the sum on the right-hand side of (6.22) is equal to a determinant of the type given in Theorem 6.21, we get

detM_ A'_ ) = (-l) V ,+

I

;

l/i(I

.

So

A = (-l) 'det(i4_X_,) ,+

fJ

To illustrate Theorems 6.20 and 6.22 consider again the graph of Fig. 6.8a. If vertex υ is the references vertex for A, then 4

-

Γ1 0

0

1 0 ]

0 0

1 01

Lo ο - ι ο

2

-u

and Γ1

A

Lo - ι

i

ι -ι

or

So τ

= det(A

2 4

=

d e t

_ A'_ ) 2

([o

2

2])

=4 and T

=

2 X 4

(-l) det{A_ A'_,) 2+3

2

-Mil:!]) = 2. In this graph the spanning 2-trees of the type T

24

{e ,e }, 4

5

{e ,e }. 3

u

and the spanning 2-trees of the type T

4

23

{ i> e

e

i )

<

{e,,e },

{e e },

4

{ 3> e

5

3

are ^4)

are

·

THE NUMBER OF DIRECTED SPANNING TREES IN A DIRECTED GRAPH

155

6.9 THE NUMBER OF DIRECTED SPANNING TREES IN A DIRECTED GRAPH

In this section we discuss a method due to Tutte [6.8] for computing the number of directed spanning trees in a given directed graph having a specified vertex as root. This method is in fact a generalization of the method given in Theorem 6.17 to compute the number of spanning trees of a graph, and it is given in terms of the in-degree matrix. The in-degree matrix Κ = [k ] of a directed graph G = (V, E) without self-loops and with V= {υ,, v ,..., v } is an η χ η matrix denned as follows: pq

2

'- ω,

if ρ Φ q and there are ω parallel edges directed from v to v ; np = q. p

d~{v ), p

n

q

Let K denote the matrix obtained by removing row i and column / from t)

K. Tutte's method is based on the following theorem. Theorem 6.23. A directed graph G = (V, E) with no self-loops and with V= {u,, v ,..., v„) is a directed tree with v as the root if and only if its in-degree matrix Κ has the following properties: 2

r

fO, [1,

ifp = r ; \ip*r.

2. d e t ( / U = 1 . Proof Necessity Suppose the given directed graph G is a directed tree with v as the root. Clearly G is acyclic. So (see Section 5.7) we can label the vertices of G with the numbers 1 , 2 , . . . ,n in such a way that (/, / ) is a directed edge of G only if i < j . Then in such a numbering, the root vertex would receive the number 1. If the ith row and the i'th column of the new in-degree matrix K' of G correspond to the vertex assigned the number i, then we can easily see that K' has the following properties:

r

k'„ = l ,

ίογρΦί,

k' =0,

\fp>q.

pq

Therefore det(*;,) = l

156

MATRICES OF A GRAPH

The matrix K' can be obtained by interchanging some rows and the corresponding columns of K. Such an interchange does not change the value of the determinant of any submatrix of K. So det(/^ ) = det(/c; ) = l . i

rr

Sufficiency Suppose the in-degree matrix Κ of the graph G satisfies the two properties given in the theorem. If G is not a directed tree, then by property 1 and Theorem 5.4, statement 5, it contains a circuit C. The root vertex v cannot be in C, for this would imply that d~(v )>0 or that d~(v)>l for some other vertex υ in C, contradicting property 1. In a similar way, we can show that r

r

1. C must be a directed circuit. 2. No edge not in C is incident into any vertex in C. Consider now the submatrix K, of Κ consisting of the columns corresponding to the vertices in C. Because of the above properties, each row of K corresponding to a vertex in C has exactly one +1 and one - 1 . All the other rows contain only zero elements. Thus the sum of the columns of is zero. In other words the sum of the columns of Κ that correspond to the vertices in C is zero. Since v is not in C, this is true in the case of the matrix K too, contradicting property 2. Hence the sufficiency. • s

r

rr

We now develop Tutte's method for computing the number τ of the directed spanning trees of a directed graph G having vertex v as the root. Assume that G has no self-loops. For any graph g let K(g) denote its in-degree matrix, and let K' be the matrix obtained from Κ by replacing its rth column by a column of zeros. Denote by 5 the collection of all the subgraphs of G in each of which d~(v ) = 0 and d~(v ) = 1 for ρ Φ r. Clearly α

r

r

p

\S\=f\d-(v ). p

#>=·

Further, for any subgraph g £ S, the corresponding in-degree matrix satisfies property 1 given in Theorem 6.23. It is well known in matrix theory that the determinant of a square matrix is a linear function of its columns. For example, if ^ = [Pi.P ,...,p' +p",...,p ] 2

/

( 1

is a square matrix with the columns />,, p ,..., 2

det Ρ = det[p,, p ,..., 2

p'„ ...,

p, + p",...,

p„] + det[p,, p ,..., 2

p„, then p",...,

p„].

THE NUMBER OF DIRECTED SPANNING TREES IN A DIRECTED GRAPH

157

Using the linearity of the determinant function and the fact that the sum of all the entries in each column of the matrix K'(G) is equal to zero, we can write det K'{G) as the sum of \S\ determinants, each of which satisfies property 1 given in Theorem 6.23. It can be seen that there is a one-to-one correspondence between these determinants and the in-degree matrices of the subgraphs in 5. Thus det K\G)=

Σ det

K'(g).

So

det Κ'„(β)=Σ

det K'„(g).

Since det K' (G) = det

K„(G),

rr

and d e t / C ( g ) = det/s: (g) rr

for all g(E 5 ,

we get d e t K „ ( G ) = Σ det

K'„(g).

From Theorem 6.23 it follows that each determinant in the sum on the right-hand side of the above equation is nonzero and equal to 1 if and only if the corresponding subgraph in 5 is a directed spanning tree. Thus we have proved the following theorem. Theorem 6.24. Let Κ be the in-degree matrix of a directed graph G without self-loops. Let the ith row of Κ correspond to vertex u, of G. Then the number r of the directed spanning trees of G having v as its root is given by d

r

T

d

where K column.

rr

=

det K„,

is the matrix obtained by removing from Κ its rth row and its rth •

We now illustrate Theorem 6.24 and the arguments leading to its proof. Consider the directed graph G shown in Fig. 6.9. Let us compute the number of directed spanning trees with vertex υ, as the root.

158

MATRICES OF A GRAPH

Figure 6.9 The in-degree matrix Κ of G is K=

1 -1 0

-1 2 -1

and 0 0 0

-1

-2 2 - 1 -1 3J

We can write det K' as 0 0 0

-2 -1 3 0 -1 -1 0 1 0 + 0 0 1 0 0 -1 0 1 0 0 -1 1

det K'

-1 2 -1

0 0 0 0 0 0

-1 1 0 -1 1 0 0

+ 0

0

1 -1

0 1 -1 -1 1 0

-1 + 3 -1 0 0 + 0 1 0 0 - 1 0 1

0 0 0

-1 1

0 1 -1

0 -1 1

The six determinants on the right-hand side correspond to the subgraphs on the following sets of edges:

{«4,«s},

{e .e }. 4

6

{e„e }. 3

Removing the first row and the first column from the above determinants, we get -1 3 1 0 + -1 1 = 5.

det K' = n

2 -1

1 Τ

1 0 I 1 0 0 1 Τ 0 1 1 0 1 + -1 1 -1

I

1 0 -1 1

-1 1

ADJACENCY MATRIX

159

The five directed spanning trees with u, as root are {«„«5}-

{e,,e3},

{e^e6},

{e4,es},

{e4, e6} .

6.10 ADJACENCY MATRIX Let G = (V, E) be a directed graph with no parallel edges. Let V = {u,, v2,..., v„}. The adjacency matrix M = [m,y] of G is an n x n matrix with m„ defined as follows: m„

(1, 0,

if(Vi,Vl)EE. otherwise.

For example, the graph of Fig. 6.10 has the following adjacency matrix: Vx

M = v2

V2

l>3

1 0 1 1 0 0 0 1 0 0 1 0

V4

0 0 1 1

In the case of an undirected graph, m/y = 1 and only if there is an edge connecting i>, and Vj. We now study some results involving the adjacency matrix. Theorem 6.25. The (1, /) entry m^ of Mr is equal to the number of directed walks of length r from v, to vr Proof. Proof is by induction on r. Obviously, the result is true for r = 1. As the inductive hypothesis we shall assume that the theorem is true for MT~\ Then

Figure 6.10

160

MATRICES OF A GRAPH

_ γ t =

/ number of directed walks of \ i \ length r - 1 from υ, to v I k

*>"

m

For each k the corresponding term in the sum in the above equation gives the number of directed walks of length r from u, to υ. whose last edge is from v to Vj. The theorem follows for M\ if we note that the number of directed walks of length r from v, to v = Σ £ (number of directed walks of length r from v, to υj whose last edge is from v to u ). • k

=1

t

k

;

For example, consider the third power of the matrix Μ of the graph of Fig. 6.10: 2 2 1 2 1 1 1 1 M = 2 1 1 1 2 1 1 1 3

The entry (1,4) in this matrix gives the number of directed walks of length 3 from y, to v . These walks are 4

(

e

6 -

e

i '

e

4 )

a

n

d

(.e e ,e ). u

4

7

We next study an important theorem due to Harary [6.9] that forms the foundation of the signal flow approach for the solution of linear algebraic equations. To introduce the theorem, we need some terminology. A 1 -factor of a directed graph G is a spanning subgraph of G in which the in-degree and the out-degree of every vertex are both equal to 1. Obviously, such a subgraph is a collection of vertex-disjoint directed circuits including self-loops of G. For example, the two 1-factors of the graph of Fig. 6.10 are shown in Fig. 6.11a and 6.11ft. Consider now a permutation;Ί, y , · · ·, /„ of the integers 1 , 2 , . . . , n. This permutation is even if an even number of interchanges are required to rearrange it as 1, 2, 3 , . . . , n. An odd permutation is defined in a similar way. For example, consider the permutation 1 3 4 2. The following sequence of two interchanges will rearrange this permutation as 1 2 3 4: 2

1. Interchange 3 and 2. 2. Interchange 3 and 4. Thus this permutation is an even permutation. Theorem 6.26. Let H,, i' = l , 2 , . . . , p , be the 1-factors of an n-vertex directed graph G. Let L, denote the number of directed circuits in //, and let

ADJACENCY MATRIX

4

<

1 6 1

6

Figure 6 . 1 1 . The two 1-factors of the graph of Fig. 6.10. Μ be the adjacency matrix of G. Then P det

Af

=

( - 1 ) "

Σ

( ~ 1 )

L

'

·

1=1

Proof. From the definition of a determinant

=Σe

det Μ

..,,,»!„,

h h

·

" J

2

,

2

m

n

)

n

,

( 6 . 2 5 )

where permutation of 1 , 2 , . . . , n. 2. € j , „ 1 if ( / ) ' permutation; otherwise it is equal to - 1 . 3 . The sum E is taken over all permutations of 1 , 2 , . . . , n. 1-

0 ) = ΟΊ> 7 2 » · · ·> /«) =

s

i s

a n

a

e v e n

y

h

( y )

A nonzero term m - m m corresponds to the set of edges (υ,, v ), (v , v ),..., (v , v ) . Each v appears exactly twice in this set, once as a first element and once as a second element of some ordered pairs. This means that in the subgraph consisting of the edges (ι>,, ), ( u , v ), ..., (υ , υ ) the in-degree and the out-degree of each vertex are both equal to 1 . Thus each nonzero term in the sum in ( 6 . 2 5 ) corresponds to a 1-factor of G. Conversely, each 1-factor corresponds to a nonzero term m · m lh

jt

2

/2

n

2h

t

nj

k

2

/2

η

Xj

2j

162

MATRICES OF A GRAPH

For example, the 1-factor in Fig. 6.116 corresponds to the term m ·m · m . We now have to fix the sign of ^ . Let C be a directed circuit in the 1-factor corresponding to / , , j , . . · , /„• Let C consist of the ω edges 1 3

m

21

32

44

h

2

(υ,,, υ, ), (u, , υ,),..., 2

2

(υ, ,

v ).

ω

it

The initial vertices of these edges form the array Ί> ' 2 » • · · » ' < » >

and their terminal vertices form the array i , i , . . . , ij . 2

3

It is easy to show that ω - 1 interchanges are sufficient to rearrange the array i , i , . . . , i , as i,, i , . . . , i . Let there be L directed circuits in the 1-factor corresponding to in Ji> · • •' in- L lengths of these directed circuits be ω,, ω , . . . , ω^. Then we need 2

3

2

e t

a

t n e

2

( , - 1 ) + (α> -1) + · · · + Κ - 1 ) = ΐ ν ω

2

interchanges to rearrange

j ,...,

as 1 , 2 , . . . , n. So,

2

;„ = ( - i r = ( - l )

n

+

t

-

Thus summarizing: 1. Each nonzero term m m ^ corresponds to a 1-factor //, of G (note that the value of this term is 1). 2. e , = (-l) ' where L, is the number of directed circuits in H,. Xh

n)

n+L

Ι\·)2·-

·>η

V

'

'

'

1

Thus (6.25) reduces to ρ

d e t M ^ - l ) " Σ (-I) <-ι

1,1

·

•

For example, consider the 1-factors of Fig. 6.11. The corresponding L,'s are

163

THE COATES AND MASON GRAPHS

Thus the determinant of the adjacency matrix of the graph of Fig. 6.10 is (-1)'+ ( - l ) = 0 . 2

This may be verified by direct expansion of the determinant. Suppose in a directed graph G, each edge (i, j) is assigned a weight Then we may define the adjacency matrix Μ = [m ] of G as follows:

w

ir

_ Γ w , if there is an edge directed from vertex i to vertex / ; i ~ {0 , otherwise . tj

m ,

Let us define the weight product w(H) of a subgraph Η of G as the product of the weights of all the edges of H. If Η has no edges, that is, Η is empty, we define w(H) = 1. Then we can obtain the following as an easy extension of Theorem 6.26. Theorem 6.27. The determinant of the adjacency matrix Μ of an n-vertex weighted directed graph G is given by det Μ = ( - 1 ) " Σ

(-l) "w(H) L

Η

where Η is a 1-factor of G, and L H. •

H

6.11

is the number of directed circuits in

THE COATES AND MASON GRAPHS

In this section we develop a graph-theoretic approach for the solution of linear algebraic equations. Two closely related methods due to Coates [6.10] and Mason [6.11] and [6.12] are discussed. 6.11.1

Coates' Method

Consider a linear system described by the set of equations (6.26)

ΑΧ=Βχ , Λ+1

where A = [a,J is an η x η nonsingular matrix, A!" is a column vector of unknown variables x ,x ,... ,x„, Β is a column vector of elements fc,, b ,..., b , and x is the input variable. We can solve for x as t

2

n

2

n+l

k

x t ,

=

Σ &A

,

dei~A~

·

iZ

where Δ, is the (i, k) cofactor of A. Α

<' > 6 27

164

MATRICES OF A GRAPH

Let A' denote the matrix obtained from A by adding - β to the right of A and then adding a row of zeros at the bottom of the resulting matrix. Let us now associate with A' a weighted directed graph G (A') as follows. G (A') has η + 1 vertices x,, x , . . . ,x„ . If a ^ 0 , then in G (A') there is an edge directed from x, to x with weight a Clearly A' is the transpose of the adjacency matrix of G (A'). And G (A') is called the Coates flow graph or simply the Coates graph associated with A'. We shall also refer to it as the Coates graph associated with the system of equations (6.26). As an example consider the set of equations: C

2

C

/(

+l

y

C

2 1 2

1 -1 1

C

jr

C

1 "1 • * R " 1" -2 x — 0 2- . i . .-1.

(6.28)

2

x

The matrix A' in this case is 2 1 A' = 2 0

1 -1 1 0

1 -f --2 0 2 1 0 0.

The Coates graph G (A') associated with the set of equations (6.28) is shown in Fig. 6.12a. Note that we may regard each vertex x,, 1 < i < n, of G (A') as representing an equation in (6.26). For example, the ith equation in (6.26) can be obtained by equating to zero the sum of the products of weights of the edges directed into x, and the variables corresponding to the vertices from which these variables originate. Furthermore, the Coates graph G (A) associated with A can be obtained by removing from G (A') the vertex x . The graph G (A) in the case of (6.28) is shown in Fig. 6.126. Since A is the transpose of the adjacency matrix of G (A) and since a matrix and its transpose have the same value of determinant, we get from Theorem 6.27 C

C

C

n + 1

C

C

C

det A =

(6.29)

(-l)^(-l) "w(H), L

where Η is a 1-factor in G (A), w(H) is the weight product of H, and L is the number of directed circuits in H. Thus we can evaluate the denominator of (6.27) in terms of the weight products of the 1-factors of G (A). To derive a similar expression for the numerator of (6.27), we need to evaluate Δ„· A 1-factorial connection Η from x, to x in G (A) is a spanning subgraph of G (A) that contains (a) a directed path Ρ from x, to x and (b) a set of vertex-disjoint directed circuits that include all the vertices of G (A) except those contained in P. As an example, a 1-factorial connection from x to x of the graph G (A') of Fig. 6.12a is shown in Fig. 6.12c. C

H

C

η

t

C

;

C

C

4

C

2

THE COATES AND MASON GRAPHS

165

Figure 6.12. (a) The Coates graph G (A')- W The graph G (A)- (<0 A 1-factorial connection H of the graph G (A'). C

C

A2

C

166

MATRICES OF A GRAPH

Theorem 6.28. Let G (A) be the Coates graph associated with an η x η matrix A. Then: C

2. Δ , , ^ - ΐ Γ ^ ί - ι ^ Μ / / , , ) ,

ivy.

where Η is a 1-factor in the graph obtained by removing vertex x, from G (A), H is a 1-factorial connection in G (A) from vertex x to vertex x , and L and are the numbers of directed circuits in Η and H, respectively. C

t)

f

C

y

H

r

Proof 1. Follows from Theorem 6.27. 2. Note that Δ,· is the determinant of the matrix obtained from A by replacing the y'th column of A by a column of zeros except for the element of the ith row, which is 1. Let the resulting matrix be denoted by A . Then the Coates graph G (A ) is obtained from G (A) by removing all the edges incident out of x (including self-loops at x ) and then adding an edge directed from x to x, with weight 1. Now from Theorem 6.27 we get a

c

a

C

;

y

;

A„ = det A

a

= (-1)"Σ(-1)Νν(# ),

(6.30)

α

"a

where H is a 1-factor in G (A ) and L is the number of directed circtuis in H . Each 1-factor H must necessarily contain the added edge with weight 1. Removing this edge from H we get a 1-factorial connection Η of G (A). Furthermore, w(H ) = w(H ). We can also see that there is a one-to-one correspondence between H in G (A ) and H in G (A) such that w(H ) = w(// ). Since the number of directed circuits in H is one less than that in H , we have L' = L -1. So we get from (6.30) a

c

a

a

a

a

a

η

C

a

tl

a

C

c

a

h

i;

a

it

a

H

Δ = (-ΐ)"-'Σ(-ΐ) Μ^)·

a

•

ι

(;

Consider now the term Σ",, 6,Δ , the numerator of (6.27). This term is equal to the determinant of the matrix obtained from A by replacing the /th column by B. We can easily relate this to detA' (where A' is the matrix obtained by removing from A' the (n + l)th row and the kth column) as follows: ιλ

n+l

k

n + i k

THE COATES AND MASON GRAPHS

Σ

b A ,

k

=(-iy

n + i )

- -\-l)dct(A' k

n + u k

167

(6.31)

).

1=1 From part 2 of Theorem 6.28 we get (-l)"

+ , +

*det(iC

l i 4

) = (-l)"

Σ

(-D "w(H L

n

+ u k

),

(6.32)

where L' is the number of directed circuits in the 1-factorial connection n i.k of G (A'). Combining (6.31) and (6.32), H

H

+

C

Σ&Α

Σ

= (-1)"

4

(-l) M// L

n

+ 1

,J.

(6.33)

From (6.29) and (6.33) we get the following theorem. Theorem 6.29. If the coefficient matrix A is nonsingular, then the solution of (6.26) is given by Σ =

(-i) MW„ ,,) l

+

^ILI

(6

.34)

Σί-ι^Μ") Η

for k = 1 , 2 , . . . , n, where 1. H is a 1-factorial connection of G (A') from vertex *„ x, 2. Η is a 1-factor of G (/4), and 3. and L are the numbers of directed circuits in H„ respectively. • n+lk

C

+ 1

to vertex

k

c

H

+lk

and H,

Equation (6.34) is called Coates' gain formula. We now illustrate an application of this formula by solving (6.28) for x lx . The 1-factors of the graph G (A) (Fig. 6.126) associated with the matrix A of (6.28) are given below along with their weight products. The different directed circuits in a 1-factor are distinguished by enclosing the vertices in each directed circuit within parentheses. 2

A

C

1-Factor ( * i ) ( *

Weight Product

) ( * 3 )

-4

(Xt)(x ,X ) (*;,)(*,, X )

-4 -2

2

2

3

3

(x )(x x ) 3

u

2

2

(*1.*3>*2)

~

4

168

MATRICES OF A GRAPH

From this we get the denominator in (6.34) as

Σ (-l) "*(H)

= (-l) (-4) + (-l) (-4) + (-l) (-2) 3

L

2

2

+ (-l) (2) + (-l)(l) + ( - l ) ( - 4 ) 2

=4-4-2+2-1+4 = 3. The 1-factorial connections from x to x of G (A') (Fig. 6.12«) are given below. The vertices that lie in a directed path are also now included within parentheses. 4

2

C

1-Factorial Connection

Weight Product -2

(x ,x ,x )(x ) 4

i

(*4>

2

}

\ >

x

*3> ^ 2 )

^

( x , * , x )(x ) 4

3

2

-4

x

1

( * 4 > * 3 > X\ > ^ 2 )

From this we get the numerator of (6.34) as Σ 1

1

η

= (-1X-2) + 4 + (-1K-4) + 1

(-l) w(Hn i.k) Li,

+

+ 1 ,lc =2+4+4+1 = 11 .

Thus we get i2 = 1 1 x 3 4

6.11.2

Mason's Method

To develop Mason's method, first we rewrite (6.26) as η X

i

=

Κ

+

ι

) ί χ

=

Σ α,*** - b,x

n + i

k

j = 1,2,..., η

,

-1

We may write this in matrix form as follows: (A'+ U )X'= n+1

X',

where A' is the (n + 1) x (n + 1) matrix defined earlier, U is the unit matrix of order n + 1, and X' is the column vector of variables n+l

169

THE COATES AND MASON GRAPHS

The Coates graph G (A' + U ) is called Mason's signal flow graph or simply the Mason graph associated with the matrix A' and it is denoted by G (A'). For example, the Mason graphs G (A') and G (A) associated with the system of equations (6.28) are shown in Fig. 6.13. We may regard each vertex in G (A') as representing a variable. If there is a directed edge from vertex JC, to x then we may consider the variable x, as making a contribution of (a,,*,) to the variable x So x is equal to the sum of the products of the weights of the edges incident into vertex x and the variables corresponding to the vertices from which these edges emanate. Thus the Mason graph is a convenient pictorial display of the flow of variables in a system. C

n + l

m

m

m

m

jt

r

t

j

*1

2 (i>)

Figure 6.13. (a) The Mason graph

G

m

( A ' ) .

(b)

The graph

G

m

( A ) .

170

MATRICES OF A GRAPH

Note that to obtain the Coates graph from a given Mason graph, we simply subtract one from the weight of each self-loop, and for each vertex of a Mason graph without a self-loop we add one with weight - 1 . This is equivalent to saying that the Coates graph G can be obtained from a Mason graph G simply by adding a self-loop of weight - 1 at each vertex. The set of such self-loops of weight - 1 added to construct G will be denoted by 5. Note that the graph G so constructed will have at most two and at least one self-loop at each of its vertices. Consider now the Mason graph G (A) associated with a matrix A, and let G be the corresponding Coates graph. Let Η be a 1-factor of G having / self-loops from the set S. Let L + / be the total number of directed circuits in H. On removing from Η the j added self-loops of the set S, we get a unique subgraph Q of G that is a collection of L vertex-disjoint directed circuits. Further, for / < n, c

m

c

c

m

c

c

Q

m

Q

(6.35)

w(H) = (-iy (Q). W

Note that if / = n, then Q is an empty subgraph of G has weight w(Q) = 1. So, in such a case,

which, by definition

m

w(//) = ( - i y .

(6.36)

We can also see that for each subgraph Q of G , which is a collection of L vertex-disjoint directed circuits, there corresponds a unique 1-factor of G that is obtained by adding self-loops (from the set 5) of weight - 1 to the vertices not in Q. From Theorem 6.27 we have m

Q c

detA = ( - l ) E ( - l ) ' ' n

c +

V(//)

Η

= (-l)"[l + Σ ( - 1 ) ^ ( 6 ) 1 · Q

(6-37)

J

The last step follows from (6.35) and (6.36). We may also rewrite (6.37) as det A = ( - 1 ) " [ l - Σ <2„ + Σ Q

l2

~ Σ Q, + • • ]

(6.38)

3

where Q is the weight product of the /th subgraph of G , which is a collection of k vertex-disjoint directed circuits. Thus we have expressed the denominator of (6.27) in terms of the weight products of appropriate subgraphs of the Mason graph. Let us refer to ( - 1 ) " det A as the determinant of the graph G (A) and denote it by Δ. Then using (6.33) and a reasoning exactly similar to that jk

m

m

THE COATES AND MASON GRAPHS

171

employed to derive (6.38), we can express the numerator of (6.27) as π

EM„

= (-irE»v(p: ,jA , + 1

1=1

;

where P'„+ is the /'th directed path from x to x in G (A), and Δ, is determinant of the subgraph of G (A'), which is vertex-disjoint from the /'th directed path P'„ . Thus we get the following. Uk

n

+

l

m

k

m

+ltk

Theorem 6.30. If the coefficient matrix A in (6.26) is nonsingular, then

Σνν(Ρ; )Δ, +Μ

*-

=-

,

i

T

(6.39)

k = l,2,...,n,

where P'„+ is the /'th directed path from x to x in G (A'), Δ is the determinant of the subgraph of G (A'), which is vertex-disjoint from the /'th directed path P'„ , and Δ is the determinant of the graph G (A). • n + }

1
k

;

m

m

+lk

m

Equation (6.39) is known as Mason's gain formula. In systems theory literature P' is referred to as a forward path from vertex x to vertex x . Furthermore the directed circuits of G (A') are called the feedback loops. We now illustrate application of (6.39) by solving (6.28) for x /x . The different collections of vertex-disjoint directed circuits of G (A) (Fig. 6.136) and their weight products are n+lk

n

k

+

l

m

2

4

m

Directed Circuit

Weight Product

CO (x )

3 3

3

(χ ,χ )

1

(x ,x )

-2

λ

2

2

3

(JC,,JC )

2

3

( x , , x , JC )

1

C*l> * 3 > * 2 )

~4

2

3

(*i)(* )

9

3

-6

(Xi)(x ,X ) 2

3

(* )(*1'*2)

3

3

From this we get the denominator in (6.39) as Δ = 1 - ( 3 +3 + 1 - 2 + 2 + 1 - 4 ) + ( 9 - 6 +3) = 3. The different directed paths of G (A') m

(Fig. 6.13a) from x to x along A

2

172

MATRICES OF A GRAPH

with their weight products are j

Weight Product

P'

4a

1

(x ,x ,x )

2 3 4

(x ,

X j , x3,

(x ,

x, 3

x)

(x ,

x$,

X],

A

4

4

4

l

-1

2

x) 2

—

2

xj 2

4 2 1

The directed circuits, which are vertex-disjoint from P and (JC,), respectively. So

l

and P , are (x ) 3

42

3

4 2

Δ, = 1 - 3 = - 2 , Δ = 1-3 = - 2 . 3

There are no directed circuits that are vertex-disjoint from P\

2

and

P\ . 2

So Δ = 1, 2

Δ = 1· 4

Thus the numerator in (6.39) is < Δ , + P\^ 2

2

+ Ρΐ, Δ< = 2 + 4 + 4 + 1 = 11

+ Ρΐ, Δ 2

3

2

So fa XA

6.12

=

11 3

FURTHER READING

Seshu and Reed [6.13], Chen [6.2], Mayeda [6.14], and Deo [6.15] are other references for the topics covered in this chapter. Chen also gives historical details regarding the results presented here. The textbook by Harary and Palmer [6.16] is devoted exclusively to the study of enumeration problems in graph theory, in particular, to those related to unlabeled graphs. See also Biggs [6.17]. The problem of counting spanning trees has received considerable attention in electrical network theory literature. Recurrence relations for counting spanning trees in special classes of graphs are available. For example, see Myers [6.18] and [6.19], Bedrosian [6.20], Bose, Feick, and Sun [6.21], and Swamy and Thulasiraman [6.22]. Berge [6.23] contains formulas for the

EXERCISES

173

number of spanning trees having certain specified properties. For algorithms to generate all the spanning trees of a connected graph see Gabow and Myers [6.24] and Jayakumar, Thulasiraman, and Swamy [6.25] and [6.26]. Chen [6.2] gives an extensive discussion of several aspects concerning the signal flow graph approach. See Robichaud, Boisvert, and Robert [6.27] for several applications of signal flow graphs. Balabanian and Bickart [6.28] may be consulted for the application of signal flow graphs in the study of electrical networks. An algorithm due to Rao [6.29] to construct graphs having specified circuit and cutset matrices is discussed in [6.30]. See also Cederbaum [6.31], Boesch and Youla [6.32], and Kim and Chien [6.33]. Swamy and Thulasiraman [6.30] give a detailed discussion of applications of graphs in electrical network theory.

6.13

EXERCISES

6.1

Let Q - [q ] be a fundamental cutset matrix and B = [b ] be a fundamental circuit matrix of a connected directed graph. Prove the following: (a) If the products (q,jq ) and (q q ) are both nonzero, then they are equal. (b) Repeat (a) for B . f

tJ

f

k)

ir

tl

kr

f

6.2

Let β be a matrix formed by any μ(β) independent rows of the circuit matrix of a connected directed graph G having m edges. Show that if F is any matrix of order p(G) x m and of rank p(G), having entries 1 , - 1 , and 0 and satisfying BF' = 0, then each row of F is a row of the cut matrix of G.

6.3

Let β be a matrix formed by any p(G) independent rows of the cut matrix of a connected directed graph G having m edges. Show that if F is any matrix of order p.(G) x m and of rank μ ( ΰ ) , having entries 1 , - 1 , and 0 and satisfying QF' = 0, then each row of F corresponds to a circuit or an edge-disjoint union of circuits of G.

6.4

Let Q(x, y) be the submatrix of the cut matrix Q of a connected nonseparable directed graph G containing only those rows of Q that correspond to cuts separating vertices χ and y. Show that Q(x, y) contains a fundamental cutset matrix of G. (See Kajitani and Ueno [6.34] for a more general result.) c

c

6.5

Let β be a submatrix of the circuit matrix B of a planar graph G containing only those rows of B that correspond to the meshes of G. Show that Β is unimodular. (See Chapter 7 for the definition of a planar graph.) c

c

174

MATRICES OF A GRAPH

6.6

Let β be a matrix formed by any /i(G) independent rows of the circuit matrix of a connected directed graph G. Show that the determinants of all the major nonsingular submatrices of Β have the same magnitude.

6.7

Prove Theorem 4.11.

6.8

Prove that for a connected directed graph G T ( G ) = det(B B' ) = d e t ( g G ' ) , /

/

/

/

where B is a fundamental circuit matrix and Q is a fundamental cutset matrix of G. f

6.9

f

Let β be a matrix formed by any p(G) independent rows of the cut matrix of a connected directed graph G, and let β be a matrix formed by any p(G) independent rows of the circuit matrix of G. Show that

is nonsingular. 6.10

Prove that the circuit and cutset subspaces W and W over GF(2) of a connected undirected graph G are orthogonal complements of the vector space W if and only if the number of spanning trees of G is odd. c

s

G

6.11 For any edge e of a graph G, let G · e denote the graph obtained from G by contracting the edge e. If G is connected, show that T(G) = T(G -

e) + T ( G · e).

6.12 Use the recursion formula in Exercise 6.11 to find the number of spanning trees in a wheel on η vertices. (See Section 8.4 for the definition of a wheel.) 6.13

Show that if e is an edge of K„, then r(K -e) n

6.14

= (n-2)n"-

3

.

(a) Let Η be an Η-vertex graph in which there are k parallel edges between every pair of adjacent vertices and let G be the underlying simple graph of H. Show that

T(H) = r~V(G). (b) Let Η be the graph obtained from an π-vertex connected graph G

EXERCISES

175

when each edge of G is replaced by a path of length k. Show that .m-n + l

τ(Η) = k!

T(G) ,

where m is the number of edges in G. 6.15

Establish Theorem 6.17 as a special case of Theorem 6.24.

6.16

A graph G with weights associated with its edges is called a weighted graph. Recall that the weight product of a subgraph of G is equal to the product of the weights of the edges of the subgraph and that if the subgraph has only isolated vertices, then its weight product is defined to be 1. Let A be the incidence matrix of a connected directed graph G, with vertex v„ as reference. Assume that the ith row of A corresponds to vertex υ,·. Let W be the diagonal matrix representing the weights of G and assume that the columns of A and W are arranged in the same edge order. Prove that (a) det(AWA') = sum of the weight products of all the spanning trees of G. (b) Δ = sum of the weight products of all the spanning 2-trees T of G, where Δ is the (/, j) cofactor of AWA'. ι;

tj

n

ί;

6.17

The out-degree matrix Κ = [k \ self-loops is defined as follows: tj

of a directed graph G with no

if i Φ j and there are ρ parallel edges directed from υ, to v ; t

Also define an in-going directed tree as a directed graph that has a vertex v to which there is a directed path from every other vertex, and its underlying undirected graph is a tree. Show that the number of in-going spanning directed trees of a graph G can be computed in a way similar to Theorem 6.24. r

6.18

The out-degree matrix Κ = [Α; ] of a weighted directed graph G without self-loops is defined as follows: (a) k = - ( s u m of the weights of the edges directed from υ, to Vj), (;

tj

(b) A:,, = sum of the weights of the edges incident out of υ,. Prove that det(AT„) is the sum of the weight products of all the in-going spanning directed trees of G having v, as root. (Note: Here by a root we mean a vertex v, such that there is a directed path to u, from every other vertex.)

176

6.19

MATRICES OF A GRAPH

Solve the following system of equations:

"*Γ = "

" 3 - 2 1" 2 0 x -1 . 3 - 2 2. *3. 2

6.20

3" 1 .-2.

Let G be the Mason graph associated with a system of linear equations. Prove the following: (a) We can remove the self-loop of weight a Φ 1 at vertex x simply by multiplying the weight of every edge incident into x by the factor 1/(1 - a ). (b) We can remove a vertex x with no self-loop by doing the following: For all ϊΦρ and k^p, add (a a ) to the weight of the edge (x , x ). m

kk

k

k

kk

p

pi

t

kp

k

6.21 Using the method in Exercise 6.20, reduce the Mason graph associated with (6.40) to a graph containing only the vertices x and x and solve for x /x . A

2

6.14

6.1 6.2 6.3

2

A

REFERENCES

F. E. Hohn, Elementary Matrix Algebra, Macmillan, New York, 1958. W. K. Chen, Applied Graph Theory, North-Holland, Amsterdam, 1971. G. Kirchhoff, "Uber die Auftosung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefiihrt wird," Ann. Phys. Chem., Vol. 72, 497-508 (1847). 6.4 A. Cayley, "A Theorem on Trees," Quart. J. Math., Vol. 23, 376-378 (1889). 6.5 J. W. Moon, "Various Proofs of Cayley's Formula for Counting Trees," in A Seminar on Graph Theory, F. Harary and L. W. Beinke, Eds., Holt, Rinehart and Winston, New York, 1967, pp. 70-78. 6.6 H. Priifer, "Neuer Beweis eines Satzes uber Permutationen," Arch. Math. Phys., Vol. 27, 742-744 (1918). 6.7 K. Sankara Rao, V. V. Bapeswara Rao, and V. G. K. Murti, "Two-Tree Admittance Products," Electron. Lett., Vol. 6, 834-835 (1970). 6.8 W. T. Tutte, "The Dissection of Equilateral Triangles into Equilateral Triangles," Proc. Cambridge Phil. Soc, Vol. 44, 203-217 (1948). 6.9 F. Harary, "The Determinant of the Adjacency Matrix of a Graph," SIAM Rev., Vol. 4, 202-210 (1962). 6.10 C. L. Coates, "Flow-Graph Solutions of Linear Algebraic Equations," IRE Trans. Circuit Theory, Vol. CT-6, 170-187 (1959). 6.11 S. J. Mason, "Feedback Theory: Some Properties of Signal Flow Graphs," Proc. IRE., Vol. 41, 1144-1156 (1953).

REFERENCES

177

6.12 S. J. Mason, "Feedback Theory: Further Properties of Signal Flow Graphs," Proc. IRE., Vol. 44, 920-926 (1956). 6.13 S. Seshu and Μ. B. Reed, Linear Graphs and Electrical Networks, AddisonWesley, Reading, Mass., 1961. 6.14 W. Mayeda, Graph Theory, Wiley-Interscience, New York, 1972. 6.15 N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Englewood Cliffs, N.J., 1974. 6.16 F. Harary and Ε. M. Palmer, Graphical Enumeration, Academic Press, New York, 1973. 6.17 N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, England, 1974. 6.18 B. R. Myers, "Number of Trees in a Cascade of 2-Port Networks," IEEE Trans. Circuit Theory, Vol. CT-14, 284-290 (1967). 6.19 B. R. Myers, "Number of Spanning Trees in a Wheel," IEEE Trans. Circuit Theory, Vol. CT-18, 280-282 (1971). 6.20 S. D. Bedrosian, "Number of Spanning Trees in Multigraph Wheels," IEEE Trans. Circuit Theory, Vol. CT-19, 77-78 (1972). 6.21 Ν. K. Bose, R. Feick, and F. K. Sun, "General Solution to the Spanning Tree Enumeration Problem in Multigraph Wheels," IEEE Trans. Circuit Theory, Vol. CT-20, 69-70 (1973). 6.22 Μ. N. S. Swamy and K. Thulasiraman, "A Theorem in the Theory of Determinants and the Number of Spanning Trees of a Graph," Proc. IEEE Int. Symp. on Circuits and Systems, 153-156 (1976). 6.23 C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. 6.24 Η. N. Gabow and E. W. Myers, "Finding all Spanning Trees of Directed and Undirected Graphs," SIAM J. Comp., Vol. 7, 280-287 (1978). 6.25 R. Jayakumar, K. Thulasiraman, and Μ. N. S. Swamy, "Complexity of Computation of a Spanning Tree Enumeration Algorithm," IEEE Trans. Circuits and Systems, Vol. CAS-31, 853-860 (1984). 6.26 R. Jayakumar, K. Thulasiraman, and Μ. N. S. Swamy, "MOD-CHAR: An Implementation of Char's Spanning Tree Enumeration Algorithm and Its Complexity Analysis," IEEE Trans. Circuits and Systems, Vol. CAS-36, 219-228 (1989). 6.27 L. P. A. Robichaud, M. Boisvert, and J. Robert, Signal Flow Graphs and Applications, Prentice-Hall, Englewood Cliffs, N.J., 1962. 6.28 N. Balabanian and T. A. Bickart, Electrical Network Theory, Wiley, New York, 1969. 6.29 V. V. Bapeswara Rao, "The Tree-Path Matrix of a Network and Its Applications," Ph.D. Thesis, Dept. of Electrical Engineering, Indian Institute of Technology, Madras, India, 1970. 6.30 Μ. N. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, 1981. 6.31 I. Cederbaum, "Applications of Matrix Algebra to Network Theory," IRE Trans. Circuit Theory, (Special supplement), Vol. CT-6, 127-137 (1959).

178

MATRICES OF A GRAPH

6.32 F. T. Boesch and D. C. Youla, "Synthesis of (n + l)-Node Resistor n-Ports," IEEE Trans. Circuit Theory, Vol. CT-12, 515-520 (1965). 6.33 W. H. Kim and R. T. Chien, Topological Analysis and Synthesis of Communication Nets, Columbia University Press, New York, 1962. 6.34 Y. Kajitani and S. Ueno, "On the Rank of Certain Classes of Cutsets and Tie-Sets of a Graph," IEEE Trans. Circuits and Sys., Vol. CAS-26, 666-668 (1979).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 7

PLANARITY AND DUALITY

In this chapter we discuss two important concepts in graph theory, namely, planarity and duality. First we consider planar graphs and derive some properties of these graphs. Characterizations of planar graphs due to Kuratowski, Wagner, Harary and Tutte and MacLane are also discussed. We then discuss Whitney's definition of duality of graphs, which is given in terms of circuits and cutsets, and relate this concept to the seemingly unrelated concept of planarity. Duality has been of considerable interest to electrical network theorists. This interest is due to the fact that the voltages and currents in an electrical network are dual variables. Duality of these variables arises as a result of Kirchhoff s laws. Kirchhoff s voltage law is in terms of circuits and Kirchhoff s current law is in terms of cutsets. Duality between circuits and cutsets was observed in Chapters 2 and 4. This duality will become obvious in Chapter 10, where we show that the set of circuits and the set of cutsets of a graph have the same algebraic structure. 7.1

PLANAR GRAPHS

A graph G is said to be embeddable on a surface 5 if it can be drawn on S so that its edges intersect only at their end vertices. A graph is said to be planar if it can be embedded on a plane. Such a drawing of a planar graph G is called a planar embedding of G. Two planar embeddings of a graph are shown in Fig. 7.1. In one of these (Fig. 7.1e) all the edges are drawn as straight-line segments, while in the other (Fig. l.\b) one of the edges is drawn as a curved line. Note that the 179

PLANARITY AND DUALITY

Figure 7.1. Two planar embeddings of a graph.

(M

edge connecting vertices a and d in Fig. 7.16 cannot be drawn as a straight line if all the remaining edges are drawn as shown. Obviously, if a graph has self-loops or parallel edges, then in none of its planar embeddings all the edges can be drawn as straight-line segments. This naturally raises the question whether for every simple planar graph G there exists a planar embedding in which all the edges of G can be drawn as straight-line segments. The answer to this question is in the affirmative, as stated in the following theorem. Theorem 7.1. For every simple planar graph there exists a planar embedding in which all the edges of the graph can be drawn as straight-line segments. •

PLANAR GRAPHS

181

This result was proved independently by Wagner [7.1], Fary [7.2], and Stein [7.3]. If a graph is not embeddable on a plane, then it may be embeddable on some other surface. However, we now show that embeddability on a plane and embeddability on a sphere are equivalent; that is, if a graph is embeddable on a plane, then it is also embeddable on a sphere and vice versa. The proof of this result uses what is called the stereographic projection of a sphere onto a plane, which is described below. Suppose that we place a sphere on a plane (Fig. 7.2) and call the point of contact the south pole and the diametrically opposite point on the sphere the north pole N. Let Ρ be any point on the sphere. Then the point P' at which the straight line joining Ν and P, when extended, meets the plane, is called the stereographic projection of Ρ onto the plane. It is clear that there is a one-to-one correspondence between the points on a sphere and their stereographic projections on the plane. Theorem 7.2. A graph G is embeddable on a plane if and only if G is embeddable on a sphere. Proof. Let G' be an embedding of G on a sphere. Place the sphere on the plane so that the north pole is neither a vertex of G' nor a point on an edge of G'. /v

Figure 7.2. Stereographic projection.

182

PLANARITY AND DUALITY

(<)

Figure 7.3. Basic nonplanar graphs, (a) K (b) K .

}

SJ

Then the image of G' under the stereographic projection is an embedding of G on the plane because edges of G' intersect only at their end vertices and there is a one-to-one correspondence between points on the sphere and their images under stereographic projection. The converse is proved similarly. • Two basic nonplanar graphs called Kuratowski's graphs are shown in Fig. 7.3. One of these is K , the complete graph on five vertices, and the other is AT . We call these graphs basic nonplanar graphs because they play a fundamental role in an important characterization of planarity due to Kuratowski (Section 7.3). The nonplanarity of these two graphs is established in the next section. Before we conclude this section, we would like to point out that Whitney [7.4] has proved that a separable graph is planar if and only if its blocks are planar. So while considering questions relating to the embedding on a plane, it is enough if we concern ourselves with only nonseparable graphs. 5

33

7.2

EULER'S FORMULA

An embedding of a planar graph on a plane divides the plane into regions. A region is finite if the area it encloses is finite; otherwise it is infinite. For example, in the planar graph shown in Fig. 7.4, the hatched region f is the infinite region; / , , / , / , and f are the finite regions. Clearly, the edges on the boundary of a region contain exactly one circuit, and this circuit is said to enclose the region. For example, the edges 5

2

3

4

EULER'S FORMULA

183

Figure 7.4 e , e , e , e , and e form the region /, in the graph of Fig. 7.4, and they contain the circuit {e,, e , e , e ). Note that in any spherical embedding of a planar graph, every region is finite. Suppose we embed a planar graph on a sphere and place the sphere on a plane so that the north pole is inside any chosen region, say, region /. Then under the stereographic projection the image of/will be the infinite region. Thus a planar graph can always be embedded on a plane so that any chosen region becomes the infinite region. Let /ι» Λ» · · •' fr regions of a planar graph with f as the infinite region. We denote by C,, 1 ^ i: :£ r, the circuit on the boundary of region/,. The circuits C,, C , . . . , C _,, corresponding to the finite regions, are called meshes. It is easy to verify that the ring sum of any k^2 meshes, say, C,, C ,...,C , is a circuit or union of edge-disjoint circuits enclosing the regions / , , / , . . . , f . Since each mesh encloses only one region, it follows that no mesh can be obtained as the ring sum of some of the remaining meshes. Thus the meshes C,, C , . . . , C _ are linearly independent. Suppose that any element C of the circuit subspace of G encloses the finite regions / , , / , . . . , f . Then it can be verified that x

8

9

10

u

g

D e

g

10

t n e

r

2

2

r

k

2

k

2

2

r

l

k

C—

φ C φ ' ' * Φ C. 2

k

184

PLANARITY AND DUALITY

For example, in the graph of Fig. 7.4, the set C = {e,, e , e , e , e , e , e , e ) encloses the regions / , , f , / , f . Therefore, 2

7

s

2

C

3

3

4

5

6

4

Cj Φ C Φ C Φ C .

=

2

3

4

Thus every element of the circuit subspace of G can be expressed as the ring sum of some or all of the meshes of G. Since the meshes are themselves independent, we get the following theorem. Theorem 7.3. The meshes of a planar graph G form a basis of the circuit subspace of G. • Following is an immediate consequence of this theorem. Corollary 7.3.1 (Euler's Formula). If a connected planar graph G has m edges, η vertices, and r regions, then η - m +r = 2. Proof. The proof follows if we note that by Theorem 7.3, the nullity μ of G is equal to r - 1. • In general it is difficult to test whether a graph is planar or not. We now use Euler's formula to derive some properties of planar graphs. These properties can be of help in detecting nonplanarity in certain cases, as we shall see soon. Corollary 7.3.2. If a connected simple planar graph G has m edges and vertices, then

>3

m < 3n - 6 . Proof. Let F= {f, f ,..., f } denote the set of regions of G. Let the degree d(f) of region f denote the number of edges on the boundary of bridges being counted twice. (For example, in the graph of Fig. 7.4, the degree of region / , is 6.) Noting the similarity between the definitions of the degree of a vertex and the degree of a region, we get from Theorem 1.1, 2

r

1

Σ <*(/,) = 2m. Since G has neither parallel edges nor self-loops and η s: 3, it follows that 'For the definition of bridge, see Exercise 1.28.

EULER'S FORMULA

185

d ( / , ) > 3 , for all i. Hence

Σ «/(/,) & 3 r . Thus 2m > 3r, that is, r< |m . Using this inequality in Euler's formula, we get η - m + \m > 2 or m < 3n - 6 .

•

Corollary 7.3.3. K is nonplanar. s

Proof. For K , n = 5 and m = 10. If it were planar, then by Corollary 7.3.2, 5

m = 1 0 < 3 n - 6 = 9; a contradiction. Thus K must be nonplanar. 5

Corollary 7.3.4. K

33

•

is nonplanar.

Proof. For K , m = 9 and η = 6. If it were planar, then by Euler's formula it has r = 9 - 6 + 2 = 5 regions. In K there is no circuit of length less than 4. Hence the degree of every region is at least 4. Thus, 3 3

3 3

r

2m = Σ ι=

d(f,)si4r 1

or rs

2m — 4 '

that is, r < 4 ; a contradiction. Hence K

33

is nonplanar.

•

Corollary 7.3.5. In a simple planar graph G there is at least one vertex of degree less than or equal to 5. Proof. Let G have m edges and η vertices. If every vertex of G has degree

186

PLANARITY AND DUALITY

greater than 5, then by Theorem 1.1, 2m>6n or w > 3« . But by Corollary 7.3.2, ms3n-6. These two inequalities contradict one another. Hence the result.

•

7.3 KURATOWSKI'S THEOREM AND OTHER CHARACTERIZATIONS OF PLANARITY

Characterizations of planarity given by Kuratowski, Wagner, Harary and Tutte and MacLane are presented in this section. To explain Kuratowski's characterization, we need the definition of the concept of homeomorphism between graphs. The two edges incident on a vertex of degree 2 are called series edges. Let e = (u, v) and e = (v, w) be the series edges incident on a vertex v. Removal of vertex ν and replacing e and e by a simple edge («, w) is called series merger (Fig. 7.5a). Adding a new vertex υ on an edge (u, w), thereby creating the edges (u, υ) and (v, w), is called series insertion (Fig. 7.5b). Two graphs are said to be homeomorphic if they are isomorphic or can be made isomorphic by repeated series insertions and/or mergers. It is clear that if a graph G is planar, then any graph homeomorphic to G is also planar, that is, planarity of a graph is not affected by series insertions or mergers. We proved in the previous section that K and K are nonplanar. Therefore, a planar graph does not contain a subgraph homeomorphic to K or K . It is remarkable that Kuratowski [7.5] could prove that the converse of this result is also true. In the following theorem, we state this celebrated characterization of planarity. Proof of this may also be found in Harary [7.6]. x

2

x

2

5

33

5

J3

Theorem 7.4 (Kuratowski). A graph is planar if and only if it does not contain a subgraph homeomorphic to K or K . • 5

3 3

See also Tutte [7.7] for a constructive proof of this theorem. As an illustration of Kuratowski's theorem, consider the graph G shown

KURATOWSKI'S THEOREM AND OTHER CHARACTERIZATIONS OF PLANARITY

187

(a)

6u>

Figure 7.5. (a) Series merger, (ft) Series insertion.

Figure 7.6. (a) Graph G. (b) Subgraph of G homeomorphic to K3 3.

in Fig. 7.6a. It contains a subgraph (Fig. 7.6ft) that is homeomorphic to K3 3 . Therefore G is nonplanar. We now present another characterization of planarity independently proved by Wagner [7.8] and by Harary and Tutte [7.9]. Theorem 7.5. A graph is planar if and only if it does not contain a subgraph contractible to K5 or Ki3. ■ Consider now the graph (known as the Petersen graph) shown in Fig. 7.7. This graph does not contain any subgraph isomorphic to K5 oi Ki3, but it is known to be nonplanar. So if we wish to use Kuratowski's criterion to

188

PLANARITY AND DUALITY

Figure 7.7. The Petersen graph. establish the nonplanar character of the Petersen graph, then we need to locate a subgraph homeomorphic to K or K . However, the nonplanarity of the graph follows easily from Theorem 7.5 because the graph reduces to K after contracting the edges e , e , e , e , and e . MacLane's characterization of planar graphs is stated next. 5

5

t

2

33

4

3

5

Theorem 7.6. A graph G is planar if and only if there exists in G a set of basis circuits such that no edge appears in more than two of these circuits. • We know that the meshes of a planar graph form a basis of the circuit subspace of the graph, and that no edge of the graph appears in more than two of the meshes. This proves the necessity of Theorem 7.6. The proof of the sufficiency may be found in MacLane [7.10]. Another important characterization of planar graphs in terms of the existence of dual graphs is discussed in Section 7.5.

7.4

DUAL GRAPHS

A graph G is a dual of a graph G, if there is a one-to-one correspondence between the edges of G and those of G, such that a set of edges in G is a circuit vector of G if and only if the corresponding set of edges in G, is a cutset vector of G Duals were first defined by Whitney [7.11], though his original definition was given in a different form. Clearly, to prove that G is a dual of G, it is enough if we show that the vectors forming a basis of the circuit subspace of G correspond to the vectors forming a basis of the cutset subspace of G,. For example, consider the graphs G, and G shown in Fig. 7.8. The edge e, of G, corresponds to the edge e* of G . It may be verified that the circuits {e,, e , e }, {e , e , e , e ), and {e , e , e } form a basis of the circuit 2

2

2

2

r

2

2

2

2

2

3

3

A

5

6

6

7

8

DUAL GRAPHS

189

subspace of G,, and the corresponding sets of edges {e*, e*, e*}, {e*, e*, e*, el}, and {e*, e*, el} form a basis of the cutset subspace of G . Thus G is a dual of G,. We now study some properties of the duals of a graph. 2

Theorem 7.7. Let G be a dual of a graph G,. Then a circuit in G corresponds to a cutset in G, and vice versa. 2

2

2

Proof. Let C* be a circuit in G and C be the corresponding set of edges in G,. Suppose that C is not a cutset. Then it follows from the definition of a dual that C must be the union of disjoint cutsets C,, C , . . . , C , k s 2 . Let C*, C * , . . . , C be the sets of edges in G that correspond to the cutsets Cj, C , . . . , C . Again from the definition of a dual it follows that Cj*, C * , . . . , C* are circuits or unions of disjoint circuits. Since C* is the union of Cj*, C*,..., C*, k ^ 2, it is clear that C* must contain more than one circuit. However, this is not possible since C* is a circuit, and no proper subset of a circuit is a circuit. Thus k = 1, or in other words, C is a cutset of G,. In a similar way we can show that each cutset in G, corresponds to a circuit in G . • 2

2

2

k

2

k

k

2

Theorem 7.8. If G is a dual of G,, then G, is a dual of G . 2

2

Proof. To prove the theorem we need to show that each circuit vector of G, corresponds to a cutset vector of G and vice versa. Let C be a circuit vector in G j , with C* denoting the corresponding set of edges in G . By Theorem 4.7, C has an even number of common edges with every cutset vector of G,. Since G is a dual of G,, C* has an even number of common edges with every circuit vector of G . Therefore, by Theorem 4.8, C* is a cutset vector of G . In a similar way we can show that each cutset vector of G corresponds to a circuit vector of G,. Hence the theorem. • 2

2

2

2

2

2

In view of Theorem 7.8, we refer to graphs G and G as simply duals if one of them is dual of the other. The following result is a consequence of Theorem 7.8 and the definition of a dual. x

2

Theorem 7.9. If G and G are dual graphs, then the rank of one is equal to the nullity of the other; that is, t

2

p(G,) = / ( G ) 1

2

190

PLANARITY AND DUALITY

and p(G ) = ( G ) . 2

M

I

•

Suppose a graph G has a dual. Then the question arises whether every subgraph of G has a dual. To answer this question we need the following result. Theorem 7.10. Consider two dual graphs G, and G . Let e = (i>,, v ) be an edge in G,, and e* = (ν*, u*) be the corresponding edge in G . Let G\ be the graph obtained by removing the edge e from G,; let G' be the graph obtained by contracting e* in G . Then Gj and G are duals, the one-to-one correspondence between their edges being the same as in G, and G . 2

2

2

2

2

2

2

Proof. Let C and C* denote corresponding sets of edges in G, and G , respectively. Suppose C is a circuit in G J. Since it does not contain e, it is also a circuit in G Hence C* is a cutset, say (V*,V ), in G . Since C* does not contain e*, the vertices v* and u* are both in V* or in V . Therefore C* is also a cutset in G . Thus every circuit in Gj corresponds to a cutset in G' . Suppose C* is a cutset in G . Since C* does not contain e*, it is also a cutset in G . Hence C is a circuit in Gp Since it does not contain e, it is also a circuit in G[. Thus every cutset in G' corresponds to a circuit in G\. • 2

P

b

2

b

2

2

2

2

2

In view of this theorem, we may say, using the language of electrical network theory, that "open-circuiting" an edge in a graph G corresponds to "short-circuiting" the corresponding edge in a dual of G. A useful corollary of Theorem 7.10 now follows. Corollary 7.10.1. If a graph G has a dual, then every edge-induced subgraph of G also has a dual. Proof. The result follows from Theorem 7.10, if we note that every edge-induced subgraph Η of G can be obtained by removing from G the edges not in H. • To illustrate Corollary 7.10.1, consider the two dual graphs G and G of Fig. 7.8. The graph Gj shown in Fig. 7.9a is obtained by removing from G, the edges e and e . The graph G' of Fig. 7.9b is obtained by contracting the edges e* and e* of G . It may be verified that Gj and G' are duals. Observing that series edges in a graph G correspond to parallel edges in a dual of G, we get the following corollary of Theorem 7.10. x

3

6

2

2

2

2

Corollary 7.10.2. If a graph G has a dual, then every graph homeomorphic to G also has a dual. •

DUAL GRAPHS

Figure 7.8. Dual graphs, (a) Graph G,. (b) Graph G . 2

191

Figure 7.9. Dual graphs, (a) Graph G[. (b) Graph G' . 2

We now proceed to develop an equivalent characterization of a dual. Let G be an η-vertex graph. We may assume without loss of generality that it is connected. Let K* be a subgraph of G, and let G' be the graph obtained by contracting the edges of K*. Note that G' is also connected. If K* has n* vertices and ρ connected components, then G' will have η - (η* - ρ) vertices. Therefore, the rank of G' is given by p(G') = « - ( « * - / , ) - ! =

p(G)-p(K*).

(7.1)

192

PLANARITY AND DUALITY

Theorem 7.11. Let G, and G be two graphs with a one-to-one correspondence between their edges. Let Η be any subgraph of G, and H* the corresponding subgraph in G . Let K* be the complement of H* in G . Then G, and G are dual graphs if and only if 2

2

2

2

(7.2)

μ(Η) = ρ(β )-ρ(Κ·). 2

Proof Necessity Let G, and G be dual graphs. Let G be the graph obtained from G by contracting the edges of K*. Then by Theorem 7.10, Η and G are dual graphs. Therefore, by Theorem 7.9, 2

2

2

2

M

(J/) = P(G;).

But by ( 7 . 1 ) ,

p(G' ) =

p(G )-p(K*).

p(H) =

p(G )-p(K*).

2

2

Hence 2

Sufficiency Assume that ( 7 . 2 ) is satisfied for every subgraph Η of G We now show that each circuit in G corresponds to a cutset in G and vice versa. Let Η be a circuit in G,. Then μ(Η) = 1. Therefore by (7.2), v

T

2

p(K*) = p ( G ) - l . 2

Since Η is a minimal subgraph of G complement of H* in G , it is clear with rank equal to p(G ) - 1. It now that H* is a cutset in G . In a similar way we can show that in G,. Thus G, and G are dual graphs. 2

2

with nullity equal to 1, and AT* is the that K* is a maximal subgraph of G follows from the definition of a cutset

2

2

2

a cutset in G corresponds to a circuit 2

•

Whitney's [7.11] original definition of duality was stated as in Theorem 7.11.

To illustrate this definition, consider the dual graphs G, and G of Fig. 7.8. A subgraph Η of G, and the complement K* of the corresponding subgraph in G are shown in Fig. 7.10. We may now verify that 2

2

p(H) =

p(,G )-p(K*) 2

PLANARITY AND DUALITY

193

Figure 7.10. Illustration of Whitney's definition of duality, (a) Graph Η, μ(Η) = \. (b) Graph K*, p(K*) = 2.

7.5

PLANARITY AND DUALITY

In this section we characterize the class of graphs that have duals. While doing so, we relate the two seemingly unrelated concepts, planarity and duality. First we prove that every planar graph has a dual. The proof is based on a procedure for constructing a dual of a given planar graph. Consider a planar graph and let G be a planar embedding of this graph. Let / j , f ,---,f be the regions of G. Construct a graph G* defined as follows: 2

r

1. G* has r vertices υ*, v*,... ,v*, vertex ν*, 1 s / < r, corresponding to region /,. 2. G* has as many edges as G has. 3. If an edge e of G is common to the regions f and / (not necessarily distinct), then the corresponding edge e* in G* connects vertices v* and v*. (Note that each edge e of G is common to at most two regions, and it is possible that an edge may be in exactly one region.) {

y

194

PLANARITY AND DUALITY

A simple way to construct G* is to first place the vertices ν*, ν*,. •., ν*, one in each region of G. Then, for each edge e common to regions / a n d / , draw a line connecting v* and υ* so that it crosses the edge e. This line represents the edge e*. The procedure for constructing G* is illustrated in Fig. 7.11. The continuous lines represent the edges of the given planar graph G and the dashed lines represent those of G*. We now prove that G* is a dual of G. Let C C , . . . , C _ , denote the meshes of G, and C*, C * , . . . , C * _ , denote the corresponding sets of edges in G*. It is clear from the procedure used to construct G* that the edges in C* are incident on the vertex v* and form a cut whose removal will separate v* from the remaining vertices of G*. By Theorem 7.3, C\, C , . . . , C _, form a basis of the circuit subspace of G, and we know that the incidence vectors C*, C*,..., C*_j form a basis of the cutset subspace of G*. Since there is a one-to-one correspondence between C,'s and C*'s, G and G* are dual graphs. Thus we have the following theorem. y

u

2

r

2

r

Theorem 7.12. Every planar graph has a dual.

•

The question that immediately arises now is whether a nonplanar graph has a dual. The answer is " n o , " and it is based on the next two lemmas. Lemma 7.1. K

3

3

has no dual.

Figure 7.11. Construction of a dual.

PLANARITY AND DUALITY

195

Proof. First observe that 1. K has no cutsets of two edges. 2. K has circuits of length 4 or 6 only. 3. K 3 has nine edges. 3 3

3 3

3

Suppose K has a dual G. Then these observations would, respectively, imply the following for G: 3 3

1. G has no circuits of two edges; that is, G has no parallel edges. 2. G has no cutsets with less than four edges. Thus every vertex in G is of degree at least 4. 3. G has nine edges. The first two imply that G has at least five vertices, each of degree at least 4. Thus G must have at least \ x 5 x 4 = 10 edges. However, this contradicts observation 3. Hence K , has no dual. • 3 3

Lemma 7.2. K has no dual. s

Proof. First observe that 1. K has no circuits of length 1 or 2. 2. K has cutsets with 4 or 6 edges only. 3. K has 10 edges. 5

5

5

Suppose K has a dual G. Then by observation 2, G has circuits of lengths 4 and 6 only. In other words all circuits of G are of even length. So G is bipartite. Since a bipartite graph with six or fewer vertices cannot have more than nine edges, it is necessary that G has seven vertices. But by observation 1 the degree of every vertex of G is at least 3. Hence G must have at least \ x 7 x 3 > 1 0 edges. This, however, contradicts observation 3. Hence K has no dual. • 5

5

The main result of this section now follows. Theorem 7.13. A graph has a dual if and only if it is planar. Proof. The sufficiency part of the theorem is the same as Theorem 7.12. We can prove the necessity by showing that a nonplanar graph G has no dual. By Kuratowski's theorem, G has a subgraph Η homeomorphic to K or K . If G has a dual, then by Corollary 7.10.1, Η has a dual. But then, by Corollary 7.10.2, K or K should have a dual. This, however, will 33

s

33

5

196

PLANARITY AND DUALITY

contradict the fact that neither of these graphs has a dual. Hence G has no dual. • Theorem 7.13 gives a characterization of planar graphs in terms of the existence of dual graphs and was originally proved by Whitney. The proof given here is due to Parsons [7.12]. Whitney's original proof, which does not make use of Kuratowski's theorem, may be found in [7.13]. From the procedure given earlier in this section it is clear that different (though isomorphic) planar embeddings of a planar graph may lead to nonisomorphic duals (Exercise 7.7). The following theorem presents a property of the duals of a graph. Theorem 7.14. All duals of a graph G are 2-isomorphic; every graph 2-isomorphic to a dual of G is also a dual of G. • The proof of this theorem follows in a straightforward manner from the definition of a dual and Theorem 1.9.

7.6

FURTHER READING

Whitney [7.4], [7.11], and [7.14] and books by Seshu and Reed [7.13], Ore [7.15], Harary [7.6], Bondy and Murty [7.16], and Chartrand and Lesniak [7.17] are recommended for further reading. Two properties of a nonplanar graph G that are of interest follow: 1. The minimum number of planar subgraphs whose union is G; this is called the thickness of G. 2. The minimum number of crossings (or intersections) in order to draw a graph on a plane; this is called the crossing number of G. For several results on the thickness and the crossing numbers of a nonplanar graph, see Harary [7.6]. See also Bose and Prabhu [7.18]. An algorithm to test the planarity of a graph will be discussed in Section 11.11 where references relating to the planar embedding problem and applications may also be found.

7.7

7.1

EXERCISES

Show that if G is a connected planar graph with m edges, η vertices, and with girth* & > 3 , then m == k(n - 2)l{k - 2). Using this result deduce that the Petersen graph is nonplanar.

'For the definition of girth, see Exercise 1.18.

EXERCISES

7.2

197

Let d,, d , • • • ,d„ be the degrees of the vertices of a simple planar graph. Prove that if η s 4 , then 2

η

Σ d, <2(n + 3 ) - 6 2 . 2

ι=

7.3 7.4

Prove that a simple planar graph with η > 4 vertices has at least four vertices with degree 5 or less. Prove or disprove: Every connected simple nonplanar graph is contractible to K or K . Let G be a simple graph with at least 11 vertices. Show that G and its complement G cannot both be planar. (In fact, a similar result can be proved with 11 replaced by 9. See Tutte [7.19].) Give an^xample of a graph G on 8 vertices with the property that both G and G are planar. Using Kuratowski's theorem prove that the Petersen graph is nonplanar. Find two nonisomorphic duals of the graph in Fig. 7.12. 5

7.5

7.6 7.7

2

1

3

3

Figure 7.12 7.8

A planar graph is a maximal planar graph if, for every pair of nonadjacent vertices u and v, the graph G U {e}, where e = (u, υ), is nonplanar. Let G be a simple maximal planar graph with « ^ 4 vertices. If p, denotes the number of vertices of degree i in G for / = 3 , 4 , . . . , k = A(G), then show that 3 p + 2p + p = 3

7.9

A

5

P l

+ 2 p + . . . + (n - 6)6p, + 12 . 8

Prove that a planar graph without self-loops is nonseparable if and only if its dual is nonseparable. 7.10 Show that the dual of a nonseparable planar graph is Eulerian if and only if the graph is bipartite.

198

PLANARITY AND DUALITY

7.11 A planar graph is self-dual if it is isomorphic to its dual. Show that if a graph G with η vertices and m edges is self-dual, then m = 2n - 2. 7.12 A one-terminal-pair graph is a graph with two vertices specially designated as the terminals of the graph. A planar one-terminal-pair graph is a one-terminal-pair graph that is planar and remains planar when an edge joining the two terminals is added to it. A series-parallel graph is a one-terminal-pair graph denned recursively as follows: (a) A single edge is a series-parallel graph. If G' and G" are series-parallel, then: (b) The series combination of G' and G" is a series-parallel graph. By the series combination of G' and G" we mean the joining of one of the terminals of G' with one of the terminals of G". (Fig. 7.13a).

Figure 7.13. (a) Series combination of G' and G". (b) Parallel combination of G' and G".

(c) The parallel combination of G' and G" is a series-parallel graph. By the parallel combination of G' and G" we mean the joining of the two terminals of G' with the two terminals of G". (Fig. 7.136). Show that a dual of a graph G is a series-parallel graph if and only if G is series-parallel. 7.8

REFERENCES

7.1

K. Wagner, "Bemerkungen zum Vierfarbenproblem," Uber. Deutsch. Math. Verein, Vol. 46, 26-32 (1936). I. Fary, "On Straight Line Representation of Planar Graphs," Acta Sci. Math. Szeged, Vol. 11, 229-233 (1948). S. K. Stein, "Convex Maps," Proc. Am. Math. Soc, Vol. 2, 464-466 (1951).

7.2 7.3

REFERENCES

7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19

199

Η. Whitney, "Non-Separable and Planar Graphs," Trans. Am. Math. Soc, Vol. 34, 339-362 (1932). C. Kuratowski, "Sur le Probleme des courbes gauches en topologie," Fund. Math., Vol. 15, 271-283 (1930). F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969. W. T. Tutte, "How to Draw a Graph," Proc. London Math. Soc, Vol. 13, 743-767 (1963). K. Wagner, "Uber eine Eigenschaft der ebenen Komplexe," Math. Ann., Vol. 114, 570-590 (1937). F. Harary and W. T. Tutte, "A Dual Form of Kuratowski's Theorem," Canad. Math. Bull., Vol. 8, 17-20 (1965). S. MacLane, "A Structural Characterization of Planar Combinatorial Graphs," Duke Math. J., Vol. 3, 340-372 (1937). H. Whitney, "Planar Graphs," Fund. Math., Vol. 21, 73-84 (1933). T. D. Parsons, "On Planar Graphs," Am. Math. Monthly, Vol. 78, 176-178 (1971). S. Seshu and Μ. B. Reed, Linear Graphs and Electrical Networks, AddisonWesley, Reading, Mass., 1961. H. Whitney, "A Set of Topological Invariants for Graphs," Am. J. Math., Vol. 55, 231-235 (1933). O. Ore, The Four Colour Problem, Academic Press, New York, 1967. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976. G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Cole, Pacific Grove, Calif., 1986. Ν. K. Bose and K. A. Prabhu, "Thickness of Graphs with Degree Constrained Vertices," IEEE Trans. Circuits and Sys., Vol. CAS-24, 184-190 (1975). W. T. Tutte, "On the Nonbiplanar Character of the Complete 9-Graph," Canad. Math. Bull., Vol. 6, 319-330 (1963).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 8

CONNECTIVITY AND MATCHING

In Chapter 1 we denned a graph to be connected if there exists a path between any two vertices of the graph. Suppose a graph G is connected. Then we may be interested in finding how "well-connected" G is. In other words we may like to know the minimum number of vertices or edges whose removal will disconnect G. This leads us to the concepts of vertex and edge connectivities of a graph. In this chapter we first develop several results relating to vertex and edge connectivities of a graph. We also discuss a classical result in graph theory, namely, Menger's theorem, which relates connectivities to the number of vertex-disjoint and edge-disjoint paths. A matching in a graph is a set of edges no two of which have common vertices. In the latter part of this chapter we develop the theory of matchings, starting our development with Hall's marriage theorem. Connectivity and matching are two extensively studied topics in graph theory. Many deep results in graph theory belong to these two areas. 8.1

CONNECTIVITY OR VERTEX CONNECTIVITY

The connectivity k ( G ) of a graph G is the minimum number of vertices whose removal from G results in a disconnected graph or a trivial graph/ K ( G ) is also called vertex connectivity to distinguish it from edge connectivity, which is introduced in the next section. For example, the connectivity of the graph of Fig. 8.1 is 2 since the removal of vertices v and v from this graph results in a disconnected graph, whereas the removal of no single vertex will disconnect the graph. x

2

'Recall (Section 1.7) that a trivial graph is a graph with just a single vertex. 200

CONNECTIVITY OR VERTEX CONNECTIVITY

201

Figure 8.1

Obviously, the connectivity of a disconnected graph is 0. Consider a graph G on n vertices. It is clear that K(G) = n — 1 if G is complete. If G is not complete, it will have at least two nonadjacent vertices vx and v2. The removal of the remaining n-2 vertices from G will then result in a graph in which υ, and v2 are not connected. Thus if G is not complete, we have K(G)
A disconnecting set in a graph G is a set of vertices whose removal from G results in a disconnected graph or a trivial graph. A graph G is said to be k-connected if K(G) a k. So, a A:-connected graph has no disconnecting set S with \S\ :£ k - 1. If a graph is connected, its connectivity is greater than or equal to 1. So connected graphs are 1-connected. If a connected graph has no cut-vertices, then its connectivity is greater than 1. So, connected graphs with no cut-vertices are 2-connected. In the following theorem we present a simple upper bound on the connectivity of a graph. Theorem 8.1. For a simple connected graph G, K(G) < S(G), where 8(G) is the minimum degree in G. Proof. Consider any vertex y in a simple connected graph G = (V, E). Let Γ(υ) denote the set of vertices adjacent to u. It is clear that Γ(ν) is a disconnecting set since the removal of the vertices in Γ(υ) results in a trivial graph or a disconnected graph in which υ is not connected to any of the remaining vertices. Thus K(G) = S | I » |

forallüGV.

Since G is simple,

\m\ = d(v),

202

CONNECTIVITY AND MATCHING

and hence K(G)*mm{d(v)} = «(G).

•

If a graph G has m edges and η vertices v v ,... from Theorem 1.1, u

,v ,

2

n

then we know

d(u,) + d(v ) + ••• + d(v„) = 2m . 2

So, «-5(G)<2m and 2m

KG)-

(8.1)

η

L

where |*J denotes the largest integer less than or equal to x. Combining (8.1) with the result of Theorem 8.1, we get the following. Theorem 8.2. For a simple connected graph G having m edges and η vertices,

Let f(k, n) denote the least number of edges that a -connected graph on η vertices must have. Of course, we assume that k< n. From Theorem 8.2 it is clear that /(*,«)> [ y j ,

(8.2)

where fjc"j denotes the smallest integer greater than or equal to x. Harary [8.1] has proved that equality holds in (8.2) by giving a procedure for constructing a /c-connected graph H which has exactly \kn/2] edges. This procedure is as follows. kn

Case 1

k even.

Let k = 2r. Then H has vertices v , v , v ,..., u _, and two vertices v, and v are adjacent if / - r s / s f + r, where addition is modulo n. H is shown in Fig. 8.2a. 2rn

f

6S

0

t

2

n

CONNECTIVITY OR VERTEX CONNECTIVITY

203

is) »o

Figure 8.2. (a) Η6Λ. (b) // 5 , 8 . (c) tf7,9

204

CONNECTIVITY AND MATCHING

k odd, η even.

Case 2

Let k = 2r + 1. H is constructed by first constructing H „ and then adding edges joining vertex v, to vertex v for i = 1, 2 , . . . , n/2 and / = ι + n/2 (mod n). / / is shown in Fig. 8.26. 2r+ln

2r

}

5 8

Case 3 & odd, η odd. Let = 2r + 1. H is constructed by first constructing H and then adding edges joining i>, and u for i = 0, 1, 2 , . . . ,(n - l ) / 2 and ; = i + (n + l ) / 2 (modn). f/ is shown in Fig. 8.2c 2r+1

n

2rn

y

7

9

It is easy to verify that the graph H constructed as described here has exactly \kn/2] edges. We now prove that H is ^-connected. kn

kn

Theorem 8.3. Graph H

is ^-connected.

kn

Proof Case 1

k = 2r.

From the symmetry of H it is sufficient to show that the vertices v and υ , α = 1 , 2 , . . . , η - 1 cannot be disconnected by removal of fewer than 2r vertices. Suppose that v and v can be disconnected by removal of 2r - 1 vertices υ,, u, 2 ,..., v . One of the two intervals [0, a], [a, n] contains at most r - 1 of these indices. Suppose it is interval [0, a ] . Then two consecutive vertices of the sequence obtained by removing υ , ν,,..., ν, from the sequence υ , υ , , υ ,...,ν are connected by an edge (because the difference between their indices is s r ) . Hence there is a path from v to v , a contradiction. 2rn

0

β

0

a

t

0

2

α

0

a

k = 2r + 1, η even.

Case 2

Suppose the vertices v and v are disconnected by the removal of 2r vertices v , υ, ,..., u, . If one of the two intervals [0, a ] , [a, n] does not contain a consecutive r number of these indices, then a path from v to v can be constructed as shown under case 1. Therefore, suppose that v and v are disconnected by the removal of v„ v ,..., v _ where 1 =£ i =£ a - r, and v , v ,...,O . where 1 < / < η - r - a. Let 0

a

2r

ti

0

0

l+1

a+j

a

+

l

+

1

a

+

J

+

r

l

t

i+r

lt

a

a

CONNECTIVITY OR VERTEX CONNECTIVITY

205

Then β G [a + j + r, η + i - 1 ] ,

β' £ [i + r, a + j - 1 ] .

There is a path from vertex v to vertex and from vertex v . to vertex v . Hence there is a path from vertex υ to vertex v since in H „ there is an edge connecting ν and υ ,. 0

p

a

0

β

Case 3

a

2r+1

β

k = 2r + 1, η odd

Proof follows as in case 2.

•

In the next theorem we present a sufficient condition for a graph to be A>connected. This result is due to Bondy [8.2]. Theorem 8.4. Let G be a simple graph of order n. Suppose we order the vertices of G so that d( )^d(v )<---^d(v ). Vl

2

n

Then G is λ-connected if d(v ) > r + k - 1 iorl
-

r

d(v„_ ) k+l

Proof. Suppose a simple graph G satisfies the conditions of the theorem. If it is not Λ-connected, then there exists a disconnecting set S such that |S| = s < * . Consider now the graph G - S that is not connected. Let Η be a component of G - S of minimum order h. Then it is clear that the degree in Η of each vertex of Η is at most h - 1. Therefore, in G, the degree of each vertex of Η is at most h + s - 1. Thus d(v )
+s-\
+k-\.

(8.3)

Therefore, by the conditions of the theorem, h>n-l-d(v _ ). n

k+1

(8.4)

Since G — S has η - s vertices and Η is a component of G — S of minimum order, we have h^ η—s—h or h + s
206

CONNECTIVITY AND MATCHING

Therefore, d(v) s A + j - l s n - A - 1 ,

ve

V(H),

(8.5)

where V(H) is the vertex set of H. Since every vertex u e V(G) - V(H) - S is adjacent to at most η - h - 1 vertices, we have d(u)

=Ξ

η - h - 1,

u e V(G) - V(H) - S .

(8.6)

From (8.5) and (8.6) we conclude that all the vertices of degrees exceeding η - h - 1 are in S. Thus there are at most s vertices of degrees exceeding η - h - 1. Therefore (8.7) Using (8.4) in (8.7), we get d(v - )
k+i

Therefore n—s
a contradiction. For example, the degrees of the graph in Fig. 8.3 satisfy the conditions of Theorem 8.4 for k = 3. Hence it is 3-connected.

Figure 8.3. A 3-connected graph that satisfies the conditions of Theorem 8.4 for k = 3.

EDGE CONNECTIVITY

8.2

207

EDGE CONNECTIVITY

Edge connectivity κ ' ( G ) of a graph G is the minimum number of edges whose removal from G results in a disconnected graph or a trivial graph. In other words κ ' ( G ) is the number of edges in a cut having the minimum number of edges. For example, for the graph of Fig. 8.4 the edge connectivity is equal to 2, since the removal of the two edges e and e disconnects the graph, and removal of no single edge results in a disconnected graph. A graph G is said to be k-edge connected if K'(G)^k. Thus at least k edges have to be removed to disconnect a Λ-edge-connected graph. Since the edges incident on any vertex υ of G form a cut of G , it follows that x

2

K'(G)<5(G). In the following theorem we relate

#c(G), κ ' ( G ) ,

and

6(G).

Theorem 8.5. For a simple graph G ,

Proof. We have already proved the second inequality. The first inequality can be proved as follows. If G is not connected, then K ( G ) = K ' ( G ) = 0. Thus in such a case the condition K ( G ) = £ K ' ( G ) is satisfied. If G is connected and κ ' ( G ) = 1, then G has a bridge e. In this case, if we remove one of the vertices on which e is incident, then a disconnected graph or the trivial graph results. Thus K ( G ) ^ κ ' ( G ) in this case too. Suppose κ ' ( G ) ^ 2. Then G has κ ' ( G ) edges whose removal disconnects it. Removal of any κ ' ( G ) - 1 of these edges results in a graph with a bridge e = (v , v ). For each of these κ ' ( G ) - 1 edges select an end vertex different from v and v . Removal of these vertices will remove from G the κ ' ( G ) - 1 edges and possibly some more. Suppose the resulting graph is disconnected. x

2

x

2

Figure 8.4. A 2-edge-connected graph.

208

CONNECTIVITY AND MATCHING

Then k ( G ) S Κ'(G) - 1. Otherwise this graph will have e as a bridge, and so removal of u, or v will result in a disconnected graph or the trivial graph. In such a case k ( G ) =S Κ'(G). Thus in all cases 2

k(G)
•

Next we present a sufficient condition for κ ' ( G ) to be equal to 3 ( G ) . This result is due to Chartrand [8.3]. Theorem 8.6. Let G be an η-vertex simple graph. If 8(G) a L"/2j then k'(G)

=

8(G).

Proof. It can be shown that G is connected (see Exercise 1.14). Therefore k'(G)>0.

Since κ ' ( G ) < 8(G), the proof will follow if we show that k\G) S 8(G). Suppose k'(G)<8(G). Then there exists a cut S = (V,, V,) such that K ' ( G ) = |S| < ^ ( G ) . Let the edges of S be incident with g vertices in V, and ρ vertices in V,. Suppose |Vj = q. Then each vertex of V, is an end vertex of at least one edge in 5. If we denote by G the induced subgraph of G on the vertex set V,, then G , has at least X

m

i

edges. Since k'(G)<8(G),

=

$q8(G)-K'(G))

we get >\(q8(G)-8(G))

mi

=

l8(G)(q-l)

>k(?-i). since 8(G) > κ'(G) s q. This is a contradiction since in a simple graph there cannot be more than q(q-$l2 edges connecting q vertices. Therefore |V,| > q. In a similar way we can prove that \V \ > p. _ If > q and |V,| > p, then there are vertices in both V, and V that are adjacent to only vertices in V, and Κ,, respectively. Thus each of Vj and V, contains at least 8(G) +1 vertices. Therefore G has at least 28(G) + 2 vertices. But X

x

>n , leading to a contradiction. Therefore there is no cut S with \S\ < 8(G).

•

GRAPHS WITH PRESCRIBED DEGREES

8.3

GRAPHS WITH PRESCRIBED

209

DEGREES

Recall that a sequence (d,, d , . . . , d„) of nonnegative integers is graphic if there exists an η-vertex graph with vertices ι>,, v ,..., v such that vertex u, has degree d,. In this section we first describe an algorithm to construct a simple graph, if one exists, having a prescribed degree sequence. We then use this algorithm to establish Edmonds' theorem on the existence of it-edgeconnected simple graphs having prescribed degree sequences. Consider a graphic sequence (d,, d , . . . , d ) with d, > d > · · • > d„. Let d, be the degree of vertex v,. "To lay off d " means to connect the corresponding vertex v to the vertices 2

2

2

n

2

n

k

k

y , , v ,...,

v,

2

\id
dk

k

or to the vertices , if d >k

v ,v ,...,v _ ,v ,...,v l

2

k

l

k+x

dk+l

.

k

The sequence (d, - 1,. . . ,d dk

1, d

d k + u

d , . . . ,d„),

. . .,d _ ,0, K

L

\ld
k+l

k

or (d, - 1 , . . . , d .

k l

- 1,0, d

k+x

- 1,..., d

- 1, d ,...,

d t + 1

d j , if d > A:

dk+2

k

is called the residual sequence after laying off d or simply the residual sequence. Hakimi [8.4] and Havel [8.5] have given an algorithm for constructing a simple graph, if one exists, having a prescribed degree sequence. This algorithm is based on a result that is a special case (where k = 1) of the following theorem due to Wang and Kleitman [8.6]. k

Theorem 8.7. If a sequence (d,, d , . . . , d„) with d, s d s · · · > d„ is the degree sequence of a simple graph, then so is the residual sequence after laying off d . 2

2

k

Proof. To prove the theorem we have to show that a graph having (d,, d , . . . , d„) as its degree sequence exists such that vertex v is adjacent to the first d vertices other than itself. If otherwise, select from among the graphs with the degree sequence (d,, d , . . . , d„) a simple graph G in which v is adjacent to the maximum number of vertices among the first d vertices other than itself. Let v be a vertex not adjacent to v in G such that 2

k

k

2

k

k

m

k

m^d

k

,

\id
210

CONNECTIVITY AND MATCHING

or m < d + 1,

if k < d .

k

k

In other words v is among the first d vertices other than v . So v is adjacent in G to some vertex v that is not among these first d vertices. Then d > d (if equality, the order of q and m can be interchanged), and hence v is adjacent to some vertex ν,, ίΦ q, ίΦ m, such that v, and v are not adjacent in G. If we now remove the edges (v , υ,) and (v , v ) and replace them by (v , v ) and (υ,, v ), we obtain a graph G' with one more vertex adjacent to v among the first d vertices other than itself violating the definition of G. • m

k

k

q

m

k

k

q

m

q

m

m

k

k

q

q

k

k

From Theorem 8.7 we get the following algorithm which is a generalization of Hakimi's algorithm for realizing a sequence D = (a ,, d ,..., d) with d > d > · · · s d„ by a simple graph. Choose any d Φ 0. "Lay off" d by connecting v to the first d vertices other than itself. Compute the residual degree sequence. Reorder the vertices so that the residual degrees in the resulting sequence are in nonincreasing order. Repeat this process until one of the following occurs: 1

2

x

n

2

k

k

k

k

1. All the residual degrees are zero. In this case the resulting graph has D as its degree sequence. 2. One of the residual degrees is negative. This means that the sequence D is not graphic. To illustrate the preceding algorithm, consider the sequence \

V

2

3

V

Z> = (4

4

V

3

5

V

©

V

2

2).

After laying off d (which is circled), we get the sequence 3

V

i

2

V

D ' = (3

3

V

2

4

S

V

0

V

1

2),

which, after reordering of the residual degrees, becomes V

\

D, = (3

2

V

2

S

4

V

3

V

φ

V

1

0).

We next lay off the degree corresponding to u and get 5

u, v

2

D[ = (2

1

v 0

s

υ 1

4

υ 0) 3

GRAPHS WITH PRESCRIBED DEGREES

211

Reordering the residual degrees in D[ we get l

V

2

V

D = (©

ι

2

4

V

ι

5

V

o

3

V

0).

Now laying off the degree corresponding to y,, we get

l/| D' = (0 2

v

2

0

v

4

0

0

1/3 0).

The algorithm terminates here. Since all the residual degrees are equal to zero, the sequence (4, 3, 3, 2 , 2) is graphic. The required graph (Fig. 8.5) is obtained by the following sequence of steps, which corresponds to the order in which the degrees were laid off: 1. Connect i> to u,, v , and v . 2. Connect v to υ, and v . 3. Connect υ, to v and v . 3

2

5

4

2

2

4

Erdos and Gallai [8.7] have given a necessary and sufficient condition (not of an algorithmic type) for a sequence to be graphic. See also Harary [8.8]. Suppose that in the algorithm we lay off at each step the smallest nonzero residual degree. Then using induction we can easily show that the resulting graph is connected if for all i

(8.8)

Σ^2(«-1).

(8.9)

d, > 1 and η

ι=

1

Note that (8.8) and (8.9) are necessary for the graph to be connected. In fact we prove in the following theorem a much stronger result. This result is due to Edmonds [8.9]. The proof given here is due to Wang and Kleitman [8.10].

Figure 8.5. A graph with degree sequence (4, 3, 3, 2, 2).

212

CONNECTIVITY AND MATCHING

Theorem 8.8 (Edmonds). A necessary and sufficient condition for a graphic sequence (d,, d , . . . , d„) to be the degree sequence of a simple /c-edgeconnected graph, for k > 2, is that each degree d, > k. 2

Proof. Necessity is obvious. We prove the sufficiency by showing that the algorithm we have just described results in a Λ-edge-connected graph when all the degrees in the given graphic sequence are greater than or equal to k. Note that at each step of the algorithm, we should lay off the smallest nonzero residual degree. Proof is by induction. Assume that the algorithm is valid for all sequences in which each degree d, s: ρ with ρ ^ k - 1. To prove the theorem we have to show that in the graph constructed by the algorithm every cutset (A, A) has at least_fe edges. Proof is trivial if |A| = 1 or |A| = 1. So assume that |A| a 2 and |A| > 2 . We claim that in the connection procedure of the algorithm, one of the following three cases will eventually occur at some step, say, step r. (Step r means the step when the rth vertex is fully connected.) Case 1 Case 2

Case 3

All the nonzero residual degrees are at least k. All the nonzero residual degrees are at least k - 1, and there is at least one edge constructed by steps 1 , . . . , r that lies in (A, A). All the nonzero residual degrees are at least k -2, and there are at least two edges constructed by steps 1,2, . . . , r that lie in (A, A).

All these three cases imply that the cutset (A, A) has at least k edges by induction. We prove this claim as follows: Let v, be the vertex that is connected at step i. Without loss of generality we may assume that υ is in A. We now show that at some step in the connection procedure case 3 must occur if cases 1 and 2 never occur. At step 1, y, is fully connected. Then: λ

a. The smallest nonzero residual degrees are k-l, because case 1 has not occurred. b. The degrees of none of the vertices in A are decreased by 1 when υ, is connected, because case 2 has not occurred. Hence v , the vertex to be connected at step 2, must be in A. (All vertices in A must have degree at least k.) Now if v is connected and case 1 does not occur and no edge connects A with A, then we shall still have (a) and (b) as before. Hence the next vertex to be connected will still be in A. Since the residual degrees are decreased at each step of the connection procedure, sooner or later there must exist an rwith υ, in A, such that when i>, is connected, an edge will connect A with A. If case 2 does not occur, 2

2

MENGER'S THEOREM

213

then one of the nonzero residual degrees among the vertices of A not yet fully connected becomes k - 2. This means, by our connection procedure, that v must connect to every vertex in A, because vertices in A all have residual degree equal to k, and_since we connect v to the vertices of the largest residual degree. Since | / l | s 2 , at this step case 3 occurs. • r

r

Wang and Kleitman [8.6] have also established necessary and sufficient conditions for a graphic sequence to be the degree sequence of a simple A;-vertex-connected graph.

8.4

MENGER'S THEOREM

We present in this section a classical result in graph theory, namely, Menger's theorem [8.11]. This theorem helps to relate the connectivity of a graph to the number of vertex-disjoint paths between any two distinct vertices in the graph. Theorem 8.9 (Menger). The minimum number of vertices whose removal from a graph disconnects two nonadjacent vertices s and t is equal to the maximum number of vertex-disjoint s-t paths in the graph. • Proof of this theorem is given in Section 12.10. Theorem 8.10. A necessary and sufficient condition that a simple graph G = (V, E), with | V | s A : + l, be Λ-connected is that there are k vertexdisjoint s-t paths between any two vertices s and t of G. Proof. Obviously, the theorem is true for k = 1. So we need to prove the theorem for k > 2. Necessity If s and t are not adjacent, then the necessity of the theorem follows from Theorem 8.9. Suppose that s and t are adjacent and that there are at most k - 1 vertex-disjoint s-t paths in G. Let e = (s, t). Consider now the graph G' = G - e. Since there are at most k - 1 vertex-disjoint s-t paths in G, there cannot be more than k-2 vertex-disjoint s-t paths in G'. Thus there exists a set ACV— {s, t} of vertices, with \A\^ k - 2 , whose removal disconnects s and t in G'. Then \V- A\ = \ V\ - \A\ > k + 1 - (k - 2) = 3 , and therefore, there is a vertex u in V- A different from s and t. Now we show that there exists an s-u path in G' that does not contain any vertex of A. Clearly, this is true if s and u are adjacent. If s and u are not adjacent, then there are k vertex-disjoint s-u paths in G, and hence

214

CONNECTIVITY AND MATCHING

there are k - 1 vertex-disjoint s-u paths in G'. Since | A\ < k - 2, at least one of these k - 1 paths will not contain any vertex of A. In a similar way we can show that in G' there exists au - t path that does not contain any vertex of A. Thus there exists in G' an s-t path that does not contain any vertex of A. This, however, contradicts that A is an s-t disconnecting set in G'. Hence the necessity. Sufficiency G is connected because there are k vertex-disjoint paths between any two distinct vertices of G. Further, not more than one of these paths can be of length 1, since there are no parallel edges in G. The union of the remaining k - 1 paths must contain at least k - 1 distinct vertices other than s and t. Hence \V\>(k-\)

+ 2> k.

Suppose in G there is a disconnecting set A with \A\ < k. Then consider the subgraph G' of G on the vertex set V- A. This graph contains at least two distinct components. If we select two vertices s and t from any two different components of G', then there are at most \A \ < k vertex-disjoint s - ί paths in G. This contradicts that any two vertices are connected by k vertex-disjoint paths in G. Hence the sufficiency. • This result is due to Whitney [8.12]. Since it is only a variation of Theorem 8.9, we shall refer to this also as Menger's theorem. Consider next two special classes of ^-connected graphs, namely, the 2-connected graphs and the 3-connected graphs. There are several equivalent characterizations of 2-connected graphs. Some of them we have already listed as exercises in Chapter 1. Tutte has given a characterization of 3-connected graphs. This characterization is given in terms of a special class of graphs called wheels. Consider a circuit C of length n. If we add a new vertex and connect it to all the vertices of C, then we get the (n + l)-wheel W . For example, W is shown in Fig. 8.6. Tutte's [8.13] characterization of 3-connected graphs is stated in the next theorem. n+ l

Figure 8.6. Wheel W . 7

7

MATCHINGS

215

Theorem 8.11. A simple graph G is 3-connected if and only if G is a wheel or can be obtained from a wheel by a sequence of operations of the following types: 1. The addition of a new edge. 2. The replacement of a vertex ν of degree ^4 by two adjacent vertices v' and υ" of degrees s 3 such that each vertex formerly adjacent to υ is now adjacent to exactly one of v' and v". • We conclude this section with a characterization of A>edge-connected graphs. This characterization is analogous to Theorem 8.9 and is also known as Menger's theorem, though it was independently discovered by Ford and Fulkerson [8.14] and by Elias, Feinstein, and Shannon [8.15]. Theorem 8.12. The minimum number of edges whose removal from a connected graph G disconnects two distinct vertices s and t is equal to the maximum number of edge-disjoint s-t paths in G. • Proof of Theorem 8.12 is also given in Section 12.10.

8.5

MATCHINGS

The discussions of this and the remaining sections of this chapter concern the following problem, known as the marriage problem, and related ones. We have a finite set of boys, each of whom has several girlfriends. Under what conditions can we marry off the boys in such a way that each boy marries one of his girlfriends? (No girl should marry more than one boy!) This problem can be posed in graph-theoretical terminology as follows: Construct a bipartite graph G in which the vertices x x ,... ,x represent the boys and the vertices >>,, y ,..., y represent the girls. An edge (χ,, y ) is present in G if and only if y is a girlfriend of x,. The marriage problem is then equivalent to finding in G a set of edges such that no two of them have a common vertex and each x, is an end vertex of one of these edges. For example, suppose there are four boys b , b , b , and 6 and four girls £i> # 2 ' # 3 ' #4> i their relationships as follows: Jy

2

t

2

n

m

t

x

a

n

o

<

w

t n

*.-»{*.}. b -*{g )

,

b -^{g ,

g) ,

b*->{gi,

g) •

2

3

2

1

2

A

2

3

4

216

CONNECTIVITY AND MATCHING *,o

b2o

-^>«,

y*

!^°*2

*3

Figure 8.7

t4o*—

og4

The bipartite graph representing this situation is shown in Fig. 8.7. It is easy to verify that it is not possible to marry off all the four boys such that each one marries one of his girlfriends. However, we can marry off as many as three boys without violating the requirement of the marriage problem. For example, two sets of such appropriate pairing are 1. (&„g,), (P2,g2), (*4.ft)· 2. (fc„g,), (&3.&)> (&4>&)· These discussions motivate the definition of a matching in a graph. Two edges in a graph are said to be independent if they do not have any common vertex. Edges e,, e 2 , . . . are said to be independent if no two of them have a common vertex. A matching in a graph is a set of independent edges. For example, in the graph of Fig. 8.8 {e,, e4} is a matching. Obviously, a maximal matching is a maximal set of independent edges. Thus in the graph of Fig. 8.8 {e1,e^} is a maximal matching, whereas {es, e6) is not. A matching with the largest number of edges is called a maximum matching. The set {el,es,e6} is a maximum matching in the graph of Fig. 8.8. The number of edges in a maximum matching of G will be called the matching number of G and will be denoted by a^G). A vertex is said to be saturated in a matching M if it is an end vertex of an edge in M. For example, the vertex v is saturated in the matching {e,, e5. e6} in the graph of Fig. 8.8. In the following a bipartite graph G = (V, E) with bipartition (X, Y) will be denoted by the triplet (X, Y, E). We say that the set X can be matched into Y in the bipartite graph (X, Y, E) if there is a matching M such that every vertex of X is saturated in M. The matching M will then be called a complete matching of X into Y. With this new terminology the marriage problem becomes equivalent to finding necessary and sufficient conditions for the existence of a complete matching of X into Y, in a bipartite graph (X, Y, E). In the next section we present several results concerning matchings in bipartite graphs, in addition to providing an answer to the marriage problem.

MATCHINGS IN BIPARTITE GRAPHS

217

Figure 8.8 8.6

MATCHINGS IN BIPARTITE GRAPHS

Consider a bipartite graph G = (Χ, Υ, E). Let S be any subset of X and let Γ(5) denote the set of vertices adjacent to the vertices in S. Suppose | 5 | > |r(S)|. Then it is clear that there is no complete matching of S into T(S), and any matching Μ in G will saturate at most |Γ(5)| of the vertices of S. So |M|<W-(|5|-|r(5)|).

(8.10)

Since (8.10) is valid for any subset 5 of X, we can conclude that for any matching M, |M|<|*|-max(|S|-|r(S)|).

(8.11)

We now define the deficiency σ ( 5 ) of any subset 5 of X and the deficiency &(G) of G as follows: σ(5) = | 5 | - | Γ ( 5 ) |

(8.12)

a(G) = m a x ( | S | - | r ( S ) | ) .

(8.13)

and

Now using (8.13), we can rewrite (8.11) as \M\x\X\-oiG).

(8.14)

The main result of this section is that in a bipartite graph G = (X,Y, E) there exists a matching with \X\- a(G) edges. In other words we show that

218

CONNECTIVITY AND MATCHING

the number of edges in a maximum matching of G is equal to \X\ - a(G). The result will be proved in two parts. We consider the case a(G) = 0 first, and the case cr(G) > 0 later. (Note that a(G) a 0, since for the empty set 0 , σ ( 0 ) = 0.) Note that the case a(G) = 0 occurs if for every subset S of X

|S|s|r(S)|, and the case a ( G ) > 0 occurs when for some subset 5 of AT |5|>|Γ(5)|. The following result is due to Hall [8.16]. Our proof is based on Halmos and Vaughan [8.17]. Theorem 8.13 (Hall). In a bipartite graph G = (Χ, Υ, E) there exists a complete matching of X into Y if and only if, for any 5 C X, |5|<|Γ(5)|. Proof Necessity From (8.14) it is clear that complete matching of X into Y requires that a(G) = 0 or, in other words, |5|<|Γ(5)|,

SCX.

Sufficiency Proof is by induction on \X\, the number of vertices in X. If |Al = 1, then obviously there is a complete matching of X into Y since the only vertex in X should be adjacent to at least one vertex in Y. Assume as the induction hypothesis that the sufficiency is true for any bipartite graph with \X\<m-\. Consider then a bipartite graph G = (Χ, Υ, E) with \X\ = m. Let for every subset S of X, \S\ < |Γ(5)|. We now show that there is a complete matching of X into Y by examining the following cases. Case 1

For every nonempty proper subset S of X, \S\ < |Γ(5)|.

Select any edge (x , y ), x EX and y Ε Y. Let G' be the graph obtained from G by removing x and y and the edges incident on them. Since the vertices of every subset S of X - {x } are adjacent to more than | 5 | vertices in y, they are adjacent to |S| or more vertices in Y - { y } . Therefore by the induction hypothesis, there exists a complete matching of X — {x } into Υ~{)>ο}· This matching together with the edge (x , y ) is a complete matching of X into Y. 0

0

0

0

0

0

Q

0

0

0

0

219

MATCHINGS IN BIPARTITE GRAPHS

Case 2

There is a nonempty property subset 5 of X such that \S \ = |Γ(5„)|· 0

0

Let G' be the subgraph of G containing the vertices in the sets S and Γ(5 ) and the edges connecting these vertices. Let G" be the subgraph of G containing the vertices in the sets X - S and Y - T(S ) and the edges connecting these vertices. We show that in G' there is a complete matching of 5 into Γ ( 5 ) , and in G" there is a complete matching of X — S into y - r ( 5 ) . These two matchings together will form a complete matching of X into Y. Consider first the graph G'. Let, for any subset S of S , Γ'(5) denote the set of vertices in Γ(5„) adjacent to those in S. It is then clear that 0

Q

0

0

0

0

0

0

0

r(S)

= r(S),

ScS . 0

Since, for every subset S of 5 , 0

| 5 | < | r ( 5 ) | = |r'(S)|,

it follows from the induction hypothesis that there is a complete matching of S into IYA). Consider next the graph G". Let, for any subset of X - S , Γ"(5) denote the set of vertices in Υ - Γ(5 ) adjacent to those in S. Now for S C X - S , 0

0

0

0

| 5 U S | = |5| + |S | 0

0

<|r(5US )| 0

= | P ( S ) | + |Γ(5 )| . 0

Since | 5 | = |Γ(5 )|, we get from the above 0

0

|5|<|Γ"(5)|,

SCX-S . 0

Thus again by the induction hypothesis there is a complete matching of X-S into Υ - Γ ( 5 ) . This completes the proof of sufficiency. • 0

0

As we observed before, Theorem 8.13 provides an answer to the marriage problem, and it is stated next. Theorem 8.14 (Hall). A necessary and sufficient condition for a solution of the marriage problem is that every set of k boys collectively have at least k girlfriends, 1 =s k s m, where m is the number of boys. • Theorem 8.13 is simply a translation into graph-theoretical terminology of Theorem 8.14.

220

CONNECTIVITY AND MATCHING

Next we prove that when a(G) > 0, the number of edges in a maximum matching is equal to \X\ - cr(G). To do so, we need the following two lemmas. Lemma 8.1. Let S, and S be any two subsets of X. Then 2

o-(S, U S ) + a(S Π S ) > σ(5,) + 2

l

a(S ).

2

2

Proof. It is a simple exercise to verify that |5, U S \ + |S, Π 5 | = | 5 , | + \S \. 2

2

(8.15)

2

Since \r(S US )\ l

\r(S )UT(S )\

=

2

l

2

and

ins.ns^lsinsjnr^)!, we get |Γ(5, U 5 ) | + |r(S, Π S )| ^ |Γ(5,) U Γ(5 )| + |Γ(5,) Π Γ ( 5 ) | 2

2

2

2

= |Γ(5,)| + | Γ ( 5 ) | .

(8.16)

2

Now subtracting (8.16) from (8.15) we get σ(5, U S ) + σ(5, Π S ) > | 5 , | + | 5 | - |Γ(5,)| - |Γ(5 )| 2

2

2

2

= σ(5,) + σ ( 5 ) . 2

•

Lemma 8.2. Let 5, and S be two subsets of X such that σ(5,) = σ ( 5 ) = σ ( ϋ ) . Then 2

2

σ(5, U 5 ) = σ(5, Π 5 ) = tf(G). 2

2

Proof. From Lemma 8.1, σ(5, U 5 ) + σ(5, Π 5 ) ^ σ ( ί , ) + σ(Ξ ) = 2a(G). 2

2

2

Since neither
2

<7(5, U 5 ) = σ(5, Π 5 ) =
2

•

Theorem 8.15. In a bipartite graph G = (Χ, Υ, E) with a ( G ) > 0 , the number of edges in maximum matching is equal to - o-(G).

MATCHINGS IN BIPARTITE GRAPHS

Proof. Let 5 , , S ,...,

S be all the subsets of X with their deficiencies equal

2

to o - ( G ) .

221

k

Let s

0

= 5,

ηs η ··· η s 2

k

.

By Lemma 8 . 2 , a ( G ) > 0 .

a(S ) = 0

Thus S is nonempty. We may now note that every subset of X having its deficiency equal to a(G) contains S . Consider any vertex x in S . Let G ' be the graph that is obtained by removing from G the vertex x and all the edges incident on it. It is clear that CT-(G') < a(G), since no subset of the set X - {x } contains S . We now show that e r ( G ' ) = a(G) - 1. Consider the set S^ = S - {x }. We have 0

Q

0

0

0

0

0

0

0

a(S')

=

\Si\-\r(S' )\ 0

= |5 |-1-|Γ(5ί)|. 0

Since a(S' )< 0

CT(G),

we obtain | 5 | - ι - | γ ( 5 ; ) | < | 5 | - | γ ( 5 ) | , 0

0

0

that is, |Γ(5;)|>|Γ(5 )|.

(8.17)

0

On the other hand, since S Q is a subset of 5 , 0

|Γ(5ί)|^|Γ(5 )|. 0

Combining

(8.17)

and

(8.18)

we get

(8.18),

TO)|

= |r(5 )|. 0

Therefore σ(5;) = | 5 | - 1 - | Γ ( 5 ) | = σ ( Ο ) - 1 . 0

0

Since t r ( G ' ) < a(G), we can conclude that CT(G') = a(G) - 1 . If we repeat this argument until an appropriate set of e r ( G ) vertices is removed from X along with the edges incident on these vertices, we get a subgraph of G with deficiency equal to zero. By Theorem 8 . 1 3 , there is a complete matching in this subgraph. This will be a matching for G contain-

222

CONNECTIVITY AND MATCHING

ing X— a(G) edges. It is clear from (8.14) that such a matching will be a maximum matching of G. • We can combine Theorems 8.13 and 8.15 into one, and because of its importance, we present it here. See Konig [8.18] and Ore [8.19]. Theorem 8.16 (Konig). The number of edges in a maximum matching of a bipartite graph G = (X, Y,E) is equal to |ΑΊ - σ(β), where a(G) is the deficiency of G. • Corollary 8.16.1. In a nonempty bipartite graph G = (Χ, Υ, E) there is a complete matching of X into Y if min {d(x)\ srmax {d(y)} . xex

yeY

Proof. Let min {d(x)} = d, and max{d(y)} = d . 2

yeY

Consider any subset A of X. Let E be the set of edges incident on the vertices in A, and E the set of edges incident on the vertices in Γ(Α). Then we have x

2

| £ , 1 > μ κ

and \E \x\T(A)\d . 2

2

Since £ , C E , we have 2

\Y{A)\d >\E \>\E \>\A\d 2

2

x

.

x

So

|r(i4)|aH,

ACX.

Thus by Hall's theorem there is a complete matching of X into Y. Next we consider two results that are related to Hall's theorem.

•

MATCHINGS IN BIPARTITE GRAPHS

223

Our first result is from transversal theory. Let Μ be a nonempty finite set and S = {£,, S , . . . , S } be a family of (not necessarily distinct) nonempty subsets of M. Then a transversal (or system of distinct representatives) of S is a set of r distinct elements of M, one from each set S For example, if Μ = {1, 2, 3, 4, 5, 6} and S, = {1, 3, 4}, S = {1, 3, 4}, 5 = {1, 2, 5}, and 5 = {5, 6}, then {1, 3, 2, 6} is a transversal of the family 5 = {5,, S , S . 5 } . On the other hand, if S, = S = {1, 3}, 5 = {3, 4), and 5 = {1, 4}, then there is no transversal for the family {5,, S , S , 5 } . The following question now arises: What are the necessary and sufficient conditions for a family of subsets of a set to have a transversal? Suppose we construct a bipartite graph G = (Χ, Υ, E) such that: 2

r

r

2

3

4

2

3

4

3

2

4

3

2

4

1. The vertex x, of X corresponds to the set 5, in S. 2. The vertex y, of Y corresponds to element i of M. 3. Edge (*,, y ) Ε £ if and only if / £ 5,. t

We can now see that the question becomes equivalent to finding a complete matching of X into Y in the bipartite graph G constructed as above. Thus we get the following theorem, which is simply a restatement of Hall's theorem in the language of transversal theory. Theorem 8.17. Let Μ be a nonempty set and 5 = {5,, 5 , . . . , 5 } a family of subsets of M. Then 5 has a transversal if and only if the union of any k, 1 < k s r, of the subsets 5, contain at least k elements of M. • 2

r

A purely combinatorial proof of Theorem 8.17 without using concepts from graph theory was given by Rado. The proof is very elegant and may be found in Rado [8.20]. The next result relates to matrices whose elements are 0's and l's. Such matrices are called (0,1) matrices. In the following, a line of a matrix refers to a row or a column of the matrix. Theorem 8.18 (Konig and Egervary). The minimum number of lines that contain all the l's in a (0,1) matrix is equal to the maximum number of l's no two of which are in the same line of M. • Given a (0,1) matrix Μ of order mx n. Suppose we construct a bipartite graph G = (Χ, Υ, E) such that: 1. The vertices x , x ,... ,x of X correspond to the m rows of M. 2. The vertices y,, y ,..., y„ of Y correspond to the η columns of M. 3. The edge (x„ yf) ε £ if the (i, j) entry of Μ is equal to 1. t

2

2

m

224

CONNECTIVITY AND MATCHING

If we now consider a vertex as covering all the edges incident on the vertex, then the Konig-Egervary theorem can be restated as follows: In a bipartite graph the minimum number of vertices that cover all the edges is equal to the number of edges in any maximum matching of the graph. In Chapter 9 (Theorem 9.2) we prove this form of the Konig-Egervary theorem.

8.7

MATCHINGS IN GENERAL GRAPHS

In this section we establish some results relating to matchings in a general graph. Consider a graph G = (V, E) and a matching Μ in G. An alternating chain in G is a trail whose edges are alternately in Μ and Ε — M. For example, the sequence of edges e,, e , e , e , e , e is an alternating chain relative to the matching Μ = {e , e , e } in the graph of Fig. 8.9. The edges in the alternating chain that belong to Μ are called dark edges and those that belong to Ε - Μ are called light edges. Thus e,, e , e are light edges, whereas e , e , e are dark edges in the alternating chain considered above. 2

2

A

3

4

7

6

6

3

2

4

7

6

Theorem 8.19. Let M, and M be two matchings in a simple graph G = (V, E). Let G' = ( V , E') be the induced subgraph of G on the edge set 2

M, ®M = (Af, - M ) U (M - M,) . 2

2

2

Then each component of G' is of one of the following types: 1. A circuit of even length whose edges are alternately in Μ and M . 2. A path whose edges are alternately in M, and M and whose end vertices are unsaturated in one of the two matchings. λ

2

Figure 8.9

2

MATCHINGS IN GENERAL GRAPHS

225

Proof. Consider any vertex ν Ε V. Case 1

ν Ε V(A/, - A/ ) and v0V(M - A/,), where V(M, - M ) denotes the set of vertices of the edges in A/, - A/ . 2

2

}

;

In this case υ is the end vertex of an edge in A/, - A/ . Since M is a matching, no other edge of M - M is incident on v. Further, no edge of M - Af, is incident on ν because υ0V(M - A/,). Thus in this case the degree of ν in G' is equal to 1. 2

x

x

2

2

2

Case 1

υ Ε V(Ai, - Αί ) and ι; Ε V(M - M,). 2

2

In this case a unique edge of Af, - A/ is incident on υ and a unique edge of Af - A/, is incident on v. Thus the degree of ν is equal to 2. Since the two cases considered are exhaustive, it follows that the maximum degree in G' is 2. Therefore the connected components will be of one of the two types described in the theorem. • 2

2

For example, consider the two matchings Af, = {e , e , e ) and M = {e,, e , e,,} of the graph G of Fig. 8.9. Then 5

7

9

2

10

Af, Φ Μ = {e,, e , e , e , e , e ) , 2

5

7

9

10

n

and the graph G' will be as in Fig. 8.10. It may be seen that the components of G' are of the two types described in Theorem 8.19. In the following theorem we establish Berge's [8.21] characterization of a maximum matching in terms of an alternating chain. Theorem 8.20 (Berge). A matching Af is maximum if and only if there exists no alternating chain between any two unsaturated vertices. Proof Necessity Suppose there is an alternating chain Ρ between two unsaturated vertices. Then replacing the dark edges in the chain by the light edges

v

7

Figure 8.10

226

CONNECTIVITY AND MATCHING

will give a matching M, with |M,| = |M| + 1. Note that Μ, = (Μ - P) U (Ρ - M). For example, in the graph of Fig. 8.9, consider the matching Μ = {e , e } . There is an alternating chain e , e , e , e , and e between the unsaturated vertices v and v . If we replace in Μ the dark edges e and e by the light edges e , e , and e , we get the matching {e , e , e } , which has one more edge than M. 2

4

s

3

5

4

n

12

2

7

u

A

12

5

u

2

1 2

Sufficiency Suppose Μ satisfies the conditions of the theorem. Let M' be a maximum matching. Then it follows from the necessity part of the theorem that Μ ' also satisfies the conditions of the theorem, namely, there is no alternating chain between any vertices that are not saturated in Μ'. We now show that |M| = \M'\, thereby proving the sufficiency. Since Μ = (Μ Π Μ') U (Μ - Μ') and Μ' = (Μ Π Μ') U (Μ' - Μ ) , it is clear that |M| = | M ' | if and only if |M - M ' | = \M' - M\. Let G' be the graph on the edge set Μ Θ Μ' = (Μ - Μ') U (Μ' - Λί). Consider first a circuit in G'. By Theorem 8.19 such a circuit is of even length, and the edges in this circuit are alternately in Μ - Μ' and M' - Μ. Therefore each circuit in G' has the same number of edges from both M-M' and M' - M. Consider next a component of G' that is a path. Again, by Theorem 8.19, the edges in this path are alternately in M-M' and M' - M. Further the end vertices of this path are unsaturated in Μ or Μ'. Suppose the path is of odd length, then the end vertices of the path will be both unsaturated in the same matching. This would mean that with respect to one of these two matchings there is an alternating chain between two unsaturated vertices. But this is a contradiction because both Μ and Μ' satisfy the condition of the theorem. So each component of G' that is a path has an even number of edges, and hence it has the same number of edges from M-M' and M'-M. Thus each component of G' has an equal number of edges from M-M' and M'-M. Since the edges of G' constitute the set (Μ - Μ') U (Μ' - Μ), we get \M-M'\ and so, |M| = |M'|.

= \M'-M\

,

•

Given a matching Μ in a graph G, let Ρ be an alternating chain between any two vertices that are not saturated in M. Then as we have seen before, Μ θ Ρ is a matching with one more edge than M. For this reason the path Ρ is called an augmenting path relative to M.

MATCHINGS IN GENERAL GRAPHS

227

Next we prove two interesting results on bipartite graphs using the theory of alternating chains. Consider a bipartite graph G = (Χ, Υ, E) with maximum degree Δ. Let X denote the set of all the vertices in X of degree Δ. If G' is the bipartite graph (X , Γ(Α",), Ε'), where E' is the set of edges connecting X and Γ(λ",), then it can be seen from Corollary 8.16.1 that there exists in C a complete matching of X into Γ(Χ ). Such a matching clearly saturates all the vertices in X . Thus there exists a matching in G that saturates all the vertices in X of degree Δ. Similarly, there exists a matching in G that saturates all the vertices in Y of degree Δ. The question that now arises is whether in a bipartite graph there exists a matching that saturates all the maximum degree vertices in both X and Y. To answer this question, we need the following result due to Mendelsohn and Dulmage [8.22]. x

x

x

x

Χ

x

Theorem 8.21 (Mendelsohn and Dulmage). Let G = (Χ, Υ, E) be a bipartite graph, and let Λ/, be a matching that matches X,CX with Y, C Y (/ = 1,2). Then there exists a matching M' C M U M that saturates X and Y . x

2

x

2

Proof. Consider the bipartite graph G' = (X L)X , Y U Y, M,UM ). Each vertex of this graph has degree 1 or 2; hence each component of this graph is either a path or a circuit whose edges are alternatively in M, and M . (See proof of Theorem 8.19.) Each vertex yE.Y -Y has degree 1 in G'. So it is in a connected component that is a path P from y to a vertex χ E X - X or to a vertex ζ G y, - Y . In the former case the last edge of P is in M and so Μ, Θ P matches X U {x) with Y U {y}. In the latter case the last edge of P is in M and so Μ, Θ P matches X with ( y , - z) U {y}. In either case Μ, Θ P saturates Υ Γ\Υ . Thus M ΦP saturates y E.Y - Y , and all the vertices in X and Y C\Y . If we let l

2

x

2

2

2

2

x

y

2

2

y

x

x

2

y

x

x

y

y

Χ

x

x

2

X

x

y

y

2

x

2

P=

U

yeY -Y,

Py,

2

then we can see that M Θ Ρ is a matching that saturates A", and Y . This is a required matching M' C M U M . • x

2

X

2

Theorem 8.22. In a bipartite graph there exists a matching that saturates all the maximum degree vertices. Proof. Consider a bipartite graph G = (Χ, Υ, E). Let X' CX and Y'CY contain all the vertices of maximum degree in G. As we have seen before, there exists a matching M, that saturates all the vertices in X' and a matching M that saturates all the vertices in Y'. So by Theorem 8.21, there exists a matching M' C M U M that saturates all the vertices in X' and Y'. 2

x

2

228

CONNECTIVITY AND MATCHING

This is a required matching saturating all the maximum degree vertices in G. U Corollary 8.22.1. The set of edges of a bipartite graph with maximum degree Δ can be partitioned into Δ matchings. Proof. Consider a bipartite graph G = (X, Υ, E) with maximum degree Δ. By Theorem 8.22 there exists a matching A/, that saturates all the vertices of degree Δ. Then the bipartite graph G' = (Χ, Υ, Ε - A/,) has maximum degree Δ - 1. This graph contains a matching M that saturates every vertex of degree Δ - 1. By repeating this process we can construct a sequence of disjoint matchings A/,, A / , . . . , Αί that form a partition of E. • 2

2

Δ

An application of Theorems 8.21 and 8.22 and Corollary 8.22.1 is considered in Section 11.5. Now we return to our discussion of matchings in general graphs. A matching saturating all the vertices of a graph G is called a perfect matching of G. We conclude this section with a theorem due to Tutte [8.23] on the conditions for the existence of a perfect matching in a graph. Our proof here is due to Anderson [8.24]. A component of a graph is odd if it has an odd number of vertices; otherwise it is even. If S is a subset of vertices of a graph G, then we shall denote by p (S) the number of odd components of G - S. 0

Theorem 8.23 (Tutte). A graph G = (V, E) has a perfect matching if and only if p (5)<|5| 0

forallSCK.

(8.19)

Proof Necessity Suppose that G has a perfect matching M. For some S C V, let G,, G , . . . ,G be the odd components of G - S. Since each G, is odd, some vertex u, of G, must be matched under A/ with some vertex v of 5. Thus S has at least k vertices, and hence 2

k

)

p (S)<|5|. 0

Sufficiency First note that if a graph G satisfies (8.19), then choosing 5 = 0 we get p (0) ^ 0. So in such a case there will be no odd components in G. In other words G will have an even number of vertices. Proof of sufficiency is by induction on /, where 2 / = |V|. We use Hall's theorem (Theorem 8.13) and the simple fact that o

p ( S ) = |S|(mod2). 0

(8.20)

229

MATCHINGS IN GENERAL GRAPHS

The case / = 1 is trivial. Assume that the result is true for all graphs with less than 21 vertices. Consider then a graph G with 21 vertices and that satisfies (8.19). Now we have two cases. Case 1

Suppose that p (S) <\S\ for all 5 , 2 < | 5 | < 21. 0

Consider any edge e = (a, b) in G. Let A = [a, b) and G = G — A. For any subset Τ of vertices of G , let p' (T) denote the number of odd components in G - T. Thenp' (T) s |T\, for if p' (T) >\T\, then we would get p (T U A) = p' (T) > I T\ = IΤ U A\ - 2, contradicting (8.19). Therefore, by induction, G , and hence G, has a perfect matching A

A

A

0

0

0

0

0

A

Case 2

Let there be a set S such that p (S) = \S\ s 2. Assume that 5 is a maximal such set. 0

First observe that there are no even components in G - S. For if there were any, we could remove a vertex ν from one and add it to 5. This would necessarily give us at least one more odd component. So p (S U v) s p (S) + 1 = \S\ + 1. Condition (8.19) requires that p (S U v) <\S\ + 1. So p (S U v) = \S\ + 1. But this would contradict the maximality of 5 . Hence there are no even components in G - S. 0

0

0

0

Let | 5 | = s, and let G , , G , . . . , G be the s odd components of G - 5 . 2

s

We now show that we can take a vertex from each one of these odd components and match it with a vertex in S. If this is not possible, then by Hall's theorem, there are k odd components that are connected in G to only h < k vertices of 5. But if Τ denotes such a set of h vertices, we would then have p (T)>k>h

= \T\

0

contradicting (8.19). Thus we can take a vertex u, from each G,, 1 < / < J , and match it with a vertex in S. Now each G\ = G, — v, has an even number of vertices. The proof will be completed if we show that each G[ has a perfect matching. If G\ contains a set R of vertices such that p" (R)> \R\, where p" (R) is the number of odd components in G ' - R, then by (8.20), p" (R) a \R\ + 2, so that 0

0

0

p (RUSU{v,})

= p" (R) +

0

0

p (S)-i 0

>|/?| + |S| + 1 =

\RUSU{v,}\-

But condition (8.19) requires that p (RUSU{ 0

u , } ) s | / l u 5 U {«,}!.

(8.21)

230

CONNECTIVITY AND MATCHING

So p (*USU{u }) = |*USU{u,}| 0

(

contradicting the maximality of S. Thus by induction, G\ has a perfect matching. • Lovasz [8.25] gives an alternative proof of Theorem 8.23 using Theorem 8.20.

8.8

FURTHER READING

Berge [8.26], Harary [8.8], Chartrand and Lesniak [8.27], and Bollobas [8.28] are general references recommended for further reading on connectivity and matching. Berge also presents a detailed discussion on the realizability of degree sequences for the cases of both directed and undirected graphs. Harary gives a historical account of Menger's theorem and its several variations. A communication network can be modeled by a graph. The concept of vulnerability arises in the study of such models. By vulnerability we mean the susceptibility of the network to attack. A network may be considered "destroyed" if after removal of some vertices or edges the resulting graph is not connected. Thus vulnerability of a network is related to the vertex and edge connectivities of the network. For example, a network TV, may be considered more vulnerable than a network N if the vertex connectivity of /V, is less than that of N . Boesch [8.29] has several papers on the design of graphs having specified connectivity and reliability properties. Some of the early papers that deal with this topic include Hakimi [8.30], Boesch and Thomas [8.31], and Amin and Hakimi [8.32]. Frank and Frisch [8.33] is recommended for further reading on this topic. For some of the more recent works, see Krishnamoorthy, Thulasiraman, and Swamy [8.34] and [8.35], Bermond, Homobono, and Peyrat [8.36], Opatrny, Srinivasan, and Alagar [8.37], and Chung and Garey [8.38]. For a detailed review of combinatorial and algorithmic aspects of reliability, see Colbourn [8.39]. The problem of testing the connectivity of a graph and the problem of finding a maximum matching are closely related to that of finding a maximum flow in a transport network. This will be discussed in Chapter 12. The papers by Mirsky and Perfect [8.40], Brualdi [8.41], and the book by Mirsky [8.42] are highly recommended for further reading on matchings and transversal theory. Applications involving the theory of matchings (in particular, the optimal assignment problem and a time-tabling problem) and related algorithms are discussed in Chapter 11. 2

2

EXERCISES

8.9

8.1

EXERCISES

Let ί ί , < ( 1 < · · · < ^ denote the degrees of a simple graph G. Show that G is &-connected, k
(a) d(v ) > r + k - 1, r <

; and

r

(b) d ( u - * , ) ^ n

8.2

.

+

Show that a simple η-vertex graph G is Α-connected if 6

8.3

231

(

G

)

a

u ± ! ^ .

Let G = (Κ, E) be a simple ^-connected graph. Let Β = [b , b ,..., b } be a set of vertices with \B\ - k. If a Ε V- Β, show that there exist k vertex-disjoint a-b paths from a to B. x

2

k

t

8.4

Let G be a simple A:-connected graph with k a: 2. Show that there exists a circuit that passes through an arbitrary set of two edges e, and e and k - 2 vertices. 2

8.5

Let G be a /c-connected graph with k > 2. Then prove that every A vertices of G lie on a circuit of G.

8.6

Prove that a simple graph G with n s 2 vertices is A-edge-connected if and only if for every two distinct vertices u and ν of G, there exist at least A: edge-disjoint u-v paths in G. (Note: This is the edge analog of Theorem 8.10.)

8.7

Show that K(H )

8.8

Find a 5-connected graph with 7 vertices and 18 edges.

8.9

Show that the edges of a simple graph G can be oriented to form a strongly connected graph if and only if G is 2-edge-connected.

8.10

The associated directed graph D(G) of an undirected graph G is the graph obtained by replacing each edge e of G by two oppositely oriented edges having the same end vertices as e. Show that: (a) There is a one-to-one correspondence between paths in G and directed paths in D(G). (b) D(G) is A:-edge-connected if and only if G is fc-edge-connected. (Note: A directed graph G is k-edge-connected if and only if at least k edges have to be removed to destroy all directed s-t paths for any two vertices s and t of G.)

8.11

Find a simple graph having the degree sequence (5, 4, 4, 4, 3, 3, 3, 3, 3) and with the largest possible edge connectivity.

kn

= K'(H ) kn

= k.

232

CONNECTIVITY AND MATCHING

8.12 Show that a sequence (d,, d , . . . , d„) of nonnegative integers is the degree sequence of a tree if and only if 2

d, > 1 for all i, and π

Σ 1=

8.13

d, =

2(n - 1 ) .

1

Show that a sequence (d,, d , . . . , d„) of nonnegative integers is the degree sequence of a graph if and only if Σ" d, is even (Hakimi [8.4]). 2

=1

8.14

Show that a sequence (d,, d , . . . , d„) of nonnegative integers is the degree sequence of a simple graph if and only if 2

(a) Σ d, is even; and 1=1

*

π

(b) Σ d,,== k(k - 1) + Σ i=l

min{fc, d,}

forl
i=*+l

(Erdos and Gallai [8.7]). 8.15

Show that a sequence (d,, d , . . . , d„) (realizable by a simple graph) is the degree sequence of a simple Λ-connected graph if and only if 2

(a) d, > k

for 1 n-k,

where m is the number of edges of G. (Wang and Kleitman [8.6]). 8.16

Prove or disprove: For every matching Μ there exists a maximum matching M' such that Μ C Μ'.

8.17

Prove that every 3-regular graph without bridges has a perfect matching.

8.18

Prove that if G is a Ar-regular bipartite graph with k > 0, then G has a perfect matching.

8.19

A nonnegative real square matrix Ρ is doubly stochastic if the sum of the entries in each row of Ρ is 1 and the sum of the entries in each column of Ρ is 1. A permutation matrix is a (0,1) matrix that has exactly one 1 in each row and each column. Show that a doubly stochastic matrix Ρ can be expressed as

EXERCISES

P = c,P, + c P + ·•• + c P 2

2

k

k

233

,

where each P, is a permutation matrix, each c, is a nonnegative real number, and Ef c, = 1. =1

8.20

Let G be a bipartite graph with bipartition (X, Y). If G has a complete matching of X into Y, show that there exists an x G X, such that for each y G Γ(χ ) at least one maximum matching uses the edge 0

0

(*<» y)8.21

A k-factor of a graph G is a Ac-regular spanning subgraph of G. Clearly a 1-factor is a perfect matching. G is k-factorable if G is the union of some edge-disjoint Α-factors. Show that K and K are 1-factorable. nn

8.22

2n

(a) Show that K can be expressed as the union of η connected 2-factors ( n ^ l ) . 2n+1

(Note: A connected 2-factor is a Hamilton circuit.) (b) Show that K is the union of a 1-factor and (η -1) 2-factors.

connected

2n

8.23

Show that a connected graph G is 2-factorable if and only if it is regular with even degree.

8.24

Let Μ and Ν be two edge-disjoint matchings of a graph G with | Λ ί | > | Ν | . Show that there are disjoint matchings M' and N' with | M ' | = | M | - 1 and |ΛΤ| = |JV| + 1, and M'l)N' = MlSN.

8.25

If G = (V, E) is a bipartite graph and A a Δ, then show that there exist A disjoint matchings M , M ,..., M of G such that x

2

k

£=M UM U"-UM 1

2

t

and for 1 < / < A,

where m is the number of edges of G. 8.26

Show that a tree Τ has a perfect matching if and only if p (v) = 1 for all vertices υ in Γ, where p (v) is the number of odd components of T-v. 0

0

8.27

Derive Hall's theorem from Tutte's theorem.

8.28

Derive Hall's theorem from Menger's theorem (see Wilson [8.43], Theorem 28d).

234

8.10

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22

CONNECTIVITY AND MATCHING

REFERENCES

F. Harary, "The Maximum Connectivity of a Graph." Proc. Nat. Acad. Sci., U.S., Vol. 48, 1142-1146 (1962). J. A. Bondy, "Properties of Graphs with Constraints on Degrees," Studia Sci. Math. Hung., Vol. 4, 473-475 (1969). G. Chartrand, "A Graph Theoretic Approach to a Communication Problem," SIA Μ J. Appl. Math., Vol. 14, 778-781 (1966). S. L. Hakimi, "On the Realizability of a Set of Integers as Degrees of the Vertices of a Graph," SIAMJ. Appl. Math., Vol. 10, 496-506 (1962). V. Havel, "A Remark on the Existence of Finite Graphs" (in Hungarian), Casopois Pest. Math., Vol. 80, 477-480 (1955). D. L. Wang and D. J. Kleitman, "On the Existence of η-Connected Graphs with Prescribed Degrees («2=2)," Networks, Vol. 3, 225-239 (1973). P. Erdos and T. Gallai, "Graphs with Prescribed Degrees of Vertices" (in Hungarian), Mat. Lapok, Vol. 11, 264-274 (1960). F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969. J. Edmonds, "Existence of λ-Edge-Connected Ordinary Graphs with Prescribed Degrees," /. Res. Nat. Bur. Stand. B., Vol. 68, 73-74 (1964). D. L. Wang and D. J. Kleitman, "A Note on κ-Edge Connectivity," SIAM J. Appl. Math., Vol. 26, 313-314 (1974). K. Menger, "Zur allgemeinen Kurventheorie," Fund. Math., Vol. 10, 96-115 (1927). H. Whitney, "Congruent Graphs and the Connectivity of Graphs," Am. J. Math., Vol. 54, 150-168 (1932). W. T. Tutte, "A Theory of 3-Connected Graphs," Indag. Math., Vol. 23, 441-455 (1961). L. R. Ford and D. R. Fulkerson, "Maximal Flow Through a Network," Canad. J. Math., Vol. 8, 399-404 (1956). P. Elias, A. Feinstein, and C. E. Shannon, "A Note on the Maximum Flow Through a Network," IRE Trans. Inform. Theory, IT-2, 117-119 (1956). P. Hall, "On Representatives of Subsets," J. London Math. Soc, Vol. 10, 26-30 (1935). P. R. Halmos and Η. E. Vaughan, "The Marriage Problem," Am. J. Math., Vol. 72, 214-215 (1950). D. Konig, "Graphs and Matrices" (in Hungarian), Mat. Fiz. Lapok, Vol. 38, 116-119 (1931). O. Ore, "Graphs and Matching Theorems," Duke Math. J., Vol. 22, 625-639 (1955). R. Rado, "Note on the Transflnite Case of Hall's Theorem on Representatives," J. London Math. Soc, Vol. 42, 321-324 (1967). C. Berge, "Two Theorems in Graph Theory," Proc. Nat. Acad. Sci. U.S., Vol. 43, 842-844 (1957). N. S. Mendelsohn and A. L. Dulmage, "Some Generalizations of the Problem of Distinct Representatives," Canad. J. Math., Vol. 10, 230-241 (1958).

REFERENCES

235

8.23 W. Τ. Tutte, "The Factorization of Linear Graphs," J. London Math. Soc, Vol. 22, 107-111 (1947). 8.24 I. Anderson, "Perfect Matchings of a Graph," J. Combinatorial Theory B, Vol. 10, 183-186 (1971). 8.25 L. Lovasz, "Three Short Proofs in Graph Theory," /. Combinatorial Theory B, Vol. 19, 111-113 (1975). 8.26 C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. 8.27 G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Cole, Pacific Grove, Calif., 1986. 8.28 B. Bollobas, Graph Theory: An Introductory Course,'Springer-Verlag, New York, 1979. 8.29 F. T. Boesch, Ed., Large-Scale Networks: Theory and Design, IEEE Press, New York, 1976. 8.30 S. L. Hakimi, "An Algorithm for Construction of Least Vulnerable Communication Networks or the Graph with the Maximum Connectivity," IEEE Trans. Circuit Theory, Vol. CT-16, 229-230 (1969). 8.31 F. T. Boesch and R. E. Thomas, "On Graphs of Invulnerable Communication Nets," IEEE Trans. Circuit Theory, Vol. CT-17, 183-192 (1970). 8.32 A. T. Amin and S. L. Hakimi, "Graphs with Given Connectivity and Independence Number or Networks with Given Measures of Vulnerability and Survivability," IEEE Trans. Circuit Theory, Vol. CT-20, 2-10 (1973). 8.33 H. Frank and I. T. Frisch, Communication, Transmission, and Transportation Networks, Addison-Wesley, Reading, Mass. 1971. 8.34 V. Krishnamoorthy, K. Thulasiraman, and Μ. N. S. Swamy, "MinimumOrder Graphs with Prescribed Diameter, Connectivity and Regularity," Networks, Vol. 19, 25-46 (1989). 8.35 V. Krishnamoorthy, K. Thulasiraman, and Μ. N. S. Swamy, "Incremental Distance and Diameter Sequences of a Graph: New Measures of Network Performance," IEEE Trans. Computer, Vol. 39, 230-237 (1990). 8.36 J. C. Bermond, N. Homobono, and C. Peyrat, "Large Fault-Tolerant Interconnection Networks," Graphs and Combinatorics, Vol. 5, 107-123 (1989). 8.37 J. Opatrny, N. Srinivasan, and V. S. Alagar, "Highly Fault-Tolerant Communication Network Models," IEEE Trans. Circuits and Systems, Vol. 30, 23-30 (1989). 8.38 F. R. K. Chung and M. R. Garey, "Diameter Bounds for Altered Graphs," J. Graph Theory, Vol. 8, 511-534 (1984). 8.39 C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, Oxford, 1987. 8.40 L. Mirsky and H. Perfect, "Systems of Representatives," J. Math. Anal. Applic, Vol. 15, 520-568 (1966). 8.41 R. A. Brualdi, "Transversal Theory and Graphs," in Studies in Graph Theory, Part II, MAA Press, 1975, Washington, D.C., pp. 23-88. 8.42 L. Mirsky, Transversal Theory, Academic Press, New York, 1971. 8.43 R. J. Wilson, Introduction to Graph Theory, Oliver and Boyd, Edinburgh, 1972.

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 9

COVERING AND COLORING

In the previous chapters we defined several useful parameters associated with a graph, namely, rank, nullity, connectivity, matching number, and so on. As we mentioned earlier, rank and nullity arise in the study of electrical networks. Connectivity arises in the study of communication nets. In this chapter we study several other useful parameters of a graph—vertex independence number, vertex and edge covering numbers, chromatic index, and chromatic number. We begin with a discussion of the vertex independence number and the vertex and edge covering numbers. We relate these numbers to the matching number defined in the previous chapter and develop equivalent formulations of Hall's theorem. Then we study the chromatic index and the chromatic number, which relate to the vertex and edge colorability properties of a graph. The parameters to be discussed in this chapter arise in the study of several practical problems such as timetable scheduling and communication net design. 9.1

INDEPENDENT SETS AND VERTEX COVERS

Consider a graph G = (V, E). A subset S of V is an independent set of G if no two vertices of 5 are adjacent in G. An independent set is also called a stable set. An independent set S of G is maximum if G has no independent set 5 ' with |S"| > The number of vertices in a maximum independent set of G is called the independence number (stability number) of G and is denoted by «o(G). 236

INDEPENDENT SETS AND VERTEX COVERS

237

b

Figure 9.1

e

For example, in the graph of Fig. 9.1, the sets {b, d), {b, /}, {a, c), and {b, d, / } are independent sets. Of these, [b, d) and {b, / } are not maximal independent sets; {a, c) is maximal but it is not maximum; {b, d, / } is maximum. A subset Κ of V is a vertex cover of G if every edge of G has at least one end vertex in K. If we consider a vertex as covering all the edges incident on it, then a vertex cover of G is a subset of V that covers all the edges of G. A vertex cover Κ of G is minimum if G has no vertex cover K' with \K'\ < \ K\. The number of vertices in a minimum vertex cover of G is called the vertex covering number of G and is denoted by β (ΰ). For example, in the graph of Fig. 9.1, the sets (a, c, e, /}, [a, c, d, e), {b,d,e,f}, and {a, c, e) are vertex covers. Of these, {a,c,e,f} and {a, c, d, e) are not minimal; {b, d, e, / } is minimal but not minimum; {a,c,e} is minimum. a (G) and fi (G) will be denoted simply as a and β , respectively, whenever the graph G under consideration is clear from the context. Recall that a , ( G ) is the number of edges in a maximum matching of G. Independent sets and vertex covers are closely related concepts, as we show in the following theorem. 0

Q

0

0

0

Theorem 9 . 1 . Consider a graph G = (V, E). A subset_S of V is an independent set of G if and only if the complement 5 of S in V (i.e., S = V- S) is a vertex cover of G. Proof. By definition 5 is an independent set of G if and only if no edge of G has both its end vertices in S. In other words S is anjndependent set if and only if every edge of G has at least one end vertex in 5, the complement of S in V. The theorem now follows from the definition of a vertex cover. • Corollary 9.1.1. For a simple η-vertex graph, a + β = η. Q

0

238

COVERING AND COLORING

Proof. Consider a maximum independent set 5* and a minimum vertex cover Κ * of a graph G = (V, E). Then \S*\ = «o

and

1*1 = A,. By Theorem 9.1, 5* = V-S* independent set. Therefore,

is a vertex cover and Έ* = V- K* is an

|5*| = | V - 5 * | = n - a > / 3 0

0

and \Κ*\ = \ν-Κ*\

=

η-β ^α . 0

0

Combining these inequalities, we get «ο

+ A) = η .

*

Consider any maximum matching M* and any minimum vertex cover K* in a graph G. Since at least \M*\ vertices are required to cover the edges of Λί*, the vertex cover K* must contain at least \M*\ vertices. Therefore, \M*\<\K*\.

(9.1)

In general, equality does not hold in (9.1). However, we show in the following theorem that \M*\ = \K*\ if G is a bipartite graph. Theorem 9.2. For a bipartite graph the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover, that is, «, = A,Proof. Let M* be a maximum matching and K* a minimum vertex cover in a bipartite graph G = (X, Y,E). Consider any subset A of X. Each edge e of G is incident on either a vertex in A or a vertex in X - A. Further, any edge incident on a vertex in A is also incident on a vertex in Γ(Α), the set of vertices adjacent to those in A. Thus the set (X - A) U Γ(Α) is a vertex cover of G, and hence \{Χ-Α)\

+

\Γ(Α)\*\Κ·\

INDEPENDENT SETS AND VERTEX COVERS

239

But from Hall's theorem (Theorem 8.13), |M*| = min {\X-A\

(9.2)

+ \T(A)\} *\K*\.

Now combining (9.1) and (9.2), we get \M*\

= \K*\.

m

Theorem 9.2 is due to Konig [9.1]. We established in Theorem 8.20 a characterization of a maximum matching using the concept of an alternating chain. Since an independent set is the vertex analog of a matching, a similar characterization may be expected to exist in the case of a maximum independent set too. This is indeed true, and in the following we develop such a characterization. We follow the treatment given in Berge [9.2]. Let us first define an alternating sequence, the vertex analog of an alternating chain. Let Y be an independent set of a graph G = (V, E). An alternating sequence relative to Y is a sequence
yx,x > 2

3 2-···} ,

of distinct vertices alternately belonging to X = Υ = V- Y and Y such that the following conditions are satisfied:

1.

JC,GX,y,EY.

2. y, is adjacent to at least one vertex in the set [x ,x ,... ,x,}. 3. JC, is not adjacent to any vertex in {*,, x ,..., JC,}. 4. is adjacent to at least one vertex in { y , y ,..., y,}. l

+ 1

2

2

t

2

An alternating sequence is maximal if no more vertices can be added to it without violating these conditions. For example, in the graph of Fig. 9.1, the sequence {/, a, b, c) is a maximal alternating sequence relative to the independent set {a, c}. Consider a tree T. Recall (Exercise 2.2) that Τ is bipartite, that is, its vertex V set can be partitioned into two subsets X and Y such that: 1. V = XL) Y. 2. χ η y = 0 . 3. X and Y are both independent sets in the tree. We shall refer to (X, Y) as a bipartition of the vertex set of T, and the vertices of X and Y as X- and Y-vertices, respectively.

240

COVERING AND COLORING

As a first step toward the development of a characterization of a maximum independent set in terms of an alternating sequence, we now show that if \X\ > | Y\ in a tree, then there is an alternating sequence relative to Y using all the vertices of Y. To do so we need the following lemma. Lemma 9.1. Let (X, Y) be a bipartition of the vertex set of a tree T. If \X\ = \Y\ + ρ, ρ > 0 , then X contains at least ρ + 1 pendant vertices of T. Proof. Proof is by induction on the number η of vertices of T. Clearly, the lemma is true for η = 2, 3, 4. Let the result be valid for all trees having fewer than η vertices, η > 5. Consider an η-vertex tree Τ with | A"| = | Y\ + p. Let υ be a pendant vertex of Τ and let T' = T-{v}

.

Note that 7" is a tree with η- I vertices. Suppose υ is an AT-vertex. If ρ = 0, then υ is the required Z-vertex of T. Otherwise, by the induction hypothesis, at least ρ pendant vertices of T' are A'-vertices. These ρ vertices are also pendant in Τ and hence they, along with υ, are the required A'-vertices for T. Proof is immediate if υ is a Y-vertex. • Lemma 9.2. Let (Χ, Y) be a bipartition of the vertex set of a tree T. 1. If |A"| = | y | or |A"| = | y | + l, then there is an alternating sequence i \> )>i> 2< y ' · • •} a t uses each vertex of Τ exactly once. 2. If |A"| > I Y| + 1, then there is a maximal alternating sequence of odd length {*,, y ,...,} that uses each vertex of Y exactly once. t n

x

x

2

x

Proof 1. The result is true for a tree with 2 vertices. Assume the result to be true for any tree with 2k vertices. Consider then a tree Γ with 2k + 1 vertices and |Al = | Y| + 1. Then, by the previous lemma, there exists in Τ a pendant vertex x E.X. Let 7" = Τ - {x , } . By the induction hypothesis, there is in T' an alternating sequence {*,, y , . . . , y ) that uses all the vertices of 7". Therefore the sequence {*,, y , . . . , y , x ) is a required sequence for T. (Note that this sequence is of odd length.) Next assume the result to be valid for any tree with 2k + 1 vertices. Consider a tree Τ with 2k+ 2 vertices and |A"| = |Y|. Then by the previous lemma, there exists in Γ a pendant vertex y E. Y. Let T' = Τ - {y }. Now, by the induction hypothesis, there is an alternating sequence {x , y , • • • ,x ) that uses all the vertices of T'. k

+ i

k

+

x

x

k

k+l

k+l

k+l

x

x

k+i

k

241

INDEPENDENT SETS AND VERTEX COVERS

Therefore the sequence {*,, y,, ... ,x , y } is a required sequence for T. 2. If | A | > | Y | + 1, remove from Γ as many pendant X-vertices as necessary until we get a tree Τ in which the number of A"-vertices is one more than the number of Y-vertices. By the previous lemma, this is always possible. Now, by the result of part 1 of the lemma, there exists a maximal alternating sequence of odd length that uses all the vertices of T' and hence all the Y-vertices of T. • k+1

k+l

For example, in the tree of Τ of Fig. 9.2, {x ,y , x, y, x, y , * , y, x> ys> * 6 > ^ 6 ' ^ 7 } ' alternating sequence that uses all the Y-vertices of T. We now state and prove a characterization of a maximum independent set in terms of an alternating sequence. l

l

2

2

3

3

4

4

s

s

Theorem 9.3. An independent set Y in a graph G is maximum if and only if in G there exists no maximal alternating sequence of odd length relative to Y. Proof Necessity Consider a graph G = (V, E). Let Y be a maximum independent set in G and X=VY. Suppose there exists a maximal alternating sequence σ of odd length relative to Y, and let σ = CTJ U

σ,

*8

*, o-

Figure 9.2

2

242

COVERING AND COLORING

where tr = σ Π Υ 2

and σ = σ ΠΧ . χ

Note that since σ is of odd length, |σ,| > \σ \, and the last vertex of σ is an A"-vertex. Furthermore, since σ is maximal, no vertex y ε Υ - σ can be added to σ without violating condition 2 in the definition of an alternating sequence. Thus no vertex y £ Υ - σ is adjacent to any vertex χ, Ε σ,. Since σ and (Υ - σ ) are themselves independent sets, it follows that (Υ - σ ) U σ is an independent set. So, 2

t

2

2

χ

2

2

|(y-a )u 2

f f l

χ

| > | ( y - ^ ) | + |a | = | y | , 2

contradicting that Y is a maximum independent set. Sufficiency Let A" be a maximum independent set and Y an independent set with | y | < |A*|. We now show that there exists an alternating sequence of odd length relative to Y. Let y = y-(A-ny), 0

χ

0

=

χ-(χηγ),

and let G be the induced subgraph of G on the vertex set (AO U Y ). If G is not connected, then let G , G ,..., G be the connected components of G with (X, U y , ) as the vertex set of G,. Note that 0

0

x

2

0

k

0

υ 1=1

and

ύ

χ, =

Υ, = y · 0

1=1

Since |AT| > \ Y \ , \X \ >\Y \. Therefore for some i, \X,\>\Y,\. Let, without loss of generality, / = 1. Now let Γ be a spanning tree of G . Clearly, (A^, Y ) is a bipartition of the vertex set of T. Since \X \ > \Y \, by the previous lemma there exists in G, a maximal alternating sequence σ (relative to Y ) of odd length that uses all the vertices of Y . Clearly σ is an alternating sequence of odd length relative to Y for G also. We now show that σ is a maximal such sequence in G. 0

0

x

X

x

X

x

x

EDGE COVERS

Note that the A'-vertices and y-vertices of σ belong to X and respectively. Consider any Y-vertex y, that is not in σ. Two cases now arise. 0

Case 1

243

Y, 0

y^YHX.

In this case y is not adjacent to A'-vertices. In particular y is not adjacent to any A"-vertex of σ. t

Case 2

t

y Ε Y. t

0

In this case y is not in G, because σ contains all the y-vertices of G,. Hence y is not adjacent to any of the vertices of G,. In particular _y is not adjacent to any A'-vertex of σ. Thus in both the cases, y cannot be used to extend σ. Therefore, for G, σ is a maximal alternating sequence of odd length relative to Y. • f

;

t

]

For example, consider the independent set Y = {a, c} in the graph of Fig. 9.1. This is not maximum, and it may be verified that {d, c, b, a, / } is a maximal alternating sequence of odd length relative to Y.

9.2

EDGE COVERS

Consider a graph G = (V, E). A subset Ρ of Ε is an edge cover of G if every vertex of G is an end vertex of at least one of the edges of P. If we consider an edge as covering its end vertices, then an edge cover is a subset of edges covering all the vertices of G. An edge cover Ρ of G is minimum if G has no edge cover P' with \P' \ < \ P\. The number of edges in a minimum edge cover of G is called the edge covering number of G and is denoted by S ^ G ) . For example, in the graph of Fig. 9.1, the set {e , e , e } is a minimum edge cover. Note that the vertex covering number and the edge covering number are denoted by β and /3,, respectively; similarly the independence number and matching number are denoted by a and a,, respectively. In the following we use the same symbol to denote an edge cover as well as the induced subgraph on the edges of the cover. Suppose that an edge cover Ρ is minimal. Then it is easy to verify that Ρ has neither circuits nor paths of length greater than 2. This means that each component of Ρ is a tree in which all the edges are incident on a common vertex. In the next theorem, which is the edge analog of Corollary 9.1.1, we relate a and β . This result is due to Gallai [9.3]. x

0

0

t

χ

3

5

244

COVERING AND COLORING

Theorem 9.4. For a simple η-vertex graph G without isolated vertices, α, + β, = η . Proof. Let M* be a maximum matching and P* a minimum edge cover of G. Let N(M*) denote the set of vertices that are not saturated in M*. Then |/V(M*)| = η - 2α,. Now, for each vertex υ in N(M*), select an edge that is incident on υ and adjacent to an edge in M*. Let P be the set of (η - 2 a , ) edges so selected. Then it is clear that the set Ρ = Μ* U P is an edge cover and a

a

\P\ = α, + η — 2a, = η - a, . Thus fl, = | P * | < | F | = n - a , , so that a, + fl, < η .

(9.3)

Now let P* have r connected components so that β ί

= |ρ·| = „ -

r

.

Select one edge from each one of the r components of P. Let Μ be the set of r edges so chosen. It is clear that Μ is a matching, and so a, = | M * | > | M | = r = n - / 3 , . Thus a, + β, > η .

(9.4)

Combining (9.3) and (9.4), we get α, + β, = η .

•

Consider a maximum independent set 5* and a minimum edge cover P* in a graph G with no isolated vertices. Since edges are required to cover the vertices of S*, any edge cover must contain at least |S*| edges. Thus |S*|<|P*|.

(9.5)

In general, equality does not hold in (9.5). However, when a graph G is

EDGE COLORING AND CHROMATIC INDEX

245

bipartite, |S*| = |P*|, and this we prove in the next theorem, which is analogous to Theorem 9.2. Theorem 9.5. In a bipartite graph G, without isolated vertices, the number of vertices in a maximum independent set is equal to the number of edges in a minimum edge cover, that is, «o =

0i

·

Proof. Let G have η vertices. Then by Corollary 9.1.1, a = η — β , and by Theorem 9.4, β, = η - α,. But by Theorem 9.2, α - β . So we get a β • Q

χ

0

0

0

ν

Theorems 9.2 and 9.5 are both equivalent formulations of Hall's theorem (Exercises 9.5 and 9.6).

9.3

EDGE COLORING AND CHROMATIC INDEX

A k-edge coloring of a graph is the assignment of k distinct colors to the edges of the graph. An edge coloring is proper if in the coloring no two adjacent edges receive the same color. A 3-edge coloring and a proper 4-edge coloring of a graph are shown in Fig. 9.3. A graph is k-edge colorable if it has a proper k-edge coloring. We shall assume, without any loss of generality, that the graphs to be considered in this section have no self-loops. The chromatic index or the edge chromatic number x'(G) of a graph G is the minimum k for which G has a proper A:-edge coloring. Graph G is k-edge chromatic if x'{G) = k. It may verified that the chromatic index of the graph of Fig. 9.3a is 4. It may be seen that a k-edge coloring of a graph G = (V, E) induces the partition ( £ , , E ,..., E ) of E, where E, denotes the subset of edges that are assigned the color i in the coloring. Similarly every partition of Ε into A: subsets corresponds to a A:-edge coloring of G. So we shall often denote an edge coloring by the partition of G it induces. If a coloring % = ( £ , , E ,..., E ) is proper, then each £, is a matching. Therefore x'(G) may be regarded as the smallest number of matchings into which the edge set of G can be partitioned. This interpretation of x'(G) will be helpful in the proof of certain useful results. Since in any proper coloring the edges incident on a vertex should receive different colors, it follows that 2

k

2

k

X'(G)*A, where Δ is the maximum degree in G.

(9.6)

246

COVERING AND COLORING a

Figure 9.3. Edge coloring (numbers indicate colors), (a) A proper 4-edge coloring, (b) A 3-edge coloring. In general x'(G) τ^Δ. However, in the case of a bipartite graph χ' = Δ. Theorem 9.6. For a bipartite graph χ' = Δ. Proof. By Corollary 8.22.1, the edge set of a bipartite graph G can be partitioned into Δ matchings. Thus x'(G) — ^ · Combining this with (9.6) proves the theorem.

•

Though Δ colors are not, in general, sufficient to properly color the edges of a graph. Vizing [9.4] has shown that for any simple graph Δ + 1 colors are sufficient. To prove Vizing's theorem we need to establish a few basic results.

EDGE COLORING AND CHROMATIC INDEX

247

We shall say that, in a coloring, color i is represented at vertex υ if in the coloring at least one of the edges incident on υ is assigned the color i. The number of distinct colors represented at vertex ν will be denoted by c(u). For example, in the coloring shown in Fig. 9.36, c(/) = 2. We now consider the question of existence of 2-edge colorings of a graph in which c(v) = 2 for every vertex υ in the graph. Lemma 9.3. Let G be a connected graph that is not a circuit of odd length. Then G has a 2-edge coloring in which both colors are represented at each vertex of degree at least 2. Proof G is Eulerian.

Case 1

If G is a circuit, then by the hypothesis of the theorem it must be of even length. In such a case it is easy to verify that G has a 2-edge coloring having the required property. If G is not a circuit, then it must have a vertex i> of degree at least 4. Let 0

ϋ

0 > \> e

V

l > 2' e

2>

V

3>

e

• • • ' m> e

V

0

be an Euler trail and let £ , = {e,|i odd}

and

E = {e\i even} . 2

(9.7)

Then ( £ , , E ) is a 2-edge coloring in which at every vertex both colors are represented because in the Euler trail considered every vertex including v appears as an internal vertex. 2

0

G is not Eulerian.

Case 2

In this case G must have an even number of odd vertices. Now construct an Eulerian graph G' by adding a new vertex v and connecting it to every vertex of odd degree in G. Let v , e,, u , , . . . , e , v be an Euler trail in G'. If £ , and E are defined as in (9.7), then ( £ , , E ) is a 2-edge coloring of G' in which both colors are represented at each vertex of G. It may now be verified that ( £ , Π Ε, E Π Ε) is a 2-edge coloring of G in which both colors are represented at each vertex of degree at least 2. • 0

0

2

m

0

2

2

A k-edge coloring £ of a graph G = (V, E) is optimum if there is no other A>edge coloring %' of G such that

248

COVERING AND COLORING

Σ

Φ) <

uev

Σ C(t>) , uev

where φ ) and c'(v) are the number of distinct colors represented at vertex υ in the colorings % and %', respectively. Clearly, in any coloring Έ, φ ) s d(v) for every vertex v. Further, for every υ, φ ) = d(v) if and only if % is a proper coloring. A useful result on the nature of an optimum coloring is presented next. Lemma 9.4. Let Έ = ( £ , , E ,..., E ) be an optimum A;-edge coloring of a graph G = (V, E). Suppose that there is a vertex u in G and there are two colors i and j such that i is not represented at u and j is represented twice at u. Let G' be the induced subgraph of G on the edge set E U E Then the component Η of G' containing the vertex u is a circuit of odd length. 2

k

t

r

Proof. If Η is not a circuit of odd length, then, by Lemma 9.3, there is a 2-edge coloring of Η in which both colors ι and / are represented at each vertex of degree at least 2 in H. If we use this 2-edge coloring to recolor the edges of Η and leave the colors of the other edges of G unaltered as they are in %, then we get a new A-edge coloring %' in which both colors i a n d a r e represented at vertex u. Hence, c'(u) = c(«) + 1 , where c'(u) is the number of distinct colors represented at vertex u in the coloring Έ'. Further, for every vertex υ Φ u, c'(v) > φ ) . Hence

Σ

c(v) >

vev

Σ

Φ),

vev

contradicting that % is an optimum edge coloring. Thus it follows that Η is a circuit of odd length. • A Α-edge coloring %' is an improvement on the Α-edge coloring % if

Σ

uev

c'(v) >

Σ

uev

Φ) ·

We now prove Vizing's theorem [9.4]. Our proof is due to Fournier [9.5]. We follow the treatment given in Bondy and Murty [9.6]. Theorem 9.7 (Vizing). If G = (V, E) is a simple graph, then either x'(G) = Δ or x'(G) = A+ 1.

EDGE COLORING AND CHROMATIC INDEX

249

Proof. Since x'{G) s= Δ, it is enough if we prove that x'(G) < Δ + 1 . Let Έ = { £ , , E ,..., E } be an optimal (Δ + l)-edge coloring of G. Assume that x'(G) > Δ + 1, that is, % is not a proper (Δ + l)-edge coloring. Then there exists a vertex u in G such that c(u),. 2

&+1

0

0

}

Figure 9.4

250

COVERING AND COLORING

(c)

Figure 9.4. (Continued) Let color i be not represented at vertex v Then i must be represented at vertex u, for otherwise we can recolor edge (u, u,) with color i and obtain a new coloring of G that is an improvement on %. Let edge (u, v ) have color i . Again not all the colors are represented at v . Suppose that color i is not represented at v . Then it must be represented at u, for otherwise we can recolor edge (u, v ) with i and edge (u,v ) with i , thereby obtaining an improvement on %. Let edge (u, v ) have color i . Repeating these arguments, we can construct a sequence υ ,υ ,... of vertices and a sequence /,, i ,... of colors such that: 2

v

2

2

2

2

2

3

2

t

2

2

3

3

3

1

2

2

1. (u, v ) has color i 2. i is not represented at v f

r

)+i

r

We can now easily see that there exists a smallest integer / such that, for some k < I, '/+ι=«'*

(Fig. 9.4a).

To establish a contradiction, we next recolor, in two different ways, some of the edges incident on u and obtain two new optimal (A+l)-edge colorings Έ' and %". = {E[, E' ,..., £ ^ } is obtained by recoloring (u, v ) with color l s / < t - l . Each one of the remaining edges of G receives in the same color as it does in % (Fig. 9.46). %" = {£", E" ,..., El ) is obtained by recoloring (u, u ) with color + 1

2

2

+l

t

;

251

VERTEX COLORING AND CHROMATIC NUMBER

1 < ; ' < / - 1, and (u, u ) with color i . Each one of the remaining edges of G receives in Έ" the same color as it does in % (Fig. 9.4c). Let G' be the subgraph of G on the edge set (E' U E') with H' denoting its component containing u. Similarly, G" is the subgraph of G on the edge set (£," U E" ) with H" denoting its component containing u. In both Έ' and color i is not represented at u and color i is represented twice at this vertex. Therefore it follows from Lemma 9.4 that H' and H" are both circuits. Now (w, v ) is the only edge of H' that is not in H". Therefore u and v are connected in G" too. Thus they lie in the same component of G", namely, the component H". The degree of v in H" is then 1, for otherwise its degree in H' will be greater than 2. This contradicts that H" is also a circuit. Thus % is a proper (Δ + l)-edge coloring. • (

k

lg

O

k

0

k

k

k

k

For a more general result than Theorem 9.7 see Exercise 9.8.

9.4

VERTEX COLORING AND CHROMATIC NUMBER

A k-vertex coloring of a graph is the assignment of k distinct colors to the vertices of the graph. A vertex coloring is proper if in the coloring no two adjacent vertices receive the same color. A proper 3-vertex coloring is shown in Fig. 9.5. A graph is k-vertex colorable if it has a proper A:-vertex coloring. We may assume, without any loss of generality, that the graphs to be considered in this section are simple. The chromatic number x(G) of a graph G is the minimum number A: for which G is Ac-vertex colorable. G is k-chromatic if x(G) = A:. For example, the chromatic number of the graph in Fig. 9.5 is 3. We shall hereafter refer to a "proper A:-vertex coloring" as simply a "A:-coloring." Similarly, "Ar-vertex colorable" will be abbreviated to "A:colorable."

1

Figure 9.5. A proper 3-vertex coloring.

252

COVERING AND COLORING

Note that a k coloring of a graph G = (V, E) induces a partition (V,, V , . . . , V ) of V, where each Vj is the subset of vertices assigned the color / and therefore is an independent set. Similarly each partition of V into k independent sets corresponds to a A-coloring of G. We proved in the previous section (Theorem 9.7) that at most Δ + 1 colors are required to properly color the edges of a simple graph. We now prove an analogous result for vertex coloring. 2

k

Theorem 9.8. If a graph G is simple, then it is Δ + 1 colorable. Proof. Given Δ + 1 distinct colors, we can obtain a Δ + 1 coloring of G as follows. Pick any vertex v and assign it to any one of the Δ + 1 colors. Then select an uncolored vertex, say υ,. Assign to υ, a color that has not been assigned to the vertices adjacent to it. This is always possible because d(v ) ^ Δ, and hence at most Δ colors will have been assigned to the vertices adjacent to i;,. Repeat this process until all the vertices are colored. The resulting coloring is clearly a proper (Δ + l)-coloring. • Q

x

Clearly, χ = Δ + 1 for complete graphs and for circuits of odd length. A very interesting result is that for all other graphs χ ^ Δ. This result due to Brooks [9.7] is proved next. Our proof here is due to Melnikov and Vizing [9.8]. For an alternate proof see Lovasz [9.9]. Theorem 9.9 (Brooks). Let G be a connected simple graph. If G is neither a circuit of odd length nor a complete graph, then x(G) ^ Δ. Proof. Clearly the theorem is valid for Δ = 0 , 1 , 2 . To prove the theorem for Δ a 3, assume the contrary, that is, there exist graphs that are not complete and for which Δ 2 : 3 and χ = Δ + 1. Then select such a graph G = (V, E) with the minimum number of vertices. Let v ε V, and G' be the graph that results after removing v from G. It follows from the choice of G that G' is Δ-colorable. This implies that d(v ) = Δ, for otherwise one of the Δ colors used to color G' can be used to color v , contradicting that x(G) = Δ + 1. Another important implication is the following. 0

0

0

0

Property 1 In any Δ-coloring of G' the vertices adjacent to v are colored differently. 0

Let «,, M , . . . , Μ be the vertices adjacent to v . Let u u , . . . , κ receive the colors 1 , 2 , . . . , Δ , respectively, in a coloring of G'. Let G(i, ;') denote the induced subgraph of G' on the vertices that are assigned colors i and /. Then: 2

δ

0

l t

2

Δ

CHROMATIC POLYNOMIALS

253

Property 2 Vertices u, and u are in the same connected component of G(i, j). ;

For otherwise, by interchanging the colors ι and / in the component containing u,, we can get a new Δ-coloring of G' in which u, and u are assigned the same color, contradicting Property 1. Let C,j be the component of G(i, j) containing u, and u Then: j

r

Property 3

C, is a path from u, to u . ;

t

Suppose that the degree of u, in C is greater than 1. Then u, is adjacent to at least two vertices of color /. Since in G', d(u,) ^ Δ - 1, we can recolor u, with a color k Φ i, j so that in the resulting new coloring w, and u receive the same color, contradicting Property 1. Similarly, the degree of u in C, is 1. The degree in C of all other vertices is 2. For otherwise, let w be the first vertex of degree (in C ) greater than 2 as we move on a path from u, to u-. If u is colored with color i, then it is adjacent to at least three vertices of color /. Since d(u) < Δ , we can then recolor u with a color k Φ i, j so that in the new coloring u, and M will be in different components, contradicting Property 2. Thus C, is a path from u, to u tl

k

y

;

/y

;

y

r

Property 4

C . and C

have no common vertex except u,.

ik

Let u Φ Uj be common to C, and C . Then u is colored with color i, and it is adjacent to at least two vertices of color / and two vertices of color k. Since d(u) Δ, there exists a color / Φ i, j , k with which we can recolor u. But this would disconnect u, and u contradicting Property 2. Now we proceed to establish a contradiction of Property 4. Since G is not a ( Δ + l)-vertex complete graph, there exist two vertices, say U] and u , which are not adjacent. Then the path C contains a vertex uΦu adjacent to u . Suppose that we interchange colors 1 and 3 in the path C (which exists because Δ ^ 3 ) , so that in the new coloring C , u receives color 3 and w receives color 1. But then the new components C[ and C' contain the common vertex uΦu , contradicting Property 4. This completes the proof of the theorem. • ;

ik

p

12

2

2

t

13

x

3

23

9.5

2

2

CHROMATIC POLYNOMIALS

In this section we discuss the question of counting the number of distinct proper λ-colorings of a graph. If a graph is λ-colorable, then it may be possible to color it with A colors

254

COVERING AND COLORING

-o

Figure 9.6 o-

b

c

in more than one way. Two colorings of a graph are considered distinct if at least one vertex of the graph is assigned different colors in the two colorings. The chromatic polynomial P(G, A) expresses for each integer λ the number of distinct λ-colorings possible for a graph G. For example, consider the graph shown in Fig. 9.6. Given λ colors, we can select any one of them to color vertex a. Vertex b can then be colored with any one of the remaining λ - 1 colors. For each coloring of vertex b, there are λ - 1 different ways of coloring vertex c. Thus the graph of Fig. 9.6 can be colored in λ(λ - l ) different ways. In other words the chromatic polynomial of the graph is λ(λ - l) . In fact, we can repeat these arguments to show that the chromatic polynomial of a path on η vertices is equal to 2

2

.Π —

A(A-l)'

As another example, consider next K„, the complete graph on η vertices v ,v ,..., v . Given λ colors, the vertex u, can be colored with any one of the λ colors, vertex v with any one of the remaining λ - 1 colors, vertex v with any one of the remaining A - 2 colors, and so on. Thus i

2

n

2

3

λ) = λ ( λ - 1 ) · · ( λ - * + Next we present a formula for determining the chromatic polynomial of a graph G. Theorem 9.10. Let u and ν be two nonadjacent vertices in a simple graph Let e = (u, v). If G · e denotes the simple graph obtained from G short-circuiting the vertices u and υ and replacing the resulting sets parallel edges with single edges, and if G + e denotes the graph obtained adding the edge e to G, then

G. by of by

P(G, λ) = P(G + e, λ) + P(G • e, λ ) . Proof. Each λ-coloring of G in different colors corresponds to Similarly, each λ-coloring of G in corresponds to a λ-coloring of G

which the vertices u and ν are assigned a λ-coloring of G + e and vice versa. which u and υ are assigned the same color • e and vice versa. Hence

P(G, A) = P(G + e, A) + P(G · e, A).

•

This result can be stated in a different form as follows. Corollary 9.10.1. If e = (u, υ) is an edge of a simple graph G, then

CHROMATIC POLYNOMIALS

255

P(G, A) = P(G - e, A) - P(G • e, λ ) , where G — e is obtained by removing e from G and G · e is as defined in Theorem 9.10. • If we repeatedly apply the formula given in Theorem 9.10 on a graph G, then the process will terminate in complete graphs, say, H , H ,..., H , so that x

2

k

P(G, A) = P(H , A) + P(H , A) + · · · + P(H , A) . X

2

k

On the other hand, if we use the formula given in Corollary 9.10.1, the process will terminate in empty graphs (i.e., graphs that have no edges) so that the chromatic polynomial is a linear combination of the chromatic polynomials of empty graphs. Both these procedures are illustrated in Fig. 9.7. Theorem 9.11. The chromatic polynomial P(G, A) of an η-vertex graph G is of degree n, with leading term A" and constant term zero. Furthermore, the coefficients are all integers and alternate in sign. Proof. The proof is by induction on the number m of edges. Clearly the theorem is valid for m = 0, since the chromatic polynomial of an n-vertex empty graph is A". Assume that the theorem holds for all graphs with fewer than m edges. Consider now an η-vertex graph G with m edges. Let e be an edge of G. Then G - e is an η-vertex graph with m - 1 edges and G · e is an (η — 1)vertex graph with m - 1 or less edges. By the induction hypothesis it now follows that there are nonnegative integer coefficients a,, a ,...,a„_, and b , b ,..., b„_ , such that 2

x

2

P(G - e, λ) = λ" - α ^ , λ " - + a^k"' 1

2

2

-•••

+ (-^"«α,λ

and P(G · e, A) = A"" - b„_ \"1

2

2

+ b^.X"^

+(-l)""^ .

By Corollary 9.10.1, P(G, A) = P(G - e, A) - P(G · e, A)

= A - - κ., + I)A— + Σ (-i)K-, + ν , μ - ' . i=2

Thus G too satisfies the theorem.

•

256

COVERING AND COLORING

Ob

*>d

O a. d

b

i

Ο

λ

) ( V

i

) (

ο

)

(**')

3

(b)

Figure 9.7. Calculation of the chromatic polynomial of a graph G. (a) Using Theorem 9.10. P(G, λ) = λ(λ - 1)(A - 2)(A - 3) + 2A(A — 1)(A — 2) + λ(λ — 1). (b) Using Corollary 9.10.1. P(G, λ) = λ — 2A + A. 3

2

THE FOUR-COLOR PROBLEM

9.6

257

THE FOUR-COLOR PROBLEM

In map making, to distinguish the different regions, one is interested in coloring the regions so that no two adjacent regions receive the same color. This problem of properly coloring the regions of a planar graph is the same as that of properly coloring the vertices of the dual of the graph. It is easy to construct an example to show that three colors are, in general, not sufficient to properly color a planar graph. In the following theorem, known as the five-color theorem, we show that five colors are sufficient. Theorem 9.12. Every planar graph is 5-colorable. Proof. We shall assume that the theorem is true for all planar graphs of order less than η and then show that it is also true for a planar graph G = (V, E) of order n. By Corollary 7.3.5, there exists in G a vertex v of degree ^ 5 . Let G' be the induced subgraph of G on V— {v }. Color G' with the five colors a , a ,...,a . (This is possible by the induction hypothesis.) Clearly, one of these colors can be assigned to v in a proper 5-coloring of G, unless d(v ) = 5, and all the 5 vertices v , v ,..., v adjacent to v are assigned different colors. Assume that u,, l < / < 5 , is assigned the color a Further, let v , v ,... ,v be arranged in clockwise order as shown in Fig. 9.8. Let G(a,, a ) be the subgraph of G on the vertices, which have been assigned the color a, or a . In G ( a , , a ) , the component that contains i>, should contain v ; for otherwise the colors a, and a could be interchanged in this component and v could be colored with a,. 0

0

x

2

s

0

Q

x

2

5

0

r

2

x

5

;

;

3

3

3

0

Figure 9.8

258

COVERING AND COLORING

Similarly, in G ( a , , a ) the component that contains i>, should also contain v . (See Fig. 9.8.) Then in G(a , a ), v and v will not be connected. If now colors a and a are interchanged in the component of G(a , a ) that contains v , then v can be colored with a . Thus G is 5-colorable. • 4

4

2

s

2

s

s

2

2

5

2

0

2

The preceding result is due to Heawood [9.10]. The question now arises as to whether the five-color theorem is the best possible. It was conjectured that every planar graph is 4-colorable. This was known as the four-color conjecture. This conjecture remained unsolved for more than 100 years and was the subject of extensive research. In 1976, Appel and Haken [9.11] proved this conjecture true. This is stated in the following theorem. Theorem 9.13 (Four-Color Theorem). Every planar graph is 4-colorable.

•

There is a vast literature on the four-color problem. See Ore [9.12], Saaty [9.13], and Saaty and Kainen [9.14] for an extensive survey of the problem with historical details. See also Whitney and Tutte [9.15].

9.7

FURTHER READING

Berge [9.2], Bondy and Murty [9.6], Harary [9.16], and Chartrand and Lesniak [9.17] are general references for the topics covered in this chapter. A classical result in graph theory due to Turan [9.18] determines the minimum number of edges than an η-vertex graph with independence number must have (Exercise 9.2). See Erdos [9.19]. This result is the starting point for a branch of graph theory known as extremal graph theory. The book by Bollobas [9.20] is devoted exclusively to the study of extremal graph problems. Extremal results form the basis of the design of communication nets having specified vulnerability and reliability properties. See Frank and Frisch [9.21], Boesch [9.22], and Karivaratharajan and Thulasiraman [9.23], [9.24]. See also Section 8.8. See Berge [9.2] (Chapter 16), Liu [9.25] (Chapter 9), and Lovasz [9.26] for an application of the independence number in information theory. Read [9.27] gives an excellent survey of results on chromatic polynomials. See also Birkhoff [9.28] and Tutte [9.29] and [9.30]. Liu [9.25] (pp. 245-246) discusses an application of coloring in the decomposition of a graph into planar subgraphs. This problem arises in the design of printed circuits. See Wood [9.31] for an application of coloring in time-tabling problems, and Christofides [9.32] for coloring algorithms. A set of vertices D in a graph G is a dominating set in G if every vertex not in D is adjacent to a vertex in D. For a review of results concerning

EXERCISES

259

dominating sets, see Cockayne and Hedetniemi [9.33]. An extremal result concerning dominating sets is given in Cockayne and Hedetniemi [9.34]. Consider a simple graph G. Let
9.8

EXERCISES

9.1

Prove or disprove: Every vertex cover of a graph contains a minimum vertex cover.

9.2

Given any two integers η and k with η ^ k > 0. Let q = [nlk\ and let r be an integer such that η = kq + r ,

0
Let G be the simple graph that consists of k disjoint complete graphs, of which r have q + 1 vertices and k — r have q vertices. Then show that every graph G with η vertices and a ( G ) ^ k that has the minimum possible number of edges is isomorphic to G„ (Turan [9.18]). An easy corollary of this result is: If G is a simple graph with η vertices and m edges and with a ( G ) = A;, then n

k

0

k

0

9.3

If G is a simple graph with η vertices and m edges, then show that 2

« (G)> 0

9.4

η 2m + η '

Show that the maximum number of edges that an w-vertex simple graph G with no triangles can have is [n /4J. Construct a graph with this property.

260

COVERING AND COLORING

9.5

Derive Hall's theorem, Theorem 8.13, from Theorem 9.2.

9.6

Derive Hall's theorem, Theorem 8.13, from Theorem 9.5.

9.7

Show that X K n) K

9.8

|„

jf is odd .

>

n

Show that if G is a graph without self-loops, then x'(G)*a

+ k,

where k is the maximum number of parallel edges between any two vertices of G (Vizing [9.4]). 9.9

Let G be a graph without self-loops. If A(G) = 3, then show that * ' ( G ) = 3 o r 4.

9.10

Show that, for a regular nonempty simple graph with odd number of vertices, * ' ( G ) = A + 1.

9.11 Show that for any arbitrary orientation of the edges of a simple graph G, there exists a directed path of length x(G) - 1. 9.12 Show that if any two circuits of odd length of a graph G have a vertex in common, then *(G)<5. 9.13

Using Corollary 7.3.5, show that every planar graph is 6-colorable.

9.14 Show that if a simple graph has degree sequence (d,, d , . . . , d„) then 2

x(G) ^ max min {d, + 1, i) (Welsh and Powell [9.39]). (Note: This result can be proved by showing that the following greedy coloring algorithm uses at most max min {d, + 1, /'} colors: Color u, with color 1, then color v with color 1 if u, and v are not adjacent and with color 2 otherwise, and so on. In other words color each vertex with the smallest color it can have at that stage.) 2

9.15 Show that for a simple graph G X(G) < 1 + max δ(Η)

2

EXERCISES

261

where the maximum is taken over all induced subgraphs Η of G (Szekeres and Wilf [9.40]). 9.16

Show that for a simple graph G * ( G ) < 1 + /(G) where /(G) denotes the length of a longest path in G.

9.17

Show that for an η-vertex simple graph G, (a) x(G)a (G)>n; (b) * ( G ) + a ( G ) < n + l; (c) x(G) + (G)^n + l. 0

0

X

9.18

Let G be a simple w-vertex regular graph of degree k. Show that

9.19

Let G be a planar graph in which each region is bounded by exactly three edges. Show that G is 3-colorable if and only if G is Eulerian.

9.20

Show that a simple graph is nonplanar if it has seven vertices and the degrees of its vertices are all equal to 4 (Liu [9.25]).

9.21

(a) Show that a graph is 2-colorable if and only if all its circuits are of even length. (b) Show that the regions of a planar graph G can be properly colored with two colors if and only if G is Eulerian.

9.22

Find the chromatic polynomial of the graph shown in Fig. 9.9.

9.23

(a) Show that an η-vertex graph G is a tree if and only if P(G, λ) = λ ( λ - 1 ) " . π

Figure 9.9

1

262

COVERING AND COLORING

(b) Show that if an η-vertex graph G is connected, then ,π-1

P(G, λ ) < λ ( λ - 1 ) ' 9.24

Show that if G is a circuit of length n, then P(G, λ) = ( λ - 1 ) " + ( - 1 ) ( λ - 1 ) . η

9.25

Show that if G is a wheel on η + 1 vertices, then P(G, A) = A ( A - 2 ) " + ( - l ) " A ( A - 2 ) .

9.26

Show that if a simple graph G has η vertices, m edges, and A: components, then (a) The coefficient of A " in P(G, λ) is - m . (b) The smallest exponent of λ in P(G, λ) with a nonzero coefficient is k. -1

9.27

A clique of a graph G = (V, E) is a set S C V such that the induced subgraph of G on S is a complete graph. Let Ρ = {Ρ,, P , . . . , P } be a partition of V into cliques. Let 2

k

0(G) = min {|P|}. Thus 0(G) is the smallest possible number of cliques into which Κ can be partitioned. Show that, for a simple graph G, a ( G ) =£ 0(G). Further, if S is an independent set and Ρ is a partition of V into cliques such that |5| = | P | , then show that 5 is a maximum independent set and Ρ is a minimum partition of V into cliques. 0

9.9

REFERENCES

9.1

D. Konig, "Graphs and Matrices," (in Hungarian), Mat. Fiz. Lapok, Vol. 38, 116-119 (1931). C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. T. Gallai, "Uber extreme Punkt und Kantenmengen," Ann. Univ. Sci. Budapest, Eotvos Sect. Math., Vol. 2, 133-138 (1959). V. G. Vizing, "On an Estimate of the Chromatic Class of a p-Graph," (in Russian), Diskret. Analiz., Vol. 3, 25-30 (1964). J. C. Fournier, "Colorations des aretes d'un graphe," Cahiers du CERO, Vol. 15, 311-314 (1973). J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976.

9.2 9.3 9.4 9.5 9.6

REFERENCES

9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25 9.26 9.27 9.28

263

R. L. Brooks, "On Colouring the Nodes of a Network," Proc. Cambridge Phil. Soc, Vol. 37, 194-197 (1941). L. S. Melnikov and V. G. Vizing, "New Proof of Brooks' Theorem," J. Combinatorial Theory, Vol. 7, 289-290 (1969). L. Lovasz, "Three Short Proofs in Graph Theory," J. Combinatorial Theory B, Vol. 19, 111-113 (1975). P. J. Heawood, "Map Colour Theorems," Quart. J. Math., Vol. 24, 332-338 (1890). Κ. I. Appel and W. Haken, "Every Planar Map is Four-Colorable," Bull. Am. Math. Soc, Vol. 82, 711-712 (1976). O. Ore, The Four-Color Problem, Academic Press, New York, 1967. T. L. Saaty, "Thirteen Colorful Variations on Guthrie's Four-Color Conjecture," Am. Math. Monthly, Vol. 79, 2-43 (1972). T. L. Saaty and P. C. Kainen, The Four-Color Problem: Assaults and Conquests, McGraw-Hill, New York, 1977. H. Whitney and W. T. Tutte, "Kempe Chains and the Four Colour Problem," in Studies in Graph Theory, Part II, MAA Press, Washington, D.C., 1975, pp. 378-413. F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969. G. Chartrand and L. Lesniak, Graphs and Digraphs, Wadsworth and Brooks/ Cole, Pacific Grove, Calif., 1986. P. Turan, "An Extremal Problem in Graph Theory," (in Hungarian), Mat. Fiz. Lapok, Vol. 48, 436-452 (1941). P. Erdos, "On the Graph Theorem of Turan," (in Hungarian), Mat. Lapok, Vol. 21, 249-251 (1970). B. Bollobas, Extremal Graph Theory, Academic Press, New York, 1978. H. Frank and I. T. Frisch, Communication, Transmission, and Transportation Networks, Addison-Wesley, Reading, Mass., 1971. F. T. Boesch, Ed., Large-Scale Networks: Theory and Design, IEEE Press, New York, 1976. P. Karivaratharajan and K. Thulasiraman, "AT-Sets of a Graph and Vulnerability of Communication Nets," Matrix and Tensor Quart., Vol. 25, 63-66, (1974), 77-86 (1975). P. Karivaratharajan and K. Thulasiraman, "An Extremal Problem in Graph Theory and Its Applications," Proc. IEEE Intl. Symp. on Circuits and Systems, Tokyo, Japan, 1979. C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968. L. Lovasz, "On the Shannon Capacity of a Graph," IEEE Trans. Inform. Theory, Vol. IT-25, 1-7 (1979). R. C. Read, "An Introduction to Chromatic Polynomials," /. Combinatorial Theory, Vol. 4, 52-71 (1968). G. D. Birkhoff, "A Determinant Formula for the Number of Ways of Coloring a Map," Ann. Math., Vol. 14, 42-46 (1912).

264

COVERING AND COLORING

9.29 W. T. Tutte, "On Chromatic Polynomials and the Golden Ratio," /. Combinatorial Theory, Vol. 9, 289-296 (1970). 9.30 W. T. Tutte, "Chromials," in Studies in Graph Theory, Part II, MAA Press, Washington, D.C., 1975, pp. 361-377. 9.31 D. C. Wood, "A Technique for Coloring a Graph Applicable to Large-Scale Timetabling Problems," Computer J., Vol. 12, 317 (1969). 9.32 N. Christofides, Graph Theory: An Algorithmic Approach, Academic Press, New York, 1975. 9.33 E. J. Cockayne and S. T. Hedetniemi, "Towards a Theory of Domination in Graphs," Networks, Vol. 7, 247-267 (1977). 9.34 E. J. Cockayne and S. T. Hedetniemi, "Optimal Domination in Graphs," IEEE Trans. Circuits and Sys., Vol. CAS-2, 41-44 (1975). 9.35 L. Lovasz, "Normal Hypergraphs and the Perfect Graph Conjecture," Discrete Math., Vol. 2, 253-267 (1972). 9.36 A. Tucker, "The Strong Perfect Graph Conjecture for Planar Graphs," Canad. J. Math., Vol. 25, 103-114 (1973). 9.37 K. R. Parthasarathy and G. Ravindra, "The Strong Perfect Graph Conjecture is True for K -free Graphs," J. Comb. Theory, Vol. 22B, 212-223 (1976). 9.38 M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980. 9.39 D. J. A. Welsh and Μ. B. Powell, "An Upper Bound for the Chromatic Number of a Graph and Its Application to Timetabling Problems," Computer J., Vol. 10, 85-87 (1967). 9.40 G. Szekeres and H. S. Wilf, "An Inequality for the Chromatic Number of a Graph," J. Comb. Theory, Vol. 4, 1-3 (1968). t 3

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 10

MATROIDS

Consider a finite set S of vectors over an arbitrary field. The collection of the independent sets of vectors of S possesses several interesting properties. For example, 1. Any subset of an independent set is independent. 2. If I and I are any two independent sets with | / | = \l \ + 1, then I together with some element of I forms an independent set of elements. p

p

p + 1

p+l

p

p + l

It is interesting to note that there are several algebraic systems that possess these two properties. For instance, the collection of subsets of edges of a graph that do not contain any circuit possesses these properties. It was while studying the properties of such systems that Whitney [10.1] introduced the concept of a matroid. In this chapter we give an introduction to the theory of matroids. We discuss several fundamental properties of a matroid. Of special interest to us is the result that the "duality" that we observed between the circuits and the cutsets of a graph is not accidental. As we shall see, this duality is a consequence of the fact that the collection of subgraphs that have no circuits and the collection of subgraphs that have no cutsets both have the matroid structure. We also study the "painting" theorem and the arc coloring lemma, which find an application in network analysis. We conclude the chapter with a discussion of the "greedy" algorithm, which is a generalization of a well-known algorithm due to Kruskal for finding a minimum cost spanning tree of a weighted connected graph. 265

MATROIDS

266

10.1

BASIC DEFINITIONS

There are several equivalent axiom systems for characterizing a matroid. We begin our discussion with what are known as the independence axioms. In Section 10.3 we derive some of the equivalent axiom systems. A matroid Μ is a finite set S and a collection 3 of subsets of S such that the following axioms, called independence axioms, are satisfied: 1-1. 0E 3. 1-2. If XE 3 and YCX, then YE 3. 1-3. If X and Y are members of 3 with |A"| = |Y| + 1, there exists xEX-Y such that Y U χ Ε 3. The elements of 5 are called the elements of the matroid M. Members of 3 are called the independent sets of M. A maximal independent set of Μ is called a base of M. The collection of the bases of Μ is denoted by S9(M) or simply 38. A subset of S not belonging to 3 is called dependent. A minimal dependent subset of 5 is called a circuit of M. An element x of S is a loop of Μ if {x} is dependent. The collection of circuits of Μ is denoted by "#(M) or simply %. The rank function ρ of Μ associates to each subset A of 5 a nonnegative integer p(A) defined by p(A) = max{\X\:XCA,XE3}

.

p(A) is called the rank of A. The rank of the matroid Μ denoted by p(M) is the rank of the set S. We now consider some examples. Let 5 be a finite subset of a vector space. As we observed earlier, the family of all the subsets of linearly independent vectors in 5 satisfy the independence axioms 1-1 through 1-3. Hence these subsets of S form the collection of independent sets of a matroid on 5. In this matroid the rank of a subset X of S is equal to the dimension of the vector space generated by X. Let G be an undirected graph with edge set E. We can define two matroids on E. First consider the collection 3 of all the subsets of Ε that do not contain any circuits. Clearly 3 satisfies 1-1 and 1-2. We can easily show that (Exercise 4.8) that 3 satisfies 1-3. Thus 3 is the collection of independent sets of a matroid Μ on E. Each base of Μ is a spanning forest of G. In this matroid the rank of any subset X of Ε is equal to the rank of the subgraph of G induced by X. Furthermore, each circuit of Μ is a circuit of G. For this reason Μ is called the circuit matroid of G.

BASIC DEFINITIONS

267

Consider next the family 3 * of all the subsets of Ε that do not contain any cutset of G. We can show (Exercise 4.9) that 3* satisfies axioms 1-1 through 1-3, and hence it is the family of independent sets of a matroid M* on E. Each base of M* is a cospanning forest of G. In this matroid the rank of any subset X oi Ε is equal to the nullity of the subgraph of G induced by X. Furthermore, each circuit of Μ * is a cutset of G. The matroid M* is called the cutset matroid or the bond matroid of G. The two matroids Μ and M* defined above possess the interesting property that the bases of one are the complements in Ε of the bases of the other. This result is true for any matroid on any finite S (not necessarily the edge set of a graph). In other words for any matroid Μ on a set 5 there exists a matroid M* on 5 with the bases of Μ * being the complements of the bases of M. We discuss this result in Section 10.4. Another example of a matroid is the matching matroid defined on the vertex set of a graph. Theorem 10.1. Let G be an undirected graph with vertex set V. Let 3 be the collection of all the subsets ICV such that the elements of / are saturated in some matching of G. Then 3 is the collection of independent sets of a matroid on V. Proof. Clearly 3· satisfies axioms 1-1 and 1-2. To show that 3 satisfies axiom 1-3, consider any two members I and I of 3 containing ρ and ρ + 1 vertices, respectively, and let X and X be any two matchings saturating the elements of I and I , respectively. Now two cases arise. p

p+l

p

p

p+l

p +X

Suppose that some element χ Ε I — l is saturated in X . Then X saturates I U χ and the axiom 1-3 is satisfied. Suppose that no element of / - I is saturated in X . Then consider the subgraph G' on the edge set X (&X =(X X )U (X - X ). By Theorem 8.19 each component of G' is either 1. a circuit whose edges are alternately in X and X or 2. a path whose edges are alternately in X and X and whose end vertices are not saturated in one of the two matchings. Since \I - l \> \I - I \, there is a path Ρ in G' from a vertex υ Ε I - I to a vertex not in I - I . Then Χ ΦΡ will be a matching saturating ν and all the elements of I . Thus I U ν is a member of 3, and axiom 1-3 is satisfied. •

Case 1

p+i

p

Case 2

p

p

p

+ 1

p

p

p

p+l

p+l

p

p

p

p

p

p+x

+ x

p

p

p + l

p+X

p+X

p+l

p

p

p+X

ρ

p

p

As an example, consider the graph G shown in Fig. 10.1a. The sets / = {u,, v , v , v } and / = { υ v , υ , v , υ , v } are saturated in the matchings X = {e,, e } and X = {e , e , e ), respectively. The subgraph on 4

2

3

4

6

4

2

1;

6

2

2

3

3

4

4

5

6

268

MATROIDS

Figure 10.1. (a) A graph G with the matchings X4 and X6 indicated, (b) Subgraph of G on the edge set X4®Xb.

the edge set XA(BX6 = {e,, e3, eA} is shown in Fig. 10.1ft. There is a path P from the vertex v5El6-IA to the vertex vb0IA-I6. The matching XA Φ P = {e2, e3, e4} saturates the set IAUvs. Two matroids Ml and Af2 on the sets S, and 5 2 , respectively, are isomorphic if there is a one-to-one correspondence between the elements of 5, and 5 2 , which preserves independence. A matroid M on 5 is called graphic if it is isomorphic to the circuit matroid of a graph G. M is called cographic if it is isomorphic to the cutset matroid of a graph G. 10.2

FUNDAMENTAL PROPERTIES

We establish in this section several fundamental properties of a matroid. All these properties are familiar in graph theory. We first consider the independent sets and the bases of a matroid M.

FUNDAMENTAL PROPERTIES

Theorem 10.2 (Augmentation Theorem). Let X and

269

Y be any two

in-

dependent subsets in a matroid M. If |A"| < | Y\, then there exists ZC.Ysuch that |A"U Z | = | Y\ and XI) Ζ is independent in M.

X

Proof. Let Z be a set such that: 1. Z CY-X. 2. X U Z is independent in M. 3. If A' U Ζ is independent in Μ for any Z C V - I , then \Xl)Z \> \xuz\. 0

0

0

0

Suppose that l A ' U Z o l ^ y l . Then there exists a set Y CY with \Y \ = \X\J Z | + 1. Since y is independent in M, by axiom 1-3 there exists an element y ε Y - (A' U Z ) such that A" U Z U >> is independent in Μ. The set Z Uy contradicts the choice of Z . Hence lA' U Z | s | y | , and the proof now follows. • 0

0

0

o

0

0

0

0

0

0

Corollary 10.2.1. All the bases of a matroid Μ on a set 5 have the same cardinality equal to the rank of M. Proof. If not, let B , B be bases with \B \ < \ B \. Then by the augmentation theorem there exists Ζ C B - B, such that B U Ζ is independent. This, however, contradicts the maximality of Β in A • x

2

X

2

2

x

λ

The following generalization of this result can also be proved using the augmentation theorem. Corollary 10.2.2. Let Μ be a matroid on a set S and ACS. Then all the maximal independent subsets of A have the same cardinality. • Another corollary of Theorem 10.2 is stated next without proof. Corollary 10.2.3. If B and B are the bases of a matroid Μ and χ ε S, - 5 , then there exists y Ε B - B, such that ( β , U )>) - χ is a base of M. • x

2

2

2

We next obtain some properties of the rank function of a matroid. Theorem 10.3. The rank function ρ of a matroid Μ on a set S satisfies the following properties, where A, Β C S and xE.S. 1. Os (A)^\A\. 2. MAC B, then p(A) < p(B). 3. p(A)
p(A) + p(B) > p(A UB) + p(A

η Β)

.

270

MATROIDS

5. If p(A U y) = p(A U JC) = p(A), then p(A U χ U y) = p(A). Proof. The first three properties follow easily from the definition of the rank function. To establish property 4, let X be a maximal independent subset of Α Π Β. By the augmentation theorem, there exists a maximal independent subset Y of AU Β containing X. Let Y= XliVUW, where VCA-B and WC B-A. Since X\JV is an independent subset of A, and X\JW is an independent subset of B, we get p(A)*\XUV\ and p(B)>\XUW\. Hence p(A) + p(B) >\XUV\

+ \XL>W\

= 2\X\ + \V\ + \W\ = | * | + ( | * | + |V| + |W|) = ρ(ΑΓ\ B) + p(AU

B).

Property 5 is an easy consequence of property 4.

•

We next consider the circuits of a matroid. Some properties, which follow directly from the definition of a circuit, are 1. Every proper subset of a circuit is independent. So, if C, and C are distinct circuits, then C gC . 2. If C is a circuit, then p(C) = |C| - 1. 3. A matroid Μ on a set 5 has no circuits if and only if all the subsets of S are independent. Thus S is the only base of such a matroid M. 2

x

2

Theorem 10.4. If C, and C are distinct circuits of a matroid Μ and χ Ε C, Π C , then there exists a circuit C such that C C (C, U C ) - x. 2

2

3

3

2

Proof. Let C = (Cj U C ) - JC. We prove the theorem by showing that C is dependent, that is, p(C')<\C'\. Since C\ and C are distinct circuits, C Π C is a proper subset of both Cj and C . Hence C, Π C is independent. Thus 2

2

2

l

2

2

p(c r\c ) x

2

=

\c,r\c \ 2

FUNDAMENTAL PROPERTIES

271

Furthermore,

and p(C )=\C \-l. 2

2

Now, using the above and the submodularity of the rank function (Theorem 10.3), we get p(C, U C ) < p ( C . ) + p ( C ) - p ( C , Π C ) 2

2

2

= 1^1-1 + 1 ^ 1 - 1 - 1 ^ 0 ^ 1 = |C,UC |-2 2

<|C'|.

(10.1)

Since C ' C Q U Q , we also have /KHspiQUCj).

(10.2)

Combining (10.1) and (10.2), we get p(C')<\C'\.

•

Corollary 10.4.1. If /I is an independent set in a matroid M on a set S, then for any χ G S , A U J : contains at most one circuit. Proof. Let, for some χ £ 5, there exist two distinct circuits C, and C such that j r £ C , n C , C QALIx, and C QA\Jx. Then by the previous theorem there exists a circuit C C (C, U C ) - x . Hence (C, U C ) - χ is dependent. But ( C , U C ) - χ C A, contradicting the independence of Α. m 2

2

{

2

2

2

2

Corollary 10.4.2. If Β is a base of a matroid Λί on a set S and xGS - B, then there exists a unique circuit C = C(*, 5 ) such that χ E. C C Β U x. Proof. Since β is a maximal independent set, Β Li χ contains a circuit. By the previous corollary, such a circuit is unique. • The circuit C(x, B) denned in the above corollary is called the fundamental circuit of χ with respect to base B. We next prove a much stronger result than Theorem 10.4.

272

MATROIDS

Theorem 10.5. If C, and C are distinct circuits of a matroid and χ Ε C, Π C , then for any element yEC -C there exists a circuit C such that 2

2

t

2

y e C C (Q U C ) - χ . 2

Proof. Suppose that C,, C , JC, y are such that the theorem is false and that | C , U C | is minimum among all pairs of distinct circuits violating the theorem. By Theorem 10.4 there exists a circuit C C (C, U C ) - x. From our choice of χ and y is it clear that y0C3. Furthermore, C Π (C - C,) Φ 0 , for otherwise C C C,, contradicting the fact that C, is a minimal dependent set. Let then zEC C$C C$. Now, for the circuits C and C we have the following: 2

2

3

2

3

2

3

3

2

2

3

1. zEC HC . 2. xEC C. 3. C U C is a proper subset of C, U C because 2

3

2

2

3

3

2

y0C L>C . 2

3

From our choice of C, and C it follows that there exists a circuit C such that 2

4

iec c(c uc )-z. 4

2

3

Consider next the circuits C, and C for which we have the following: 4

1. x e c , n c . 4

2. y e C, - C , because y e'Cz U C . 3. C, U C is a proper subset of C, U C , because ζ &C U C . 4

3

4

2

Again from our such that y Ε C Since C, U C such that y e C

choice of C, and C it follows that there exists a circuit C C (C, U C ) - JC. is a proper subset of C, U C , we have found a circuit C C (Cj U C ) - jt, contradicting our choice of C, and C . • 2

5

5

10.3

4

X

4

5

4

5

2

2

2

EQUIVALENT AXIOM SYSTEMS

In this section we establish some alternative axiom systems for defining a matroid. Such equivalent characterizations would help us in gaining more insight into the structure of matroids. We begin with an equivalent set of independence axioms. Theorem 10.6 (Independence Axioms). A collection J of subsets of a set S is the set of independent sets of a matroid on 5 if and only if 3 satisfies axioms s

EQUIVALENT AXIOM SYSTEMS

273

1-1, 1-2, and 1-3'. If A is any subset of S, then all the maximal subsets Y of A, with Y G 3, have the same cardinality. Proof. The proof is left as an exercise. See Corollary 10.2.2.

•

Theorem 10.7 (Base Axioms). Let 38 be the set of bases of a matroid. Then B-l. 38 ¥=0, and no set in 38 contains another properly. B-2. If β , , B e 38 and χ e β , , then there exists y £ f l ( β , - JC) U y G 38. 2

2

such that

Conversely, if 38 is a collection of subsets of a set S satisfying B-l and B-2, then 3 = {I\I C B, for some Β G 38} is the collection of independent sets of a matroid on S. Proof. We have already proved that the bases of a matroid satisfy B-l and B-2. (Recall the definition of a base and see Corollary 10.2.3.) Given 38 satisfying B-l and B-2, we now show that 3 as defined in the theorem is the collection of independent sets of a matroid on 5. First note that B-l and B-2 imply that for * G β , - B there exists y G B - B, such that ( β , - χ) U y is a member of 38. Clearly, 3 satisfies axioms 1-1 and 1-2. To prove axiom 1-3, first we show that all the members of 38 have the same cardinality. Suppose there exists fl„ fi G38 with | β , | > | β | . Then by repeated application of β-2 we can remove β - β , from β and replace it it by a set ACB -B with |^4| = | β - β , | and such that β = ( β Π β , ) U A is a member of 38. But β C β , , contradicting B-l. Thus all the members of 38 have the same cardinality. Consider next any two distinct members X and Y of 3 with | Y| = |AT| + 1. Let XC β , , Y C β , and β , , β 6 38. Let 2

2

2

2

2

i

2

2

2

3

2

3

2

2

X — {x , x ,... x

&i

~

i i> x

,x } ,

2

2>

x

k

· · •

°i,

b ,..., 2

b) , q

B = {yi> y >- • • - y*. y * + i . c , , c , . . . 2

2

2

,c _,}. q

Now consider B -b . By B-2 there exists zE.B such that B' = ( β , - Z> ) U ζ is a member of 38. If ζ G Y, then ( ^ U z ) C B ' , and hence it is a member of 3. So 1-3 is satisfied in this case. 1

?

q

2

274

MATROIDS

liz^Y, then consider B"= B' - b „ Again by B-2 there exists z'Ε B such that B" = (B' - b _ )L)z' is a member of 38. If z' Ε Y, then XLiz'C B". Hence (XL) z')E3, and axiom 1-3 is satisfied. If z ' 0 Y , remove b _ from B", and so on. Since |{6,, b ,..., b }\> |{c,, c , . . . , c , } | , after at most q steps we arrive at a situation where we replace some b, by an element of Y, thereby satisfying axiom 1-3. • q

q

v

2

l

q

2

2

q

2

Theorem 10.8 (Rank Axioms). The rank function ρ of a matroid Μ on a set 5 satisfies the following: R-l. 0 p(A U Β) + p(A Π Β). Conversely, if an integer-valued function ρ defined on a finite set S satisfies R-l, R-2, and R-3, then the set 3 = {I\ (I) P

=

\I\,ICS}

is the collection of independent sets of a matroid on 5. Proof. We have already proved in Theorem 10.3 that the rank function of a matroid satisfies R-l, R-2, and R-3. To prove the converse, suppose that ρ satisfies R-l, R-2, and R-3. It follows from R-l that p ( 0 ) = 0. So 0 E # , and axiom 1-1 is satisfied. Let BE3 and ACB. Then by R-3, we get p(A) + p(B - A) > p(A U (Β - A)) + p(A η (Β - A)) = p(B) + p ( 0 ) = |β|.

(10.3)

If p ( A ) < | A | , then by R-l, p(A) + p(B - A)<\A\

+ \B - A\

"1*1, contradicting (10.3). Hence p(A) = \A\, and thus A is also a member of J, thereby satisfying axiom 1-2. To prove 1-3, first note that R-2 and R-3 imply the following: R-3'. If p(A U ^ ) = p(A Uy) = p(A), then p(A U χ U y ) = p(A).

EQUIVALENT AXIOM SYSTEMS

275

Now let X and Y be two distinct members of $ with \X\ = k and | Y| = k + 1. Suppose that, for every y Ε Υ - X, p(X Uy) = k. Then by repeated application of R-3' we get p(XD(Y-X))

= p(XU Y) = k,

contradicting that p(Y) = k + 1. So there exists an element yEY-X the property p(XL) y) = k + 1, and thus axiom 1-3 is satisfied. •

with

Theorem 10.9 (Circuit Axioms). Let <β be the set of circuits of a matroid on a set S. Then: C-l. 0 ^ ^ and no set in contains another properly. C-2. If Cj, C G C, Φ C , and χG C, Π C , then there exists such that C C (C, U C ) 2

2

3

CE € <

2

J

2

Conversely, if is a collection of subsets of a set 5 satisfying C-l and C-2, then the set 3 = {l\C$Zl, for all C G €) is the collection of independent sets of a matroid on 5. c

Proof. We have proved earlier that the circuits of a matroid satisfy C-l and C-2 (see Theorem 10.4.) Given satisfying C-l and C-2, we now show that the set $ denned as in the theorem is the collection of independent sets of a matroid on S. We do so by showing that 3> satisfies axioms 1-1, 1-2, and 1-3' (see Theorem 10.6.) First note that $ is the collection of all the subsets of 5 that do not contain any member belonging to %. Therefore it satisfies axioms 1-1 and 1-2. To prove that $ satisfies axiom 1-3', consider any subset A of 5. Let 5, and 5 be distinct maximal subsets of A that belong to Assuming that | 5 | > | S , | , we establish a contradiction. Since Sj and S are distinct, 5, - Ξ Φ0 and S - S, Φ 0. Let χ G S - 5 , . Then clearly S^x^^. Hence there exists C £ « with xECCS Ux. Further, C Π (5, - S ) Φ 0, as otherwise C C 5 , contradicting that 5 £ $>. Let then x' G C Π (S, - 5 ) and 5 = (5, - x') U x. Note that | 5 | = | 5 , | . We first prove: 2

2

2

2

2

2

x

2

2

2

3

2

3

1. 5 G J f . 3

Clearly S, - x' G 3. If S £ Ά then there exists C'E<€ with χ Ε C C S ES \Jx. Further C Φ C as x'gC. Thus C and C are distinct members of % such that 3

3

x

(a) xECnC. (b) C C S , U x and C ' C J . U x .

276

MATROIDS

Then it follows from C-2 that there exists C"E <€ with C " C ( C U C ) - χ C Sj Ε 3. This contradicts that no member of 3 contains a member of Therefore, S G 3. 3

We next prove the following: 2. S E3 is a maximal subset of A. 3

If not, let 5 ' G 3 be a maximal subset of A with S C 5'. Now x' 0S', as otherwise S, C S \ contradicting the maximality of 5, in A. Then S ' U J C ' ^ J * . Hence there exists C " ' G < £ with JC' G C"'C 5 ' U J C ' . Further, C " n ( 5 ' - S ) ^ 0 , as otherwise C " ' C S , . Let J C " G C " ' n ( S ' 5,). Then we can show (as in the proof of 1) that ( 5 ' - JC") U Χ' Ε 3. But S, C ( 5 ' - JC") U Χ' Ε 3, contradicting the maximality of 5, in A. Thus S is a maximal subset of A. 3

1

3

Note that S is constructed from S, by replacing an element JC' G 5, - 5 by an element JC G 5 - 5,. Since | 5 | > l i j , we can repeat this construction a finite number of times to obtain a maximal subset S of A with S Ε 3 and S„ C S , a contradiction. • 2

3

2

2

n

n

2

10.4

MATROID DUALITY AND GRAPHOIDS

In this section we first introduce the concept of matroid duality and discuss some results relating a matroid and its dual. We then establish the "painting" theorem, which would lead to the definition of a graphoid. We conclude with the development of Minty's self-dual axiom system for matroids, which is based on graphoids. We begin with a theorem due to Whitney [10.1], which defines the dual of a matroid. Theorem 10.10. Let 38 be the set of bases of a matroid Μ on a set S. Then 3 8 * = {S - B\B G 3 8 } is the set of bases of a matroid M* on S. Proof. Clearly, 3 8 * satisfies axiom B-l. To show that axiom B-2 is also satisfied, consider any two members B* and B\ of 9 8 * with B* = S - B and B* = S - B . Let xE Β* — B*. Then χ Ε B — B . By Exercise 10.4 there exists yEB -B such that B = (B, - y) U JC is a base of Μ. Now y G Β* - B* and 1

2

x

2

2

1

3

(B*-x)Uy

= = 5-β

S-((B,-y)Ux) 3

= β*.

Thus axiom B-2 is satisfied and hence 38 * is the set of bases of a matroid M* on 5. •

MATROID DUALITY AND GRAPHOIDS

277

The matroid M* denned in Theorem 10.10 is called the dual matroid or simply the dual of M. It is easy to see that Μ is the dual of Μ *. So we shall refer to Μ and M* as dual matroids. The circuit matroid and the cutset matroid of a graph defined in Section 10.1 may now be seen to be dual matroids. A base of M* is called a cobase of Μ. Similarly a circuit of M* is a cocircuit of M, a loop of Μ * is a coloop of M, the rank function of M* is the corank function of M, and so on. The rank function of Μ * is denoted by p*. If Β is a base of M, then the cobase S - Β is denoted by B*. The collection of cobases of Μ is denoted by 38*(Λί) and the collection of cocircuits is denoted by <£*(M). Now we establish some results relating a matroid and its dual. Many of these results are well known in graph theory (see Chapter 2). It is clear from the definition of a dual that (10.4)

p*(Af)=|S|-p(Af).

The following theorem generalizes this relationship between ρ and p*. Theorem 10.11. If Μ and M* are dual matroids on a set S, then for all ACS p*(A) = \A\ +

(10.5)

p(S-A)-p(M).

Proof. Consider ACS and let β* be a base of M* such that |J3* Π is maximum. Then β is a base of Μ with \B Π (5 - A)\ maximum. It follows from the definition of a rank function that p*(A) =

(10.6)

\B*DA\

and p(S - A)

(10.7)

\BD(S-A)\.

Now \B*nA\

\A\-\BHA\

and BH(S-

Α)\ =

\Β\-\ΒΠΑ\

= ρ(Μ)-\ΒΠΑ\

.

278

MATROIDS

So we get from (10.6) and (10.7) p*(A) = \A\ + p(S-A)-p(M).

Μ

Note that for each statement involving the circuits, bases, and so on, of a matroid there is a dual statement involving the cocircuits, cobases, and so on, of the matroid. This is because the cocircuits, cobases, and so on, are themselves are circuits, bases, and so on, of a matroid. In all the lemmas and theorems of this section we include the dual statements wherever applicable. The proofs of these dual statements follow in the obvious manner. Lemma 10.1 Let Μ be a matroid on a set S. If A C S is independent in M, then 5 - A contains a cobase of M. Dually, if A* C S is independent in M*, then S - A* contains a base of M. Proof. There exists a base Β of Μ with A C B. So S - A contains the corresponding cobase B*. • Theorem 10.12. Let Μ be a matroid on a set S. If A and A* are subsets of S with AD A* = 0, A independent in M and A* independent in Λί*, then there exists a base β of Μ with AC Β and A* CB*. Proof. By the previous lemma S- A* contains a base of M. Since AC S - A* is independent in M, there exists, by Theorem 10.2, a base Β of Μ with AC BC S -A*, and hence A* C B*. • Lemma 10.2. Let Μ be a matroid on a set S. Then each base Β of Μ has nonnull intersection with every cocircuit of M. Dually, each cobase of Μ has nonnull intersection with every circuit of M. Proof. If, for any cocircuit C* of M, C* Π Β = 0 , then Β* = S - Β would contain C*, contradicting the independence of B* in M*. • Let β* be any cobase of a matroid on a set S. By the dual of Corollary 10.4.2, for any χ Ε Β, Β* U χ contains exactly one cocircuit. This cocircuit is called the fundamental cocircuit of χ with respect to B. Note that this cocircuit contains exactly one element from B, namely, the element x. Lemma 10.3. Let Μ be a matroid. For any independent set A of Μ there exists a cocircuit having exactly one element from A. Further, if \A\< p(M), then there exists a cocircuit having null intersection with A. Dually, for any independent set A* of M* there exists a circuit having exactly one element from A*. Further, if \ A*\< p*(M*), then there exists a circuit having null intersection with A*.

MATROID DUALITY AND GRAPHOIDS

279

Proof. Let β be a base of Μ with A C B, and let C* be the fundamental cocircuit of an element χ Ε Β with respect to B. Then C* Π A = {x} if χ Ε A, and C* Π A = 0 if jr G β - Λ. Thus follows the proof. • Theorem 10.13. Let Μ be a matroid on a set 5. A subset A"of 5 is a base of Μ if and only if A" is a minimal subset having nonnull intersection with every cocircuit of M. Dually, a subset X* of 5 is a cobase of Μ if and only if it is a minimal subset having nonnull intersection with every circuit of M. Proof. Necessity follows easily from Lemmas 10.2 and 10.3. To prove the sufficiency, let A' be a minimal subset of 5 having nonnull intersection with every cocircuit of Μ. Then S — X is a maximal subset having no cocircuit of Μ. So, by definition, 5 - X is a cobase of M, and hence A is a base of Μ. • r

Next we obtain some new characterizations of circuits and cocircuits. Theorem 10.14. Let Μ be a matroid on a set S. A subset AOf S is a circuit of Μ if and only if it is a minimal subset having nonnull intersection with every cobase of M. Dually, a subset X* of 5 is a cocircuit of Μ if and only if it is a minimal subset having nonnull intersection with every base of M. Proof. See proof of Theorem 2.13.

•

Let C* be a cocircuit of a matroid Μ on 5. By the previous theorem, C* is a minimal subset having nonnull intersection with every base of M. So 5 — C* is a maximal subset containing no base of M. Thus we have the following. Lemma 10.4. If C* is a cocircuit of a matroid Μ on a set 5, then for any χ G C*, (S — C*) U χ contains a base Β of M. Dually, if C is a circuit of a matroid Μ on a set S, then for any χ EC, (S - C)Lix contains a cobase of M. • Theorem 10.15. Let Μ be a matroid on a set S. A subset AOf 5 is a circuit of Μ if and only if it is a minimal subset with l A T l C l ^ l for every cocircuit C* of M. Dually, a subset A'* of 5 is a cocircuit of Μ if and only if it is a minimal subset with \X* Π C| Φ 1 for every circuit C of M. Proof Necessity Proof is by contradiction. For any proper subset C of a circuit C, there exists, by Lemma 10.3, a cocircuit C* such that |C* Π C'\ = 1. Suppose that |C Π C*| = 1 for some circuit C and some cocircuit C*, and

280

MATROIDS

let C Π C* = {*}. Consider now 5 ' = 5 - C* and C = C - x. Clearly, C C 5'. By Lemma 10.4, S' U χ contains a base. Let then Β C 5 ' U χ be a base with C C β. Note that xEB. So the circuit C = C U χ is contained in B, a contradiction. Sufficiency If A is a subset of S with | Α Π C*| ^ 1 for every cocircuit C*, then A should contain a circuit, for otherwise X would be independent in M, and so by Lemma 10.3 there would exist a cocircuit having exactly one element from X. Let then C be a circuit contained in A". It is clear from the necessity part of the theorem that |C Π C*| ^ 1 for every cocircuit C*. Therefore A = C, for otherwise the minimality of A would be contradicted. • We now introduce the notion of a painting of a finite set 5 and establish the "painting" theorem. A painting of a set S is a partitioning of S into three subsets R, G, and β such that \G\ = 1. For easy visualization we regard the elements in R as being "painted red," the element in G as being "painted green," and the elements in Β as being "painted blue." Theorem 10.16 (Painting Theorem). Let Af be a matroid on a set S. For any painting of S, there is either 1. a circuit C of Μ consisting of the green element and no blue elements or 2. a cocircuit C* of Μ consisting of the green element and no red elements. Proof. We shall assume that the green element is in some circuit, for otherwise it will be a coloop (why?), and the theorem will follow trivially. Suppose, for some painting of S, statement 1 is not true. Then consider the subset S' = RU G. Clearly, any circuit contained in S' consists of only red elements. Let R' be a minimal subset of red elements such that the set S" = S' - R' contains no circuit, and let β be a base with S"CB. Then we can easily see that the fundamental circuit of any element yER' with respect to Β consists of only red elements. Now, let C* be the fundamental cocircuit of the green element with respect to B. Suppose C* contains a red element x. Then xER' and |C* Π C \ = 1, where C is the fundamental circuit of χ with respect to B. This contradicts Theorem 10.15, and hence C* is a cocircuit containing the green element and no red elements. Thus statement 2 is true. • x

x

Theorems 10.15 and 10.16 lead us to the definition of a graphoid introduced by Minty [10.2]. A graphoid is a structure (5, ^,3)) consisting of a finite set S and two

MATROID DUALITY AND GRAPHOIDS

collections conditions:

281

and 3 of nonempty subsets of S satisfying the following

G-l. If C e ^ a n d DE3, then | C n D | * l . G-2. For any painting of 5, there is either a. a member of elements or b. a member of elements. G-3. No member of no member of 3

% consisting of the green element and no blue 3 consisting of the green element and no red contains another member of properly; contains another member of 3 properly.

Theorem 10.17. Let Μ be a matroid on a set 5. Then (5, <€(Μ), <€*(M)) is a graphoid. Proof. Follows from Theorems 10.15 and 10.16.

•

We now proceed to establish the converse of Theorem 10.17. Theorem 10.18. Let ( 5 , ^ , 2 ) ) be a graphoid. Then BCS is a maximal subset containing no member of % if and only if Β contains no member of % and S - Β contains no member of 9). Proof Necessity Let β be a maximal subset of 5 containing no member of <€. Suppose S - Β contains a member D of 3. Let χ Ε D. Then Β U χ contains a member C of and χ EC. So CC\D = {x}, contradicting G-l. Thus S - Β contains no member of 3). Sufficiency Let Β C 5 contain no member of ^ and S - Β contain no member of 3). We now show that, for any xESΒ, Β Ux contains a member of <€. Consider a painting of 5 in which all the elements of Β are red, χ is green, and all the remaining elements of S - Β are blue. Clearly, there is no member of 3 containing only green and blue elements. Thus by G-2, BU χ contains a member of '4. • Theorem 10.19. Let (5, containing no member of no member of 3.

3) be a graphoid. If β is a maximal subset of 5 then 5 - Β is a maximal subset of S containing

Proof. Follows from Theorem 10.18 and its dual.

•

Theorem 10.20. Let ( 5 , <β, 3) be a graphoid. Then % is the collection of circuits of a matroid Μ on 5 and 3 is the collection of cocircuits of M.

282

MATROIDS

Figure 10.2 Proof. We first show that <
2

2

2

2

3

3

2

2

x

2

2

The equivalence of the graphoid axiom system and the circuit axiom system follows from Theorems 10.17 and 10.20. This result is due to Minty [10.2] who has given an elegant exposition of the theory of matroids based on graphoids. 10.5

RESTRICTION, CONTRACTION, AND MINORS OF A MATROID

Consider a graph G with edge set E. Relative to any subset Τ of E, we can define two graphs denoted by G\T and G- T. The graph G|T, called the restriction of G to T, is obtained from G by deleting (open-circuiting) the edges belonging to Ε - Τ and then removing the resulting isolated vertices. In fact G\T is the induced subgraph of G on T. The graph G · T, called the contraction of G to T, is obtained by contracting the edges belonging to

RESTRICTION, CONTRACTION, AND MINORS OF A MATROID

283

W

Figure 10.3. (a) Graph G. (b) G\T, restriction of G to T; T= {e , e , e , e }. (c) G · T, contraction of G to T. x

6

7

g

Ε — T. Figure 10.3 shows a graph G and the graphs G\ Τ and G · Τ for Τ — {e , e , e , Now it is easy to verify the following: x

6

7

1. A subset X of edges is an acyclic subgraph of G\ Τ if and only if it is an acyclic subgraph of G. 2. A subset X of edges is an acyclic subgraph of G · Τ if and only if there exists a subset Y of Ε - Τ such that Y is a spanning forest of G\(E — T) and XU Y is an acyclic subgraph of G. These ideas motivate the introduction of two submatroids induced on a matroid by a subset of the elements of the matroid. If J(M) is the set of independent sets of a matroid Μ on S and Τ C S, let f(M\T)

= {X\X QT,XE

J(M)}

.

We can easily show that J(M\ T) is the set of independent sets of a matroid

MATROIDS

284

on Τ. This matroid is denoted by Μ \ Τ and is called the restriction of Μ to T. The following observations are obvious: 1. A C Τ is independent in M\T if and only if X is independent in M. 2. A C Γ is a circuit of M\T if and only if it is a circuit of M. 3. If λ is the rank function of M\T, then for any A C T , A(A) = p ( A ) .

(10.8)

4. If M(G) is the circuit matroid of a graph G with edge set E, then for any ΤC E, M(G)\T = M(G\T). Next we define the submatroid Μ · Τ. Let 3(M · T) be the collection of all those subsets X of Τ such that there exists a maximal independent subset Y of 5 - Γ with A U y e i ( M ) . Theorem 10.21. J(M · T) is the collection of independent sets of a matroid on T. Proof. Let A, and A be two members of $(Μ·Τ) such that | A | = I A, I + 1. Then there exist maximal independent subsets 7, and Y of S - Τ with A ' = A, U y, and A" = A U Y independent in Μ. Now |A"| = |A'| + 1. So there exists, by axiom 1-3, an element χ EX"- X' such that A ' U χ is independent in M. Note that A" - A ' = (A - A,) U (Y - Y ). Further, χ e' y - y, because y, is a maximal independent subset of S - T. So χ ε A - A, and (A, U JC) C Τ is a member of J?(M · T). Thus ^(Λί · Γ) satisfies axiom 1-3. It is easy to see that axioms 1-1 and 1-2 are also satisfied. • 2

2

2

2

2

2

2

x

2

2

The matroid defined in Theorem 10.21 is called the contraction of Μ to Τ and is denoted by Μ · Τ. Note that if M(G) is the circuit matroid of a graph G with edge set Ε then, for any Τ C E, M(G · T) = M(G) • T. Let ρ be the rank function of Μ • Τ. Then from the definition of Μ · Τ we get, for any ACT, T

(10.9)

p (A) = p(AU(S-T))-p(S-T). T

We observed in Chapter 7 that open-circuiting or removing of an edge and the contracting of an edge of a graph are dual operations. In particular, if G and G' are dual graphs and Τ and T' are the corresponding subsets of edges of G and G', respectively, then G\T is the dual of G'-T' (see Theorem 7.10). The following theorem is the matroid analog of this relationship. Theorem 10.22. If Μ is a matroid on S and TCS,

then:

REPRESENTABILITY OF A MATROID

285

1. (M\T)* = M* • T. 2. (M-T)* = M*\T. Proof 1. Let A be the rank function of M\T and A* be the rank function of (M\T)*. Then from (10.5) we have, for any XC.T, X*(X) = \X\ X(T) + p(T-X). Let ρ be the rank function of Μ · Τ and (p*) be the rank function of Μ * · Τ. Then from (10.9) and (10.5) we get T

(p*) (X) T

= p*(X U (5 - T)) + p*(S - T) =

p*(S-(T-X))-p*(S-T)

= \S\-\T\

+ \X\-p(S)

-(\S\-\T\-p(S) = \X\-p(T)

+

p(T-X)

+ p(T))

+

p(T-X)

= |A'|-A(7') + A ( 7 - A ' ) ,

by (10.8)

,

= λ*(Α-). Since (Λ/| Τ)* and Μ* · Τ have the same rank function, we get (M\T)* = Μ* · Τ . 2. Putting Μ for M* in 1 and taking the duals, we get 2.

•

If Μ is a matroid on 5 and Τ C 5, a matroid Ν on Τ is called a minor of Μ if Ν is obtained by a succession of restrictions and/or contractions of M. This topic has been extensively developed by Tutte [10.3]. Using the terminology developed thus far, we can state in matroid terminology many of the results relating to planar graphs and dual graphs. For example, Theorem 7.5 can be stated as follows: A graph is planar if and only if the circuit matroid M(G) of the graph does not contain either of the matroids M(K ) or M(K ) as a minor. 5

10.6

33

REPRESENTABILITY OF A MATROID

Let Μ be a matroid on a set 5. Let Β = {6,, b ,..., b ) be a base of Μ and let B* = S - Β = {e,, e ,..., e ) be the corresponding cobase of M. Recall that Β* Π C(e ) = e j = 1 , . . . , q, where C(e ) is the fundamental circuit of 2

2

y

r

q

;

r

286

MATROIDS

e Similarly Β Π C*{b ) = b , j = 1 , . . . , r, where C*(/> ) is the fundamental cocircuit of b . As in the case of undirected graphs (Chapter 6), we may now define the fundamental cocircuit matrixL D and the fundamental circuit matrix L / ^ a i i u l i l t j unuurnctuui ttrtuti ΙΠΜΙΙΛ D* of Μ with respect to the base Β Clearly, these two matrices will be of the form r

t

;

}

f

f

D], where U is a unit matrix whose columns correspond to the elements of B. A matrix of this form is called a standard representation of Μ with respect to the base B. Now consider a graph G. Suppose we arbitrarily assign orientations to the edges of G and let G' be the resulting directed graph. Then by Theorem 6.9 the incidence matrix of G' is a representation of the circuit matroid M(G) over any field F. We can also see (Theorem 6.10) that any fundamental cutset matrix of G' is a standard representation of M(G) over any field F. Further by Theorem 6.11 any fundamental circuit matrix of G' is a standard representation of the cutset matroid M*(G) of G over any field F. Thus we have the following. Theorem 10.23. The circuit matroid and the cutset matroid of a graph G are both representable over any field F. • We now prove the following main theorem of this section. Theorem 10.24. If a matroid Μ on 5 is representable over a field F, then the dual matroid M* on S is also representable over F. Proof. Suppose Μ is of rank r and has η elements. Let the r x η matrix A be a representation of Μ over F. Let X be the set of all those column vectors χ such that Ax = 0; AT is called the null space of A. It is well known in linear algebra that the dimension of X is equal to η - r. Now select from X a set of η - r linearly independent column vectors, and with these vectors as columns form an η x (η - r) matrix B. Note that AB = 0. We now show that B' is a representation over F of the dual matroid M*. We do so by proving that any r columns of A are linearly dependent if and only if the complementary set of η - r columns of B' are linearly dependent. For this purpose we shall choose the first r columns of A. Clearly, this involves no loss of generality. The first r columns of A are linearly dependent if and only if there exists a nonzero column vector χ = [ J C , , x ,...,x , 0, 0 , . . . ,0]' belonging to X. Such a vector xGX exists if and only if there exists a column vector y Φ0 2

r

BINARY MATROIDS

287

with η - r entries and such that x= By . Now writing Β as

β] where β, is r x ( « - r) and B is (n - r) x (n - r), we can see from the above equation that B y = 0. Since y Φ 0, it follows that B is singular. Thus the rows of B , and hence the last η - r columns of B' are dependent, and the theorem is proved. • 2

2

2

2

An easy corollary of Theorem 10.24 is the following result. Corollary {s , s ,... l

2

10.24.1. Let Μ be a matroid of rank ,s }. If Μ has the standard representation

r on a set 5 =

n

l

S

2

S

[

' ' ' r

r+1

S

Ur

n

' ' '

S

I

S

Λ

],

then M* has the standard representation l 2 ''' r

S

[

S

r+l ' ' ' n

S

-A'

S

I

where U is the kx k identity matrix. k

S

u -A n

•

The relationship (6.13) between the fundamental circuit matrix and the fundamental cutset matrix of a connected graph follows easily from the corollary. For further discussions on the matroid representability problem see Welsh [10.4].

10.7

BINARY MATROIDS

A matroid is binary if it is representable over GF(2), the field of integers modulo 2. Clearly, the circuit matroid M(G) and the cutset matroid M*(G) of a graph G are binary. We have proved in Theorem 4.6 that each circuit of M(G) can be expressed as a ring sum of some of the fundamental circuits of

288

MATROIDS

G. A similar result is true in the case of the cutsets of G. This property of circuits and cutsets (cocircuits) holds true in the case of any binary matroid. However, it is not, in general, true in the case of arbitrary matroids. As an example, let S = {1,2,3,4} and let M have as its circuits all subsets of three elements of S. Then the ring sum of the circuits {1,2,3} and {1,2,4} is {3,4}, which is an independent set of M. In this section we establish several properties of a binary matroid, which would lead us to the development of alternative characterizations of such a matroid. Let M be a matroid on a set S. Let B = {/>,, b2,..., br) be a base of M and let B* = S - B = {ex, e2,..., eq} be the corresponding cobase of M. Recall that B* Π C(et) = ey, /' = 1 , . . . , q, where C{ej) is the fundamental circuit of er Similarly B Π C*(fty) = ft;, / = 1, , r, where C*(fty) is the fundamental cocircuit of br As in the case of undirected graphs (Chapter 6), we may now define the fundamental cocircuit matrix Df and the fundamental circuit matrix D* of M with respect to the base B. Clearly, these two matrices will be of the form b1b2--br Df = [

D}=[

ele2---eq

Ur

|

F

],

G

I

I/,

].

Note that Df = [d,t] and D* = [d,*] are both (0,1) matrices. Suppose the matrix M is binary. Then it has a standard representation over GF(2) of the form bxb2---br

[

exe2---eq

U,

|

A

].

(10.10)

Let the fundamental circuit C(ey) be {e;, bx, b2,...,bk). Then the modulo 2 sum of the columns of the matrix in (10.10) corresponding to the elements of C(e;) is zero. Since the modulo 2 sum of vectors corresponds to the ring sum of the corresponding sets, we can see that _fl, »-[θ,

α

if ft, e CÎ*,). ìib&Qe,).

In other words A = G'.

(10.11)

Thus the matrix [Ur G'] is a standard representation over GF(2) of M with respect to the base B. Starting from a standard representation for the dual matroid M *, we can

BINARY MATROIDS

289

show in a similar manner that [F' U ] is a standard representation over GF(2) of M*. Since a matroid has a unique standard representation with respect to a given base, it follows from Corollary 10.24.1 that q

(10.12)

F=G'. Thus we have proved the following theorem. Theorem 10.25. Let Μ be a binary matroid on a set S.

1. The fundamental cocircuit matrix of Μ with respect to any base is a standard representation of M. 2. The fundamental circuit matrix of Μ with respect to any base is a standard representation of M*. • From (10.12) we also get the following important result. Theorem 10.26. Let Μ be a binary matroid. Let D and D* be the fundamental cocircuit matrix and the fundamental circuit matrix of Μ with respect to a common base. Then f

= o.

D (D*y f

(io.i3)

m

Given a circuit C of a matroid M, we can associate with C a (0,1) row vector, each entry of which corresponds to an element of M, and an entry in this vector is 1 if the corresponding element is in C. For example, the rows of D* are the vectors that correspond to the fundamental circuits. The matrix of all the circuits vectors of Μ is called the circuit matrix of Μ and is denoted by D*(M). In a similar way the cocircuit matrix D(M) of Μ is defined. Suppose χ is the vector corresponding to a circuit C of a binary matroid M. Since D is a standard representation of M, the ring sum of the columns of D corresponding to the elements of C is zero. In other words f

f

(10.14)

D x' = 0. f

Similarly if χ is a cocircuit vector, we have (10.15)

D* x' = 0. f

Note that we have assumed that the columns of D *, D , and χ are arranged in the same element order. f

Theorem 10.27. Let Μ be a binary matroid on S. For any base Β and circuit

MATROIDS

290

C of M, if C - Β = { J C , , x ,..., JC, with respect to B, then

x } and if C ( J C , ) is the fundamental circuit of

2

k

C = C ( J C , ) Φ C ( J C ) Φ · · · Φ C(JC J

.

2

Proof. Proof is based on (10.12) and (10.14). See proof of Theorem 6.7. • To prove the converse of Theorem 10.27, we need the following. Theorem 10.28. Let Μ be a matroid. For any base Β and circuit C of M, let C = C ( J C , ) Φ C ( J C ) Φ · • · Φ C ( J C J , where {JC,, J C , . . . , J C * } = C - Β and C ( J C , ) is the fundamental circuit of JC, in B. Then the ring sum C , φ C of any two distinct circuits of Μ contains a circuit of M. 2

2

2

Proof. Suppose C , φ C does not contain a circuit and let C , Π C = {JC, , x ,...,x }. Then C , ® C = ( C , U C )- { J C , , . . . ,jc } is independent in M. So there exists a base β D C , φ C . But then C - B = C - B = { J C , , . . . , jc }, so that by the hypothesis of the theorem, 2

2

p

2

p

2

2

2

x

2

p

C , = C ( J C , ) © C ( J C ) Φ · · · Φ C ( J C ) = C2 2

contradicting the fact that C\ ¥= C .

P

'

•

2

Theorem 10.29. Let Μ be a matroid. For any base Β and circuit C of M, let C = C ( J C , ) © C ( J C ) ® - - - ® C ( x ) , where {JC,, J C , . . . ,x } = C - B a n d C ( J C , ) is the fundamental circuit of JC, in β. Then Μ is binary. 2

k

k

2

Proof. Let Β = {6,, b ,..., b ) be any base of Μ and let S-B = {e,, e , . . . , e }. The fundamental circuit matrix D*(M) of Λί with respect to B is of the form 2

r

2

b b --b x

{

2

f

Λ

We prove the theorem by showing that the matrix b, b< ---b 2

e e ---e

U

r

r

l

I

2

q

A'

(10.16)

}.

is a standard representation of Μ over GF(2). Let C be a circuit of M. If C-B = {e , e hypothesis of the theorem H

e, }, then by the ,...,e, k

BINARY MATROIDS

291

c=c(e .)ec( , )®---ec(e ). I

e 2

l4

It now follows from the definition of A that the modulo 2 sum of the columns of the matrix in (10.16) corresponding to the members of C is zero. Thus these columns are linearly dependent over GF(2). Suppose W= { f t , , . . . , b„ e , , . . . , e } is a circuit of the matroid M' induced on S by the linear dependence of the corresponding column vectors of the matrix in (10.16). Then p

W= C ( e , ) © C(e )θ· 2

· · θ C(e ) . p

By Theorem 10.28, W contains a circuit C" of Μ. But C cannot be a proper subset of W, for then it would contradict that W is a circuit of Μ'. Thus W= C. Hence the matrix in (10.16) is a representation of Μ over GF(2). • Further characterizations of a binary matroid are given in the following theorem. Theorem 10.30. For a matroid Μ the following statements are equivalent: 1. For any circuit C and any cocircuit C* of M, \CD C*\ is even. 2. The ring sum of any collection of distinct circuits of Μ is the union of disjoint circuits of Μ. 3. For any base Β and circuit C οί M, if C - Β = {e,, e ,..., e ] and C(e,) is the fundamental circuit of e, in the base B, then C= C( )®C(e )®---®C(e ). 4. Μ is binary. 2

ei

2

q

Proof 1φ2 Let C,, C ,...,C be distinct circuits of M, and let A = C ®C ®---®C . We may assume without loss of generality that A contains no loops. Since for any cocircuit C*,\C* Π C,\ is even for 1 ^ i ^ k, it follows easily that Π C*| is also even. Suppose that A is not dependent. Then a contradiction results because by Lemma 10.3 there exists a cocircuit having exactly one element from A. Thus A is dependent and contains a circuit C. If A = C, the theorem is proved. If not, let A = A ® C. Note that for any cocircuit C*, \ C* Π Α \ is even. So we may now repeat the argument on A,. Since A, is finite and A, = A - C, this process will eventually terminate, yielding a finite collection of disjoint circuits whose union is A. 2

l

2

k

k

1

x

λ

2φ3

See proof of Theorem 4.6

3=p4

Same as Theorem 10.29.

292

MATROIDS

4φ1 By Theory 10.27 each circuit C can be expressed in terms of fundamental circuits as C = C{x )Θ C(x )Θ · · · Θ x

2

C(x ). k

By (10.15), |C* Π C(x,)\ is even for all 1 < i < k. So it now readily follows that |C* Π C| is even. • Obviously, alternative characterizations of a binary matroid in terms of cocircuits can be obtained by replacing circuits by cocircuits in Theorem 10.30.

10.8

ORIENTABLE MATROIDS

A matroid Μ is orientable if it is possible to assign negative signs to some of the nonzero entries in the circuit matrix D* = D*(M) and in the cocircuit matrix D = D(M) in such a way that the inner product of any row of the signed circuit matrix with any row of the signed cocircuit matrix is zero over the ring of integers. Clearly, the circuit matroid and the cutset matroid of a graph are both orientable. A painting of an orientable matroid Μ is a partitioning of the elements of Μ into three sets R, G, and Β and the distinguishing of one element of the set G. We can visualize this as coloring of the elements of Μ with three colors, each element being painted red, green, or blue, and exactly one green element being colored dark green. Note that the dark green element is also to be treated as a green element. The main result of this section is the "arc coloring lemma" due to Minty [10.2]. Theorem 10.31 (Arc Coloring Lemma). Let Μ be an orientable matroid. For any painting of the elements of M, exactly one of the following is true: 1. There exists a circuit containing the dark green element but no blue elements, in which all the green elements are similarly oriented (i.e., all have the same sign in the signed circuit matrix). 2. There exists a cocircuit containing the dark green element but no red elements, in which all the green elements are similarly oriented. Proof. Proof is by induction on the number of green elements. If there is only one green element, the result follows by axiom G-2. Suppose the result is true when the number of green elements is m. Consider then a painting in which there are m + 1 green elements. Choose a green element χ other than the dark green element (see Fig. 10.4).

ORIENTABLE MATROIDS

293

Figure 10.4 Color the element Λ: red. In the resulting painting there are m green elements. If now there is a cocircuit of type 2, the theorem is proved. Suppose we color χ blue. If in the resulting painting there is a circuit of type 1, the theorem is proved. Suppose neither of the two occurs. Then by the induction hypothesis we have the following: a. There is a cocircuit of type 2 when χ is colored blue. b. There is a circuit of type 1 when χ is colored red. Now let the corresponding rows of the signed circuit and cocircuit matrices be as shown: Cocircuit Circuit

dg +1 +1

R 00---0 0 -1 1 - 0 - 1

Β 1 - 1 · ·-01 0 0-00

G 11-10 01--10

χ ? ?

Here we have assumed without loss of generality that +1 appears in the dark green position of both vectors. By definition, the inner product of these two vectors is zero. The contribution to this inner product from the dark green element is 1; from all the red and blue elements, zero; from the green elements, a nonnegative integer p; and from x, an unknown integer q, which must be 0, 1, or —1. Thus we have 1 + ρ + q = 0. This equation is satisfied only for ρ = 0 and q = - 1 . Therefore in one of the two vectors the question mark is + 1 and in the other it is - 1 . Choosing the vector in which the question mark is 1, we get the desired circuit or cocircuit. Thus either statement 1 or statement 2 occurs. Both cannot occur simultaneously, for then the inner product of the corresponding circuit and cocircuit vectors will not be zero. • The arc coloring lemma in the special case of graphs ([10.5] and [10.6]), is obvious.

294

10.9

MATROIDS

MATROIDS AND THE GREEDY ALGORITHM

Consider a set S whose elements s, have been assigned nonnegative weights w(s,). The weight of a subset of S is defined as equal to the sum of the weights of all the elements in the subset. Let J be a collection of subsets of S. Many problems in combinatorial optimization reduce to the following problem: Determine a maximum weight member of For example, finding a maximum weight spanning tree of a weighted connected graph G reduces to the above problem, where J is the collection of all spanning trees of G. The following algorithm, called the greedy algorithm, naturally suggests itself to solve such a combinatorial optimization problem. Algorithm 10.1 The Greedy Algorithm 51. Choose an element i , such that { i , } G $> and such that H > ( i , ) > w(s) for all s with {s} G J. If no such J, exists, stop. 52. Choose an element s such that {J, , i } G $ and such that w(s ) a w(s) for all s Φ s with {s,, s) G If no such element s exists, stop. Sk. Choose an element s distinct from s , s ,... ,s _ such that {i,, s , . . . , s _ , s ) G $ and such that w(s ) is a maximum over all such s. If no such s exists, stop. • 2

2

2

x

2

k

2

k

l

x

k

2

k

x

k

k

Clearly, when the greedy algorithm terminates, it will have picked a maximal member of 3>. But this member may not be of maximum weight in For example, if S = {a, b,c, d) , tv(a) = 4 ,

w(b) = 3,

w(c) = 2,

w(d) = 2,

and J = {{a},{a,

c),{b,

c, d),{b,

d}},

then the algorithm will pick {a,c}. But {b, c, d) is a maximum weight member of S. However, suppose we modify the weights so that w(a) = 6,

w(b) = 3,

w(c) = 2,

w(d) = 2 .

The algorithm will again pick {a,c}, which is now a maximum weight member of J. Next we investigate the relationship between the greedy algorithm and the structure of J.

MATROIDS AND THE GREEDY ALGORITHM

295

Consider a matroid Μ on a set S. Let $ be the collection of independent sets of M. Let Λ = {<*!' « 2 . ·

• • . « « }

and

/ = {ft,,ft ,...,ft„} 2

2

be two independent sets whose elements are arranged in the order of nonincreasing weights. Thus w(a ) s: w(a ) > · · · > w(a ) and w(ft,) > w(b ) > · · · > »v(i» ). Then / , is lexicographically greater than / if there is some k such that w(a,) = vf(ft,) for 1 £ i s k - 1 and > w(b ) or else νν(α,) = w(ft,) for 1 n. A set that is not lexicographically less than any other set is said to be lexicographically maximum. From this definition it should be clear that a lexicographically maximum independent set must be a base. A set Β Ε J is Gale optimal in 3> if for every / ε 3> there exists a one-to-one correspondence between / and Β such that, for all a El, w ( a ) < w(ft), where b is the element of Β that corresponds to a. Clearly, only bases can be Gale optimal. Also if a base is Gale optimal, then it must have maximum weight. In the following we assume that the elements of each set are arranged in the order of nonincreasing weights. x

2

m

n

2

2

k

1

Theorem 10.32. Let β be the collection of independent sets of a matroid on S and Β a base of M. For any nonnegative weighting of the elements of S, the following are equivalent: 1. β is lexicographically maximum in J. 2. Β is Gale optimal in β. 3. β is a maximum weight member of β. Proof 1=£>2 Let β = {ft,, b ,..., b ) be a lexicographically maximum base of M. Suppose β is not Gale optimal. Then there exists an independent set / = {a,, a ,...,a ) such that νν(α,) = iv(ft,) for 1 < / < A: - 1, and w(a )> w(b ). Consider then the independent sets B _ = {ft,, b ,... ,b _ ) and / = {a,, a ,..., a ). By axiom 1-3 there exists α . ε / such that 2

2

r

k

k

k

k

2

x

2

k

l

;

k

/' =

{b ,b ,...,b _ ,a } l

2

k

l

J

is an independent set. But then / ' is lexicographically greater than Β because νν(α,)2= w(a )> w(b ), which contradicts that Β is lexicographically maximum in $. k

k

296

matroids

2=>3

Obvious.

3=pl Let B = {b , b ,...,b ) be a lexicographically maximum base and B' = [b[, b' ,..., b' ) be a base of maximum weight. Since 1 Φ 2 , it follows that 1

2

2

r

r

w{b,) > w{b[)

for all 1 < i < r .

As B' is a maximum weight base, it follows that w(b,) = tv(6,). Hence B' is also lexicographically maximum. • It is easy to show (as in the proof of 1 Φ 2 in Theorem 10.32) that the greedy algorithm will pick a lexicographically maximum base that, by Theorem 10.32, has maximum weight. Thus we have the following. Theorem 10.33. Let $ be the collection of independent sets of a matroid on 5 whose elements have been assigned nonnegative weights. The greedy algorithm when applied on $ will pick a maximum weight member of S. • In view of Theorem 10.33 it is clear that a maximum weight base will be selected if we choose the elements of the matroid in the order of nonincreasing weights, rejecting an element only if its selection would destroy the independence of the set of chosen elements. The greedy algorithm to pick a minimum weight base is obvious. As an example, consider the weighted graph G in Fig. 10.5. The weights of the edges are as shown in the figure. The greedy algorithm to select a maximum weight spanning tree of G will first arrange the edges in the order of nonincreasing weight. Thus the edges will be ordered as a, b, e, f, d, c,g,h

.

The algorithm will pick the first three edges a, b, and e because they do not a

6

Figure 10.5. A weighted graph.

MATROIDS AND THE GREEDY ALGORITHM

297

contain any circuit. The edge / will be rejected because the set {a, b, e) U {/} contains a circuit. The edge d will then be picked, c will be rejected because it forms a circuit with e and d, which have already been picked. For the same reason g and h will also be rejected. Thus the greedy algorithm picks {a, b, e, d}, which is a maximum weight spanning tree of G. We now prove the converse of Theorem 10.33. Theorem 10.34. Let 3> be the collection of subsets of a set S with the property that A G J and Β C A implies that BE.3>. Then for all nonnegative weightings of the elements of S, the greedy algorithm when applied on $• picks a maximum weight number of $ only if 3> is the collection of independent sets of a matroid on 5. Proof. It is clear from the independence axioms that we have only to show that if A = {a,, a , . . . ,a ) £ $ and Β = {b , b ,..., b } £ 3>, then there exists a b,0 A such that A U b £ J . For this purpose let us define a weighting of the elements of S as follows: 2

k

x

2

k+l

1

t

w(o,) = l ,

ls/
w(b,) = x, w(e) = 0 ,

b,(EB-A, eES-(AU

Β) ,

where 0 < J C < 1 . Then the greedy algorithm first selects the elements a,, a ,..., a . If no b, exists with {b„ a , , . . . , a ) £ β, the algorithm will thereafter select from members of S - (A U B). So when it terminates, it will have picked a set whose weight is equal to the weight of A. If \BC\A\ = t, then 2

k

k

w(A) = A: and w(B) = t + (k + \-t)x

.

Clearly, we can select 0 < χ < 1 such that w(A) < w(B). But then the greedy algorithm will not have selected a maximum weight member of $. This is a contradiction. • The greedy algorithm for the problem of finding a maximum weight spanning tree was first discovered by Kruskal [10.7]. The extension of this algorithm to matroids was independently discovered by Rado [10.8], Edmonds [10.9], Gale [10.10], and Welsh [10.11]. For further discussions of the maximum weight spanning tree problem see Section 11.3.

298

10.10

MATROIDS

FURTHER READING

Welsh [10.4], Randow [10.12], and Recski [10.13] are excellent texts that contain a wealth of information on matroid theory. Berge [10.14] has a chapter on matroids. Wilson [10.15] gives an elegant introduction to matroids. He brings out clearly the power of the generality of a matroid by including simple proofs of two theorems on edge-disjoint spanning trees in a graph. In addition to Whitney's original paper [10.1], we highly recommend for further reading the papers by Rado [10.16], Lehman [10.17], Tutte [10.18] through [10.21], Mirsky [10.22], Harary and Welsh [10.23], Wilson [10.24], Edmonds [10.25], and Edmonds and Fulkerson [10.26], as well as the books by Mirsky [10.27], Lawler [10.28], Papadimitriou and Steiglitz [10.29], and Graver and Watkins [10.30]. Tutte [10.3], [10.18], and [10.19] develops the theory of chain groups and matroids. He defines a matroid as regular if it is isomorphic to the matroid of a regular chain group. It can be shown that a matroid is regular if and only if it is representable over every field. Another characterization is that a matroid is regular if and only if it is orientable [10.2]. Tutte also develops necessary and sufficient conditions for a matroid to be graphic. A binary matroid is graphic if and only if it does not contain as minor the Fano matroid or its dual or M*(K ) or Μ*(K ) . For more discussions on this question see [10.20] and [10.4]. (See Exercise 10.22 for a definition of the Fano matroid.) For a theory of connectivity in matroids see Tutte [10.21]. For a theory of orientability that is applicable to general matroids, see Bland and Las Vergnas [10.31] and Folkman and Lawrence [10.32]. Lehman [10.17] gives a solution of the Shannon switching game using matroid concepts. See Edmonds [10.33], Bruno and Weinberg [10.34], and Welsh [10.4]. A beauty of matroid theory is its unifying nature, which has yielded simple proofs of several results in transversal theory and graph theory. See Edmonds [10.25] and [10.33] and Wilson [10.15]. For matroid algorithms see Lawler [10.28], Papadimitriou and Steiglitz [10.29], and Recski [10.13]. See also Knuth [10.35], Edmonds [10.36], and Welsh [10.4]. Matroid theory is now being increasingly used in the study of electrical network problems. Recski [10.13] gives a detailed treatment of applications of matroids in electrical network theory. See also Duffin [10.37], Duffin and Morley [10.38], Narayanan [10.39], Bruno and Weinberg [10.40], [10.41] and [10.42], Iri and Tomizawa [10.43], Recski [10.44] and [10.45], and Peterson [10.46]. Bruno and Weinberg [10.40] also give a good introduction to matroid theory. For network-theoretic applications of the arc coloring lemma see Vandewalle and Chua [10.47], Chua and Greene [10.48], and Wolaver [10.49]. S

3

3

EXERCISES

299

10.11

EXERCISES

10.1

Let Μ be a matroid on 5 with ACS. Define J ' to be the collection of subsets AOf 5 such that X is independent in Μ and Χ Π A = 0 . Prove that 3' is the collection of independent sets of a matroid on S.

10.2

Let S be a set having η elements. Show that the collection J of all subsets of S having k or fewer elements is the set of independent sets of a matroid. This matroid is called the uniform matroid of rank k and is denoted by U „.

1

k

10.3

Let Μ be a matroid on S, and let B B be distinct bases of M. Prove that there exists a one-to-one correspondence between β, and B such that for all eEB (B - e ' ) U e is a base of M, where e' Ε Β corresponds to e. u

2

u

2

2

2

10.4

If β , , B are bases of a matroid Μ and A' C β , , prove that there exists X C B such that ( β , - Χ ) U X and ( β - X ) U AT, are both bases of Μ (Greene [10.50]). (See also [10.12], Chap. V.) 1

2

2

2

λ

2

2

2

10.5

Let 3 be a collection of nonnull subsets of S such that for any two distinct members Χ, Y of 3 with Λ Γ Ε Α Ί Ί Κ , yEXY there exists Ζ Ε 3 such that y Ε Ζ C (Α' U Y) - x. Then prove that the collection 3' of minimal members of 3) is the set of circuits of a matroid (Tutte [10.3]).

10.6

If C is a circuit of a matroid Μ and a EC, prove that there exists a base β such that C = C(a, B).

10.7

Prove that if β is a base of a matroid Μ and xE B, there is exactly one cocircuit C* of Μ such that C* Π ( β - χ) = 0 .

10.8

If C is a circuit of a matroid Λ/ and x, y are distinct elements of C, prove that there exists a cocircuit C* containing χ and .y but no other element of C (Minty [10.2]).

10.9

Let Μ be a matroid on S and let x, y, ζ be distinct elements of 5. If there is a circuit C, containing χ and y and a circuit C containing y and z, then prove that there exists a circuit C containing χ and z. 2

3

10.10

A set A C S is closed in a matroid Μ on 5 if for all J C E 5 - A, p(A U x) = p(A) + 1. Show that the intersection of two closed sets is a closed set.

10.11.

Let Μ be a matroid on a set S. The closure σ(Α) of a subset A of 5 is the set of all elements χ of S with the property that p(A Ux) = p(A). Prove the following: (a) If Λ: is contained in a(AL)y) but not in σ(Α), then y is contained in σ(Α U x).

300

MATROIDS

(b) An element JC belongs to σ(Α) if and only if JC £ A or there exists a circuit C of Μ for which C - A = { J C } . 10.12 A hyperplane of a matroid Μ on S is a maximal proper closed subset of S. Show that Η is a hyperplane of the matroid Μ if and only if S - Η is a cocircuit of Μ (see Welsh [10.4] for a matroid axiom system in terms of hyperplanes). 10.13 Show that if Μ is a matroid on 5 and ACS, then the contraction Μ · Τ of Μ to Γ is the matroid whose cocircuits are precisely those cocircuits of Μ which are contained in A. 10.14

If Μ is a matroid on S and TCXCS, prove the following: (a) M\T=(M\X)\T. (b) Μ· Τ=(Μ· Χ)· Τ. (c) (M\X)-T = (M-(S-(X-T)))\T. (d) (M · Jf)l Γ = (M\S -(XT)) • T.

10.15 A matroid Μ on S is connected or nonseparable if for every pair of distinct elements JC and y of S there is a circuit of Μ containing χ and y. Otherwise it is disconnected or separable. Show that a matroid Μ is connected if and only if its dual Μ * is connected. (Note: If G is a graph, then M(G) is connected if and only if G is 2-connected.) 10.16 Show that a matroid Μ on 5 is not connected if and only if there exists a proper subset A of 5 such that p(A) + p(S-A)

= p(S)

(Whitney [10.1]). 10.17

Prove or disprove: Graph G is contractible to Η if and only if M(G) contains M(H) as a contraction minor.

10.18

Prove that the uniform matroid U except GF(2).

10.19

Prove that a matroid is binary if and only if for any circuit C and cocircuit C*, \CΠ C * | ^ 3 .

2 4

is representable over every field

10.20 Let 9 be a family of finite nonempty subsets of a set S. A transversal of a subfamily of is called a partial transversal of S*. Show that if 9* is a collection of finite nonempty subsets of a set S, then the collection of partial transversals of 9 is the set of independent sets of a matroid on 5. (See Section 8.6 for the definition of a transversal.) Find the rank function of this matroid. A matroid Μ on 5 is called a transversal matroid if there exists some family 9 of subsets of S such s

s

EXERCISES

301

that S(M) is the family of partial transversals of 5*. Find the rank function of a transversal matroid (see Theorem 8.15). 10.21

Show that every fc-uniform matroid is a transversal matroid.

10.22

The Fano matroid F is the matroid defined on the set 5 = {1, 2, 3, 4, 5, 6, 7} whose bases are all those subsets of S containing three elements except {1, 2, 3}, {1, 4, 5}, {1, 6, 7}, {2, 4, 7}, {2, 5, 6}, {3, 4, 6}, and {3, 5, 7}. Show that F is (a) binary, (b) nontransversal, (c) nongraphic, and (d) noncographic.

10.23

Show that the circuit matroid of K is not a transversal matroid.

10.24

A matroid Μ on S is Eulerian if S can be expressed as the union of disjoint circuits. A matroid is bipartite if every circuit of Μ contains an even number of elements. Prove that a matroid is bipartite if and only if M* is Eulerian.

10.25

Let D be a directed graph without self-loops and let X and Y be two disjoint sets of vertices of D. A subset A of X is called independent if there exist \A\ vertex-disjoint chains from A to Y. Show that these independent sets form the independent sets of a matroid on X. (Such a matroid is called a gammoid.) (See Mason [10.51].)

10.26

A matroid Μ on S is base orderable if for any two bases β , , B of Μ there exists a one-to-one correspondence between B, and B such that for each xE.B , both (B - x) U x ' and (B - x')Ux are bases of Μ where x' is the element of B that corresponds to x. Show the following: (a) M(K ) is not base orderable. (b) If Μ is base orderable on S and TC.S, then M\T is base orderable. (c) If Μ is base orderable, then any minor of Μ is base orderable.

A

2

2

x

x

2

2

A

10.27

Let M and M be two matroids on a set S. (a) Show that the set of all unions / U / of an independent set / of M, and an independent set J of Μ form the independent sets of a new matroid. (This matroid is called the union of M and M and is denoted by M U M .) (b) If p, and p denote the rank functions of matroids Μ, and M on a set 5, show that x

2

2

x

t

2

2

2

2

p(A) = mm { (X) Pl

+ p (X) + \A2

X\}

302

MATROIDS

where ACS and ρ is the rank function of A/, U M [10.15], pp. 154-158).

2

10.28

(Wilson

Let Af be a matroid on S. Prove the following: (a) Ai contains k disjoint bases if and only if, for any / I C S , p(A) +

\S-A\>kp(.S).

(b) S can be expressed as the union of not more than k independent sets if and only if for any ACS, kp(A)>\A\. [Hint: Consider the union of k copies of A/ and use the result of Exercise 10.27 (Wilson [10.15], pp. 154-158).] 10.29 Show that when the greedy algorithm has chosen A: elements, these k elements are of maximum weight with respect to all independent sets of k or fewer elements. 10.30 A number of jobs are to be processed by a single machine. All jobs require the same processing time. Each job has assigned to it a deadline. (a) Show that the collection of all subsets of jobs that can be completed on time forms the independent sets of a matroid. (b) Suppose each job has a penalty that must be paid if it is not completed by its deadline. In what order should these jobs be processed so that the total penalty is minimum? 10.31 Let A/ be a matroid whose elements have been assigned nonnegative weights. Prove the following: (a) No element of a maximum weight base is the smallest element of any circuit of M. (b) Each element of a maximum weight base is the largest element of at least one cocircuit of M. Using (b), design a procedure for constructing a maximum weight base of a matroid. (Prim [10.52] describes such a procedure for constructing a minimum weight spanning tree of a connected graph.) 10.32 Let Μ be a matroid on S with nonnegative weights assigned to the elements of S. Let S3 be the collection of bases of Μ and %* the collection of cocircuits of A/. Prove min max w(e) — max min Best

(EB

C'et'eec

w(e).

REFERENCES

10.12 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21

303

REFERENCES Η. Whitney, "On the Abstract Properties of Linear Dependence," Am. J. Math., Vol. 57, 509-533 (1935). G. J. Minty, "On the Axiomatic Foundations of the Theories of Directed Linear Graphs, Electrical Networks and Network Programming," /. Math, and Mech., Vol. 15, 485-520 (1966). W. T. Tutte, Introduction to the Theory of Matroids, American Elsevier, New York, 1971. D. J. A. Welsh, Matroid Theory, Academic Press, New York, 1976. G. J. Minty, "Monotone Networks," Proc. Roy. Soc, A, Vol. 257, 194-212 (1960). G. J. Minty, "Solving Steady-State Non-Linear Networks of 'Monotone' Elements," IRE Trans, Circuit Theory, Vol. CT-8, 99-104 (1961). J. B. Kruskal, "On the Shortest Spanning Subgraph of a Graph and the Travelling Salesman Problem," Proc. Am. Math. Soc, Vol. 7, 48-49 (1956). R. Rado, "Note on Independence Functions," Proc. London Math. Soc, Vol. 7, 300-320 (1957). J. Edmonds, "Matroids and the Greedy Algorithm," Math. Programming, Vol. 1, 127-136 (1971). D. Gale, "Optimal Assignments in an Ordered Set: An Application of Matroid Theory," J. Combinatorial Theory, Vol. 4, 176-180 (1968). D. J. A. Welsh, "Kruskal's Theorem for Matroids," Proc. Cambridge Phil. Soc, Vol. 64, 3-4 (1968). Rabe von Randow, Introduction to the Theory of Matroids, Springer Lecture Notes in Mathematical Economics, Vol. 109, Springer-Verlag, Berlin, 1975. A. Recski, Matroid Theory and Its Applications in Electric Network Theory and Statics, Springer-Verlag, Berlin, 1989. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. R. J. Wilson, Introduction to Graph Theory, Oliver and Boyd, Edinburgh, 1972. R. Rado, "A Theorem on Independence Relations," Quart, J. Math (Oxford), Vol. 13, 83-89 (1942). A. Lehman, "A Solution of the Shannon Switching Game," SIAM J. Appl. Math., Vol. 12, 687-725 (1964). W. T. Tutte, "A Homotopy Theorem for Matroids-I and II," Trans. Am. Math. Soc, Vol. 88, 144-174 (1958). W. T. Tutte, "Lectures on Matroids," J. Res. Nat. Bur. Stand., Vol. 69B, 1-48 (1965). W. T. Tutte, "Matroids and Graphs," Trans. Am. Math. Soc, Vol. 90, 527-552 (1959). W. T. Tutte, "Connectivity in Matroids," Canad. J. Math., Vol. 18, 13011324 (1966).

304

MATROIDS

10.22 L. Mirsky, "Application of the Notion of Independence to Combinatorial Analysis," J. Combinatorial Theory, Vol. 2, 327-357 (1967). 10.23 F. Harary and D. J. A. Welsh, "Matroids versus Graphs," in The Many Facets of Graph Theory, Springer Lecture Notes, Vol. 110, Springer-Verlag, Berlin, 1969, pp. 155-170. 10.24 R. J. Wilson, "An Introduction to Matroid Theory, Am. Math. Monthly, Vol. 80, 500-525 (1973). 10.25 J. Edmonds, "Minimum Partition of a Matroid into Independent Subsets," /. Res. Nat. Bur. Stand., Vol. 69B, 67-72 (1965). 10.26 J. Edmonds and D. R. Fulkerson, "Transversals and Matroids Partition," /. Res. Nat. Bur. Stand., Vol. 69B, 147-153 (1965). 10.27 L. Mirsky, Transversal Theory, Academic Press, London, 1971. 10.28 E. L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976. 10.29 C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982. 10.30 J. E. Graver and M. Watkins, Combinatorics with Emphasis on the Theory of Graphs, Springer-Verlag, New York, 1977. 10.31 R. G. Bland and M. Las Vergnas, "Orientability of Matroids," /. Combinatorial Theory B, Vol. 24, 94-123 (1978). 10.32 J. Folkman and J. Lawrence, "Oriented Matroids," J. Combinatorial Theory B, Vol. 25, 199-236 (1978). 10.33 J. Edmonds, "Lehman's Switching Game and a Theorem of Tutte and Nash-Williams," J. Res. Nat. Bur. Stand., Vol. 69B, 73-77 (1965). 10.34 J. Bruno and L. Weinberg, "A Constructive Graph-Theoretic Solution of the Shannon Switching Game," IEEE Trans. Circuit Theory, Vol. CT-17, 74-81 (1970). 10.35 D. E. Knuth, "Matroid Partitioning," Stanford University Rep. STAN-CS73-342, 1-12 (1973). 10.36 J. Edmonds, "Matroid Partition," in Lectures in Appl. Math., Vol. 11: Mathematics of Decision Sciences, 1967, pp. 335-346. 10.37 R. J. Duffin, "Topology of Series-Parallel Networks," J. Math. Anal. Appl., Vol. 10, 303-318 (1965). 10.38 R. J. Duffin and T. D. Morley, "Wang Algebra and Matroids," IEEE Trans. Circuits and Syst., Vol. CAS-25, 755-762 (1978). 10.39 H. Narayanan, "Theory of Matroids and Network Analysis," Ph.D. Thesis, Indian Institute of Technology, Bombay, India, 1974. 10.40 J. Bruno and L. Weinberg, "Generalized Networks: Networks Embedded on a Matroid, Part 1," Networks, Vol. 6, 53-94 (1976). 10.41 J. Bruno and L. Weinberg, "Generalized Networks: Networks Embedded on a Matroid, Part 2," Networks, Vol. 6, 231-272 (1976). 10.42 L. Weinberg, "Matroids, Generalized Networks and Electric Network Synthesis," J. Combinatorial Theory B, Vol. 23, 106-126 (1977). 10.43 M. Iri and N. Tomizawa, "A Unifying Approach to Fundamental Problems

REFERENCES

10.44 10.45 10.46 10.47 10.48 10.49 10.50 10.51 10.52

305

in Network Theory by Means of Matroids," Electron. Commun. in Japan, Vol. 58-A, 28-35 (1975). A. Recski, "On Partitional Matroids with Applications," in Coll, Math. Soc. J. Bolyai, Vol. 10: Infinite and Finite Sets, North-Holland-American Elsevier, Amsterdam, 1974, pp. 1169-1179. A. Recski, "Matroids and Independent State Variables," Proc. 2nd European Conf. Circuit Theory and Design, Genova, 1976. B. Petersen, 'Investigating Solvability and Complexity of Linear Active Networks by Means of Matroids," IEEE Trans. Circuits and Syst., Vol. CAS-26, 330-342 (1979). J. Vandewalle and L. O. Chua, "The Colored Branching Theorem and Its Applications in Circuit Theory," IEEE Trans. Circuits and Systems, Vol. CAS-27, 816-825 (1980). L. O. Chua and D. M. Greene, "Graph-Theoretic Properties of Dynamic Nonlinear Networks," IEEE Trans. Circuits and Systems, Vol. CAS-23, 292-312 (1976). D. A. Wolaver, "Proof in Graph Theory of the No Gain Property of Resistor Networks." IEEE Trans. Circuits and Systems, Vol. CAS-17, 436-437 (1970). C. Greene, "A Multiple Exchange Property for Bases," Proc. Am. Math. Soc, Vol. 39, 45-50 (1973). J. H. Mason, "On a Class of Matroids Arising from Paths in Graphs," Proc. London Math. Soc, Vol. 25, 55-74 (1972). R. C. Prim, "Shortest Connection Networks and Some Generalizations," Bell Sys. Tech. J., Vol. 36, 1389-1402 (1957).

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 11

GRAPH ALGORITHMS

Graphs arise in the study of several practical problems. The first step in such studies is to discover graph-theoretic properties of the problem under consideration that would help us in the formulation of a method of solution to the problem. Usually solving a problem involves analysis of a graph or testing a graph for some specified property. Graphs that arise in the study of real-life problems are very large and complicated. Analysis of such graphs in an efficient manner, therefore, involves the design of efficient computer algorithms. In this and the following chapter of the book we discuss several graph algorithms. We consider these algorithms to be basic in the sense that they serve as building blocks in the design of more complex algorithms. These algorithms will also help us provide the reader with further insight on some of the topics discussed in the previous chapters as well as introduce new concepts. While our main concern is to develop the theoretical foundation on which the design of the algorithms is based, we also develop results concerning the computational complexity of these algorithms (in particular, in Chapter 12). In certain cases the computational complexity depends crucially on the computational complexity of certain basic operations such as finding union of disjoint sets. In such cases we provide adequate references for the interested reader to pursue further. The computational complexity of an algorithm is a measure of the running time of the algorithm. Thus it is a function of the size of the input. In the case of graph algorithms, complexity results will be in terms of the number of vertices and the number of edges in the graph. In the following a function g(n) is said to be 0( /(n)) if and only if there exist constants c and 306

TRANSITIVE CLOSURE

307

n such that \g(n)\ ^ c\f(n)\ for all η > n . Furthermore, all our complexity results will be with respect to the worst-case analysis. There are different methods of representing a graph on a computer. Two of the most common methods use the adjacency matrix (Section 6.10) and the adjacency list. Adjacency matrix representation is not a very efficient one in the case of sparse graphs. In the adjacency list representation, we associate with each vertex a list that contains all the edges incident on it. A detailed discussion of data structures for representing a graph may be found in some of the references listed at the end of this chapter. 0

11.1

0

TRANSITIVE CLOSURE

We may recall (Section 5.2) that a binary relation on a set is a collection of ordered pairs of the elements of the set. The transitive closure of a binary relation R is a relation R* defined as follows: χ R* y if and only if there exists a sequence x — x, x , x ,..., 0

x

x —y

2

k

such that k>0 and x R x , x, R x ,... ,x _, R x. Clearly, if χ R y, then χ R* y. Hence R C R*. Further, it can be easily shown that R* is transitive. In fact, it is the smallest transitive relation containing R. So if R is transitive, then R* = R. As we have already pointed out in Section 5.2, a binary relation can be represented by a directed graph. Suppose that G is the directed graph representing a relation R. The directed graph G* representing the transitive closure R* of R is called the transitive closure of G. It follows from the definition of R* that the edge (x, y), χ Φ y, is in G* if and only if there exists in G a directed path from the vertex χ to the vertex y. Similarly the self-loop (x, x) at vertex χ is in G* if and only if there exists in G a directed circuit containing x. For example, the graph shown in Fig. 11.1ft is the transitive closure of the graph of Fig. 11.1a. Suppose that we define the reachability matrix of an η -vertex directed graph G as an η x η (0,1) matrix in which the (i, ; ) entry is equal to 1 if and only if there exists a directed path from vertex i to vertex j when i Φ j , or a directed circuit containing vertex i" when ι = /. In other words the (i, /) entry of the reachability matrix is equal to 1 if and only if vertex / is reachable from vertex i through a sequence of directed edges. It is now easy to see that the adjacency matrix of G* is the same as the reachability matrix of G. The problem of constructing the transitive closure of a directed graph arises in several applications. For examples, see Gries [11.1]. In this section we discuss an elegant and computationally efficient algorithm due to Warshall [11.2] for computing the transitive closure. We also discuss a variation of WarshalFs algorithm given by Warren [11.3]. 0

x

2

t

k

308

GRAPH ALGORITHMS <j

»

ο

la)

Figure 11.1. (a) Graph G. (b) G*, transitive closure of G.

Let G be an η-vertex directed graph with its vertices denoted by the integers 1 , 2 , . . . , n. Let G° = G. Warshall's algorithm constructs a sequence of graphs so that G ' C G ' , 0 S ( < n - l , and G" is the transitive closure of G. The graph G', i a 1 , is obtained from G ' by processing vertex i in G'~\ Processing vertex i in G'~ involves addition of new edges to G'~ as described now. Let, in G'~ , the edges (i, k), (i, / ) , (/, m),... be incident out of vertex i. Then for each edge (;', i) incident into vertex i, add to G the edges (/, k), (/, 0 , (/»»»),... if these edges are not already present in G . The graph that results after vertex i is processed is denoted as G'. Warshall's algorithm is illustrated in Fig. 1 1 . 2 . It is clear that G' C G \ 1 ^ 0 , To show that G" is the transitive closure of G we need to prove the following result. + 1

- 1

l

l

l

, _ 1

l _ 1

i+

Theorem 11.1 1 . Suppose that, for any two vertices s and t, there exists in G a directed path Ρ from vertex ί to vertex / such that all its vertices other than s and r are from the set { 1 , 2 , . . . , / } . Then G' contains the edge (s, t). 2 . Suppose that, for any vertex *, there exists in G a directed circuit C containing s such that all its vertices other than s are from the set { 1 , 2 , . . . , i}. Then G' contains the self-loop (s, s).

TRANSITIVE CLOSURE

309

(s)

04

(M

(e)

Figure 11.2. Illustration of Warshall's algorithm. (a) G", (b) G' = G2. (c) G3 = G4.

Proof 1. Proof is by induction on i. Clearly the result is true for G1 since Warshall's construction, while processing vertex 1, introduces the edge (s, t) if G° (=G) contains the edges (s, 1) and (1, t). Let the result be true for all Gk, k< i. Suppose that / is not an internal vertex of P. Then it follows from the induction hypothesis that G' _1 contains the edge (s, t). Hence G' also contains (s, t) because G' - 1 C.G'. Suppose that 1 is an internal vertex of P. Then again from the induction hypothesis it follows that G , _ 1 contains the edges (s, i) and (i, t). Therefore, while processing vertex i in G'~\ the edge (s, r) is added to G'. 2. Proof follows along the same lines. ■

GRAPH ALGORITHMS

310

As an immediate consequence of this theorem we get the following. Corollary 11.1.1. G" is the transitive closure of G.

•

We next give a formal description of Warshall's algorithm. In this description the graph G is represented by its adjacency matrix Μ and the symbol ν stands for Boolean addition. Algorithm 11.1. Transitive Closure (Warshall) 51. 52. 53. 54. 55. 56.

(Initialization)Μ is the adjacency matrix of G. Do S3 for i = 1 , 2 , . . . ,n. Do S4 for j = 1 , 2 , . . . , n. If M(j, i) = 1, do S5 for A: = 1 , 2 , . . . , n. M(j, k) = M(j, k) ν M(i, k). HALT. (Ai is the adjacency matrix of G*.) •

Note that the matrix Μ (when the algorithm begins to execute S3 with ί = ρ) is the adjacency matrix of G ~\ Further, processing a diagonal entry does not result in adding new nonzero entries. A few observations are now in order: p

1. Warshall's algorithm transforms the adjacency matrix Μ of a graph G to the adjacency matrix of the transitive closure of G by suitably overwriting on M. It is for this reason that the algorithm is said to work "in place." 2. The algorithm processes all the edges incident into a vertex before it begins to process the next vertex. In other words it processes the matrix Μ columnwise. Hence we describe Warshall's algorithm as column-oriented. 3. While processing a vertex no new edge (i.e., an edge that is not present when the processing of that vertex begins) incident into the vertex is added to the graph. This means that while processing a vertex we can choose the edges incident into the vertex in any arbitrary order. 4. Suppose that the edge (;', i) incident into the vertex i is not present while vertex / is processed, but that it is added subsequently while processing some vertex k, k> i. Clearly this edge is not processed while processing vertex i. Neither will it be processed later since no vertex is processed more than once. In fact, such an edge will not result in adding any new edges. 5. Warshall's algorithm is said to work in one pass since each vertex is processed exactly once.

TRANSITIVE CLOSURE

311

Suppose that we wish to modify Warshall's algorithm so that it becomes row-oriented. In a row-oriented algorithm, while processing a vertex, all the edges incident out of the vertex are to be processed. The processing of the edge (/', ;') introduces the edges (i, k) for every edge (/, k) incident out of vertex Therefore new edges incident out of a vertex may be added while processing a vertex "rowwise". Some of these newly added edges may not be processed before the processing of the vertex under consideration is completed. If the processing of these edges is necessary for the computation of the transitive closure, then such a processing can be done only in a second pass. Thus, in general, a row-oriented algorithm may require more than one pass to compute the transitive closure. For example, consider the graph G of Fig. 11.3a. After processing rowwise the vertices of G we obtain the graph G' shown in Fig. 11.36. Clearly, G' is not the transitive closure of G since the edge (1,2) is yet to be added. It may be noted that the edge (1,3) is not processed in this pass ο 1

>

ο

»

4

ο

•———ο

3

2

la I

(c)

Figure 11.3. An example of row-oriented transitive closure algorithm, (a) G. (b) G'. (c) G*.

312

GRAPH ALGORITHMS

because it is added only while processing the edge (1,4). The same is the case with the edge (4,2). Suppose we next process the vertex of G'. In this second pass the edge (1,2) is added while processing vertex 1 and we get the transitive closure G* shown in Fig. 11.3c. Thus in the case of the graph of Fig. 11.3a two passes of the row-oriented algorithm are required. Now the question arises whether two passes always suffice. The answer is in the affirmative, and Warren [11.3] has demonstrated this by devising a clever two-pass row-oriented algorithm. In this algorithm, while processing a vertex, say vertex /', in the first pass only edges connected to vertices less than i are processed, and in the second pass only edges connected to vertices greater than i are processed. In other words the algorithm transforms the adjacency matrix Μ of the graph G to the adjacency matrix of G* by processing in the first pass only entries below the main diagonal of Μ and in the second pass only entries above the main diagonal. Thus during each pass at most n(n - l ) / 2 edges are processed. A description of Warren's modification of Warshall's algorithm now follows. Algorithm 11.2. Transitive Closure (Warren) 51. 52. 53. 54. 55. 56. 57. 58. 59. S10.

Μ is the adjacency matrix of G. Do S3 for i = 2 , 3 , . . . , n. Do S4 for ; = 1 , 2 , . . . , i - 1. If M(i, j) = 1, then do S5 for k = 1 , . . . , n. M(i, k) = M(i, k) ν M(j, k). Do S7 for ι = 1 , 2 , . . . , η - 1 . Do S8 for / = ί + 1, ι + 2 , . . . , n. If M(i, j) = 1, then do S9 for k = 1 , 2 , . . . , n. M(i, k) = M{i, k) ν M(j, k). HALT. (M is the adjacency matrix of G*.) •

Note that in this algorithm steps S2 through S5 correspond to the first pass and steps S6 through S9 correspond to the second pass. As an example, consider again the graph shown in Fig. 11.3a. At the end of the first pass of Warren's algorithm we obtain the graph shown in Fig. 11.4a, and at the end of the second pass we get the transitive closure G* shown in Fig. 11.46. The proof of correctness of Warren's algorithm is based on the following lemma. Lemma 11.1. Suppose that, for any two vertices s and t, there exists in G a directed path Ρ from s to t. Then the graph that results after processing vertex s in the first pass (steps S2 through S5) of Warren's algorithm contains an edge (s, r), where r is a successor of s on Ρ and either r > s or r = t.

TRANSITIVE CLOSURE

1

4

3

313

2

(b)

Figure 11.4. Illustration of Warren's algorithm. Proof. Proof is by induction on s. If 5 = 1, then the lemma is clearly true because all the successors of 1 on Ρ are greater than 1. Assume that the lemma is true for all s < k and let s = k. Suppose (s, *',) is the first edge on P. If i, > s, then clearly the lemma is true. If i, < s, then by the induction hypothesis the graph that results after processing vertex i, in the first pass contains an edge (/,, i ) , where i is a successor of i, on Ρ and either i > i, or i = t. If i Φ t and i < s, then again by the induction hypothesis the graph that results after processing vertex i in the first pass contains an edge ( i , i ) where i is a successor of i on P, and either i > i or i = t. If i 5^ r and i < s, we repeat the arguments on i until we locate an i such that either i > s or i = i. Thus the graph that we have before the processing of vertex s begins contains the edges CM]), (i,, i ) , . . . , ( i _ ! , / ) such that the following conditions are satisfied: 2

2

2

2

2

2

2

2

3

2

3

3

3

m

3

m

m

1. i is a successor of i _ p

p

2. 'm-i > ' - 2 > 'm-3 3. i = r or i > s. m

m

3

3

m

m

2

2

>

l

on Ρ, ρ 2 2.

"'' > «Ί. and i < s for k Φ m. k

m

We now begin to process vertex s. Processing of (s, /,) introduces the edge (s, i ) because of the presence of (/',, i ) . Since i > i',, the edge (s, i ) is subsequently processed. Processing of this edge introduces (s, i ) because of the presence of ( i , i ), and so on. Thus when the processing of s is completed, the required edge (s, i ) is present in the resulting graph. • 2

2

2

2

3

2

3

m

Theorem 11.2. Warren's algorithm computes the transitive closure of a graph G.

314

GRAPH ALGORITHMS

Proof. We need to consider two cases: Case 1

For any two distinct vertices s and t there exists in G a directed path Ρ from s to t.

Let (i, / ) be the first edge on Ρ (as we proceed from s to t) such that i > j . Then it follows from the previous lemma that the graph that we have, before the second pass of Warren's algorithm begins, contains an edge (/', k), where A; is a successor of i on Ρ and either k = t or k > i. Thus after the first pass is completed there exists a path P ' : s, /,, i ,...,i , t such that s /,, the edge (s, i ) is processed subsequently. This, in turn, introduces the edge (s, t ) , and so on. Thus when the processing of s is completed, we have the edge (s, t) in the resulting graph. 2

m

m

i

2

;

J+i

2

2

2

2

3

Case 2

There exists in G a directed circuit containing a vertex s.

In this case we can prove along the same lines as before that when the processing of vertex s is completed in the second pass, the resulting graph contains the self-loop (s, s). • Clearly both Warshalls and Warren's algorithms have the worst-case complexity 0(n ). Warren [11.3] refers to other row-oriented algorithms. For some of the other transitive closure algorithms, see [11.4] through [11.12]. Syslo and Dzikiewicz [11.13] discuss computational experiences with several of the transitive closure algorithms. Melhorn [11.14] discusses the transitive closure problem in the context of general path problems in graphs. Several additional references on this topic can also be found in [11.14]. 3

11.2

SHORTEST PATHS

Let G be a connected directed graph in which each directed edge is associated with a real number called the length of the edge. The length of an edge directed from a vertex i to a vertex is denoted by w(i, j). If there is no edge directed from vertex i to vertex /', then w(i, j) = oo. The length of a directed path in G is the sum of the lengths of the edges in the path. A minimum length directed s-t path is called a shortest path from s to t. The length of a shortest directed s-t path, if it exists, is called the distance from s to t, and it is denoted as d(s, t). In this section we consider the following two problems:

SHORTEST PATHS

315

1. Find the shortest paths from a specified vertex s to all other vertices in G. 2. Find the shortest paths between all the ordered pairs of vertices in G. These two problems arise in several optimization problems. For example, finding a minimum cost flow in a transport network involves finding a shortest path from the source to the sink in the network [11.15].

11.2.1 Shortest Paths from a Specified Vertex s to All Other Vertices in a Graph We first develop an algorithm due to Dijkstra [11.16], which finds shortest paths from a specified vertex s to all the other vertices in an n-vertex connected directed graph G. We assume that all the lengths are nonnegative. The following ideas form the basis of Dijkstra's algorithm. Let V denote the vertex set of G, and let S be a subset of V such that J G S . Let 5 denote the complement of S in V. Thus 5 = V- S. Among all the directed paths from s to the vertices in S, let Ρ be a path with minimum length. The length of Ρ is then called the distance d(s, S) from s to S. Let P: s,...,u, v. Then clearly νG S and « G S . Further, the s-u section of Ρ must be a shortest s-u path with all its vertices in S. Therefore, d(s, S) = d(s, u) + w(u, v). From this we can see that the distance d(s, S) can be computed using the formula d(s, S) = min {d(s, u) + w(u, v)} .

(11.1)

ues

ves

Now, if ν is the vertex in 5 such that d(s, S) = d(s, u) + w(u, υ) for some u G S, then clearly (11.2)

d(s, v) = d(s, S).

Dijkstra's algorithm constructs a sequence S = {s}, S,, 5 , . . . of subsets of V such that the following conditions are satisfied: 0

2

1. If s = u , M , . . . , u _ ! are the vertices of V such that d(s, «,) s d(s, u ) < d(s, u ) < · · · < d(s, «„_,), then S, = {s, u „ u , . . . , u,} for 2

0

n

3

2

2

;>o. 2. When the set S, has been determined, the shortest paths from s to u,, M , . . . , u will be known. 2

t

316

GRAPH ALGORITHMS

If the sets 5, are defined as stated, then by (11.2) we have d( ,u ) S

(11.3)

= d(s,S,).

i+l

Thus determining S from 5, involves computing d(s, S ). The subsets S,, S ,...,S _ can be constructed as follows: By (11.1), (

i+l

2

n

}

d(s, S ) = min {d(s, u) + w(u, v)} = min {w(s, v)} . 0

" fo vel

es

e

v

0

0

Hence by (11.3), M, is a vertex with the property d(s, w,) = min {w(s, v)}.

(11-4)

ves

0

If P, denotes a shortest path from s to the vertex u,, then clearly P,: s, u . Suppose that the subsets S , S ... ,S, and the paths P , , P , . . . _ , P, have been determined. Now to determine _ S , we first compute d(i,S,) using (11.1). By (11.3), u is a vertex in 5, with the property x

0

2

lt

1+1

i + 1

d(s, u ) l+i

=

d(s,S ). l

By (11.1), there is a vertex y G 5, such that Φ.

= d(s, u ) + w(uj, «, f

+

).

1

Therefore we can obtain P by adjoining the edge (u , «,+,) to the path P . If we are interested only in a shortest path from a specified vertex s to another specified vertex t, then we can terminate the procedure after we have determined the first set 5, which contains t. It is clear that in this procedure we need to compute the minimum in (11.1]) at each stage. If this minimum were to be computed from scratch at each stage, then determining S, from 5,_, would require (i - 1) x (n - i) additions and {i(n - i) - 1} comparisons. The total number of operations for the entire algorithm would then be m

;

;

Σ {(2/-l)(n-i)-l> resulting in an overall complexity of Oi[n ). However, many of these additions and comparisons would be repeated unnecessarily. Dijkstra, in his algorithm, avoids such repeated additions and comparisons by storing computational information from one stage to the next. This is achieved by a labeling procedure that, as we shall see, improves the complexity of the algorithm to 0(n ). The following ideas lead to Dijkstra's 3

2

SHORTEST PATHS

317

labeling procedure. Suppose we do the labeling so that for ί = 0 , 1 , 2 , . . . the label /,(u) for vertex ν satisfies the following: 1. l (s) = 0 and / (i>) = °° for all υ Φ s. 2. For i > l , 0

0

l,(v) = d(s, v)

for all υ Ε S,_, ,

l.(v) = min {d(s, u) + w(w, U)} for all ν

Ε

5 ,.

Clearly, then, u, is a vertex with the property d(s, «,) = φ , Now we can compute

= min { / » } .

from /,(υ) as follows:

1. For υ Ε 5,., 2. For υ Ε 5,,

= /,(«) = φ ,

υ).

= min {d(s, u) + w(u, υ)}

+

= min {/,.(u), d(s, «,) + w(u„ v)} = min { / » , / , ( « , ) +w{u„ v)}. Then

(11.5)

is to be chosen such that d(s, ι ι

ι + Ι

) = φ,5,) = min{/ (i;)}. ves.

(11.6)

I+1

Note that the label of u does not change after the set S, has been determined. Thus Dijkstra's algorithm begins with the label /„(s) = 0 and l (v) = » for all υ Φ s. As the algorithm progresses, the labels are modified according to (11.5). The labels /„_,(u) would give the distance from s to v. _ It is clear that determining M involves computing / (i>) for all ν GS, and then finding the minimum of these labels. For i > 1 the former computations (11.5) require ( n - i - 1 ) additions and ( n - i - 1 ) comparisons, whereas the latter computations (11.6) require {{n-i)-2\ comparisons. Thus, clearly, the complexity of Dijkstra's algorithm is 0(n ) . A description of Dijkstra's algorithm is presented next. In this description LABEL is an array in which the current labels of the vertices are stored. A vertex becomes permanently labeled when it is set equal to u, for some i. We use an array PERM to indicate which of the vertices are permanently t

0

i + 1

1+1

318

GRAPH ALGORITHMS

labeled. If P E R M ( U ) = 1, then ν is a permanently labeled vertex. Note that in such a case the label of ν is equal to d(s, v). We start with P E R M ( s ) = 1

and P E R M ( u ) = 0 for all υ Φ s.

PRED is an array that keeps a record of the vertices from which the vertices get permanently labeled. If a vertex υ is permanently labeled, then u, PRED(u), PRED(PRED(v)),

...,s

are the vertices in a shortest directed s-v path. Algorithm 11.3. Shortest Paths (Dijkstra) 50. 51.

G is the given directed graph with lengths associated with its edges. Shortest paths from vertex s to all other vertices in G are required. (Initialize.) Set L A B E L ( J ) = 0, PERM(s) = 1, and PRED(s) = s. For all ν ¥• s, set L A B E L ( U ) = oo, PERM(u) = 0, and PRED(i;) = v.

52. 53.

54.

55.

Set i = 0 and u = s. (w is the latest vertex permanently labeled. Now it is s.) [Compute L A B E L ( U ) and update the entries of the PRED array.] Set ι = / + l. Do the following for each vertex ν that is not yet labelled permanently: 1. Set Μ = min{LABEL(u), LABEL(u) + w(u, v)}. 2. If Λί ), then set LABEL(u) = Μ and P R E D ( U ) = H. (Identify vertex «,). Find among all vertices, which are not yet permanently labeled, a vertex w with the smallest label. (In case of a tie the choice can be made arbitrarily.) Set PERM(w) = 1 and u = w. (u, = w, and it is the latest vertex labeled permanently.) If i < η - 1, then go to S3. Otherwise HALT. [All the shortest paths are found. The vertex labels give the lengths of the shortest paths. Now v, PRED(i;), PRED(PRED(u)),... ,s are the vertices in a shortest directed s-υ path.] •

Note that in a computer program °° is represented by as high a number as necessary. Further, if the final label of a vertex υ is equal to oo, then it means that there is no directed path from s to υ. To illustrate Dijkstra's algorithm, consider the graph G in Fig. 11.5 in which the length of an edge is shown next to the edge. In Fig. 11.6 we have shown the entries of the LABEL and PRED arrays. For any i the circled entries in the LABEL array correspond to the permanently labeled vertices, namely, the vertices in 5,. The entry with the mark * is the label of the latest vertex permanently labeled, namely, the vertex «,. The shortest paths from s and the corresponding distances are obtained from the final values of the entries in the PRED and LABEL arrays.

SHORTEST PATHS

319

14

<

Ί

Χ

S

'2

D^-^'

8

8

E

12'

*5

6 \

\ Λ 1 9

6"

<

7

N.

X*

H

10

Figure 11.5

Vertices i

Λ

0 1 2 3 4

©* © © © © © © ©

5 6 7

B

c

D

E

F

00

00

00

00

00

7

»

8 8

00

00

00

00

00

00

CD* CD CD CD CD CD

00

®* ® ® ® ®

00

« 00

«

©*

©· © ® (g)

00 00

@· @

<7

H

00

00

©· © © © © © ©

00 00 00

21

©* © ©

(«) PRED(£)«2) PRED(F)-# PRED(G)-/1 PREEK # ) - / >

PRED(/l)«/4 PRED(Ä)-/< PRED(C)-C PREEK D)~A From

To

Λ Λ A A A A A

B C D E F G H

Shortest Path A, B No path A,D A,D,E A,D,H,F A,G A,D,H (b)

ure 11.6. Illustration of Dijkstra's algorithm. The LABEL array is shown in (a).

320

GRAPH ALGORITHMS

In our discussions thus far we have assumed that all the lengths are nonnegative. Dijkstra's algorithm is not valid if some of the lengths are negative. (Why?) We next present a shortest path algorithm due to Ford [11.17] that allows the presence of edges with negative lengths but does not allow directed circuits of negative length. This algorithm is also attributed to Bellman [11.18] and Moore [11.19]. Note that for convenience in subsequent discussions of the algorithm, we have chosen to use λ(ι>) instead of LABEL(u) to denote the label of a vertex. Algorithm 11.4. Shortest Paths in Graphs with Negative Length Edges (Bellman-Ford-Moore)

50. G is the given directed graph with lengths associated with its edges. Shortest paths from vertex s to all other vertices in G are required. 51. (Initialize.) Set A(s) = 0 and PRED(s) = s. For all ν Φ s, set λ(υ) = °° and PRED(u) = v. 52. If there exists no edge β = («,•, v ) for which λ ^ ) > λ(υ,) + w(e), then HALT. (The current vertex label values represent the lengths of the shortest paths.) 53. Select an edge e = (v,,v ) for which λ(ι> ) > λ(υ,) + w(e) and set A(u ) = λ(υ,) + w(e) and PRED(u ) = υ,. Go to S2. • t

;

l

y

;

For our purposes in the above algorithm oo is not greater than oo + k, even if k is negative. Also, the algorithm will not terminate if there is a directed path from s to a vertex on a directed circuit of negative length because in such a case going around this circuit will decrease label values, and the process can be repeated indefinitely. We now prove that in all other cases the algorithm will terminate in a finite number of steps and at termination, for every vertex υ, λ(ν) will give the length of a shortest directed path from J to v. Lemma. 11.2. If the graph G has no directed circuits of negative lengths, then, if at any stage of Algorithm 11.4 λ(υ) is finite, there is a directed path from J to w whose length is λ(υ). Proof. We prove the result by displaying a directed path from s to v. The construction is backward from v. Let u denote the vertex that gave υ its present label λ(υ). If A ' ( M ) represents the label of u at the time it gave ν the label λ(ι>), then X(v) = A ' ( M ) + w(e), where e = (Μ, V). Now, continue from u to the vertex that gave it the label A ' ( « ) , and so on. In this process no vertex will be encountered more than once because each step in the process refers to an earlier time in the execution of the algorithm, and a vertex can decrease its own label only by going through a directed circuit of negative length. Thus

SHORTEST PATHS

321

there is a directed path from s to ν such that the label A(i/) is the sum of the lengths of the edges on this path, and the required result follows. • Recall that d(s, υ), the distance from s to u, denotes the length of a shortest directed path from s to v. Theorem 11.3. For a directed graph G with no directed circuits of negative length, the Bellman-Ford-Moore Algorithm (Algorithm 11.4) terminates in a finite number of steps, and upon termination, A(u) = d(s, v) for all v. Proof. Consider any vertex ν in G. By Lemma 11.2, at any stage of the Bellman-Ford-Moore algorithm, A ( u ) represents the length of a directed path from s to v. Since there are only a finite number of such paths, it follows that the number of possible values for A(i>) is also finite. Clearly, upon termination of the algorithm, A(u) 2 d(s, v). If A(u)> d(s, υ), then let P: s = v , v ,..., v = ν be a shortest directed path from s to υ and let e, denote the edge (υ,·_ι> i>,) on this path. For every i = 0, 1, 2 , . . . , k, we have 0

x

k

d(s, υ,)

= Σ w(e,) •

Let v, be the first vertex on this path for which, upon termination of the algorithm, λ(υ,) > d(s, v ). Since λ(υ,_,) = d(s, υ , . , ) we have d(s, v ) = A(y,_j) + w(e ) and so upon termination A ( u ) > A ( u , _ , ) + w(e ). This is a contradiction because when the algorithm terminates there is no edge e = (M, W) with A ( H » ) > A(w) + w(e). Thus, for every vertex υ, on Ρ, λ(ι>,) = d(s, v,). In particular, X(v ) = A(i>) = d(s, v) and the theorem follows. • t

t

(

:

l

k

Note that the number of directed paths from J to a vertex could be exponential in the number of vertices of the graph. So, if in the implementation of Algorithm 11.4 edges are selected in an arbitrary manner, it is possible that the algorithm may perform an exponential number of operations before terminating. We can show that the following implementation of the algorithm will achieve a complexity of 0(mri) where m and η are, respectively, the number of edges and the number of vertices of G. Order the edges as e,, e ,...,e . Perform S3, examining edges in this order and updating the vertex labels whenever required. After one such scan or sweep of all the edges, perform additional sweeps until an entire sweep produces no changes in the vertex labels. If the number of edges in a shortest directed path from s to υ is k, then it can be shown by induction that by the end of the kth sweep, ν will have its final label. Since k η - 1 and each sweep requires 0(m) operations, this implementation of the Bellman-Ford-Moore algorithm requires 0(mn) time. Sometimes we may be interested in getting the second, third, or higher 2

m

322

GRAPH ALGORITHMS

shortest paths. These and related problems are discussed in Dreyfus [11.20], Hu [11.21], Frank and Frisch [11.21], Christofides [11.23], Lawler [11.24], Gordon and Minoux [11.25], Minieka [11.26], and Spira and Pan [11.27]. For algorithms designed for sparse networks see Johnson [11.28] and Wagner [11.29]. See also Johnson [11.30], Edmonds and Karp [11.31], and Fredman [11.32]. For a shortest path problem that arises in solving a special system of linear inequalities and its applications see Comeau and Thulasiraman [11.33], Liao and Wong [1.34], and Lengauer [11.35]. 11.2.2 Shortest Paths between All Pairs of Vertices Suppose that we are interested in finding the shortest paths between all the n{n - 1) ordered pairs of vertices in an η-vertex directed graph. A straightforward approach to get these paths would be to use Dijkstra's algorithm η times. However, there are algorithms that are computationally more efficient than this. These algorithms are applicable even when the lengths are negative, but there are no negative-length directed circuits. Now we discuss one of these algorithms. This algorithm, due to Floyd [11.36], is based on Warshall's procedure (Algorithm 11.1) for computing transitive closure. Consider an η-vertex directed graph G with lengths associated with its edges. Let the vertices of G be denoted as 1, 2 , . . . , n. Assume that there are no negative-length directed circuits in G. Let W = [ H ' ] be the η x η matrix of direct lengths in G, that is, w is the length of the directed edge (/, ;') in G. We set w = °° if there is no edge (i,;') directed from ι to /. We also set w„ = 0 for all i. Starting with the matrix W = W, Floyd's algorithm constructs a sequence W \W of η x η matrices so that the entry in W would give the distance from i to ; in G. The matrix = [w]^] is constructed from the matrix = [ w | * ] according to the following rule: i /

tl

(j

( 0 )

(

2

(

n

)

M

_I)

<

)

= m i n { < - " , w r

,

)

+ < - '

)

} .

(H.7)

Let Pf^ denote a path of minimum length among all the directed i-j paths, which use as internal vertices only those from the set { 1 , 2 , . . . , /c}. The following theorem proves the correctness of Floyd's algorithm. Theorem 11.4. For 0 < k < n,

is equal to the length of P$\

Proof. Proof follows along the same lines as that for Theorem 11.1.

•

Usually, in addition to the shortest lengths, we are also interested in obtaining the paths that have these lengths. Recall that in Dijkstra's algorithm we use the PRED array to keep a record of the vertices that occur in the shortest paths. This is achieved in Floyd's algorithm as described next.

SHORTEST PATHS

323

As we construct the sequence W , W \..., W "\ we also construct another sequence Z , Z , . . . , Z of matrices such that the entry z'*' of Z gives the vertex that immediately follows vertex i in P{*\ Clearly, then (0)

( 0 )

( 1 )

(1

l

< n )

( t )

z Given Z * rule: Let (

0

( 0 )

j, = 0,

if νν Φ οο ; ifw„ = o o . η

ί - ) 1 1

8

= [z, * ° ] , Z *' = [z,^] is obtained according to the following (

(

M =min{<-",w,r

+

i )

<- } 1)

Then iz**" , 0

ifM=H'<*-

^· - U - « .

1 )

«*<•*-».

(

η

·

9

)

This rule is similar to the one given in S3 of Algorithm 11.3 for updating the PRED array. The justification for this rule is as follows. If

then Length of />{*> = length of

.

Therefore zj*' is the same as z**" '. On the other hand, if Μ < w^' , then P ^ is the concatenation of the paths Pf ' and P ~ in that order. So 1

x)

x)

k

k

x)

k

_<*)

=

7

(*-D

It should be clear that the shortest i-j path is given by the sequence /', ι,, h< • • • > ' p ' J °f vertices, where =

/=

ϊ , - z g ,

Note that in (11.7) = if simple observation is made use of in algorithm. We have also incorporated presence of a negative-length directed

(11.10)

w<* or wj£ is equal to °°. This the following description of Floyd's in this algorithm a test to detect the circuit. υ

0

Algorithm 11.5. Shortest Paths between All Pairs of Vertices (Floyd)

SI. W= [w,j] is the η x η matrix of direct lengths in the given directed graph G. Here w = 0 for all i = 1, 2 , . . . , η. Ζ = [z ] is an η x η lt

iy

324

GRAPH ALGORITHMS

matrix in which V S2. Set k =

0,

if w = y

oo.

0.

53. Set k = k + 1. For all ι ^ k such that w # oo, and all / Φ k such that w Φ oo, do the following: 1. Set Μ — min{n' y, w + w ). 2. If Μ < w , then set z = z and w = M. rt

kj

I

t]

lk

k)

tj

ik

tj

54. 1. If any w„<0, then vertex i is in some negative-length directed circuit, and so HALT. 2. If every w s O and k = n, then [ w j gives the lengths of all the shortest paths, and z gives the first vertex after vertex i in a shortest directed i-j path. HALT. 3. If all w„ > 0 but k < n, then go to S3. • u

it

It is easy to see that Floyd's algorithm is of complexity 0 ( n ) . This algorithm is also valid for finding shortest paths in a network with negative lengths provided the network does not have a directed circuit of negative length. Dantzig [11.37] proposed a variant of Floyd's algorithm that is also of complexity 0 ( n ) . For some of the other shortest path algorithms see Tabourier [11.38], Yen [11.39], and Williams and White [11.40]. Pierce [11.41] and Deo and Pang [11.42] give exhaustive bibliographies of algorithms for the shortest path and related problems. A discussion of complexity results for shortest path problems can be found in Melhorn [11.14] and Tarjan [11.43]. Discussions of shortest path problems in a more general setting can be found in [11.14] and [11.25], Carre [11.44], and Tarjan [11.45]. For some recent developments see Frederickson [11.46] and Moffat and Takoka [11.47]. 3

3

11.3

MINIMUM WEIGHT SPANNING TREE

Consider a weighted connected undirected graph G with a nonnegative real weight w(e) associated with each edge e of G. The weight of a subgraph of G will refer to the sum of the weights of the edges of the subgraph. In this section we discuss the problem of constructing a minimum weight spanning tree of G. Recall that this problem was discussed earlier in Section 10.9 in the general setting of a matroid, namely, determining a maximum or minimum weight independent set of a matroid. As noted earlier, the greedy algorithm presented in that section was first proposed by Kruskal in [11.48] in the context of the minimum weight spanning tree problem. Our interest in this section is to present a unified treatment of two algorithms for this

325

MINIMUM WEIGHT SPANNING TREE

problem, namely, Kruskal's algorithm and the one due to Prim [11.49]. The next two theorems provide the basis of these algorithms. Theorem 11.5. Consider a vertex υ in a weighted connected graph G. Among all the edges incident on v, let e be one of minimum weight. Then, G has a minimum weight spanning tree that contains e. Proof. Let T be a minimum weight spanning tree of G. If T does not contain e, then consider the fundamental circuit C of T^,, with respect to e. Let e' be the edge of C that is adjacent to e. Clearly e' £ T . Also 7" = T - e' + e is a spanning tree of G. Since e and e' are both incident on v, we get w(e)^w(e'). But w ( e ) > w ( e ' ) because w(T') = w(T ) - w(e') + w(e) > w(T ). So, w(e) = w(e') and w(T') = w(T ). Thus Γ is a minimum weight spanning tree. The theorem follows since T' contains e. U mm

mm

m i n

min

mtn

mia

min

Theorem 11.6. Let Γ be an acyclic subgraph of a weighted connected graph G such that there exists a minimum weight spanning tree containing T. If G' denotes the graph obtained by contracting the edges of T, and T ^ is a minimum weight spanning tree of G', then T' U Τ is a minimum weight spanning tree of G. in

min

Proof. Let r be a minimum weight spanning tree of G containing T. If T = Τ U T', then clearly T' is a spanning tree of G'. Therefore m i n

m i n

(11.11)

w(T')>w(T' J. m

It is easy to see that T'

mm

U Τ is also a spanning tree of G. So

w(T; ur)>w(T i n

m i n

)

= νν(Γ) + νν(Γ').

(11.12)

From this we get H>(r;, )>.v(r). n

(11-13)

Combining (11.11) and (11.13) we get w(T' ) = w(T'), and so w(T' U T) = w{T' U T) = w(T ). Thus T' U Τ is a minimum spanning tree of G. • mia

min

min

We now present Kruskal's algorithm. Algorithm 11.6. Minimum Weight Spanning Tree (Kruskal)

SO. G is the given nontrivial η-vertex weighted connected graph.

min

326

GRAPH ALGORITHMS

5 1 . Set i = 1 and E = 0. 0

52. Select an edge e, of minimum weight such that e,&E,_ and the edge-induced subgraph (E,_iU{e,}) is acyclic, and then define T, = ( £ , _ , U {e,}) and £, = U {e,}. If no such edge exists, then let T = Γ,., and HALT. 53. Set i = i + 1 and go to S2. • l

m i n

Kruskal's algorithm essentially proceeds as follows. Edges are first sorted in the order of nondecreasing weights and then examined, one at a time, for inclusion in a minimum weight spanning tree. An edge is included if it does not form a circuit with the edges already selected. For an illustration of Kruskal's algorithm see Section 10.9. Next we prove the correctness of Kruskal's algorithm. Theorem 11.7. Kruskal's algorithm constructs a minimum weight spanning tree of a weighted connected graph. Proof. Let G be the given nontrivial weighted connected graph. Clearly, when Kruskal's algorithm terminates, the edges selected will form a spanning tree of G. Thus the algorithm terminates with i = n, and T„_ is a spanning tree of G. We next establish that Γ„_, is indeed a minimum weight spanning tree of G by proving that every 7", constructed in the course of Kruskal's algorithm is contained in a minimum weight spanning tree of G. Our proof is by induction on /'. Clearly by Theorem 11.5 G contains a minimum weight spanning tree that contains the edge e In other words T is contained in a minimum weight spanning tree of G. As inductive hypothesis assume that T„ i s 1 is contained in a minimum weight spanning tree of G. Let G' be the graph obtained by contracting the edges of T Again, by Theorem 11.5, the edge e is contained in a minimum weight spanning tree T' of G'. By Theorem 11.6, T, U T' is a minimum weight spanning tree of G. More specifically T = T U {e } is contained in a minimum weight spanning tree of G and the correctness of Kruskal's algorithm follows. • t

v

x

r

i+l

min

mm

i+l

t

i+1

We next present another algorithm due to Prim [11.49] to construct a minimum-weight spanning tree of a weighted connected graph. As we shall see soon, this algorithm is also based on Theorems 11.5 and 11.6 and can be viewed as a variant of Kruskal's algorithm. Algorithm 11.7. Minimum Weight Spanning Tree (Prim)

50. G is the given nontrivial η-vertex weighted connected graph. 51. Set i = 1 and E = 0. Select any vertex, say v, of G and set V = {v}. 52. Select an edge e, = (p, q) of minimum weight such that e, has exactly 0

x

OPTIMUM BRANCHINGS

one end vertex, say p, in V Define V _ = V \J {q}, E , = {e,}, and Γ, = ( £ , _ , U e,). S3. If i < η - 1, set i = / + 1 and return to S2. Otherwise, let T = and HALT. • r

iJI

x

i

327

U

i

m i n

T_ n

1

As in Kruskal's algorithm, Prim's algorithm also constructs a sequence of acyclic subgraphs 7 * , . . . , and terminates with T _ , a minimum weight spanning tree of G. The subgraph T is constructed from T, by adding an edge of minimum weight with exactly one end vertex in T . This construction ensures that all T,'s are connected. If G' denotes the graph obtained by contracting the edges of T,, and v' denotes the vertex of G', which corresponds to the vertex set of 7",·, then e , is in fact a minimum weight edge incident on v' in G'. This observation and Theorems 11.5 and 11.6 can be used (as in the proof of Theorem 11.7) to prove that each T, is contained in a minimum weight spanning tree of G. This would then establish that the spanning tree produced by Prim's algorithm is a minimum weight spanning tree of G. For complexity results relating to the minimum weight spanning tree enumeration problem see Kerschenbaum and Van Slyke [11.50], Yao [11.51], and Cheriton and Tarjan [11.52]. For sensitivity analysis of minimum weight spanning trees and shortest path trees see Tarjan [11.53]. See Papadimitriou and Yannakakis [11.54] for a discussion of the complexity of restricted minimum weight spanning tree problems. For a history of the minimum weight spanning tree problem see Graham and Hall [11.55]. 2

n

1

l + 1

t

+1

11.4

OPTIMUM BRANCHINGS

Consider a weighted directed graph G = (V, E). Let w(e) be the weight of edge e. Recall that the weight of a subgraph of G is defined to be equal to the sum of the weights of all the edges in the subgraph. A subgraph G of G is a branching in G if G has no directed circuits and the in-degree of each vertex of G, is at most 1. Clearly each component of G is a directed tree. A branching of maximum weight is called an optimum branching. In this section we discuss an algorithm due to Edmonds [11.56] for computing an optimum branching of G. Our discussion here is based on Karp [11.57]. An edge e = (i, ;') directed from vertex i to vertex ;' is critical if s

s

s

1. w ( e ) > 0 . 2. w(e) > w(e') for every edge e' = (k, j) incident into j . A spanning subgraph Η of G is a critical subgraph of G if

328

GRAPH ALGORITHMS

1. Every edge of Η is critical. 2. The in-degree of every vertex of Η is at most 1. A directed graph G and a critical subgraph Η of G are shown in Fig. 11.7. It is easy to see that 1. Each component of a critical subgraph contains at most one circuit, and such a circuit will be a directed circuit. 2. A critical subgraph with no circuits is an optimum branching of G. Consider a branching B. Let e = (/, /) be an edge not in B, and let e' be the edge of Β incident into vertex j . Then e is eligible relative to Β if B' =

(BUe)-e'

is a branching. For example, the edges {e,, e , e , e , e , e } form a branching Β of the graph of Fig. 11.7. The edge e , not in B, is eligible relative to Β since 2

3

4

7

g

6

(B U e ) - e 6

7

is a branching of this graph. The following two lemmas are easy to prove, and they lead to Theorem 11.8, which forms the basis of Karp's proof of the correctness of Edmonds' algorithm. In the following the edge set of a subgraph Η will also be denoted by H. Lemma 11.3. Let Β be a branching, and let e = (i, j) be an edge not in B. Then e is eligible relative to Β if and only if in Β there is no directed path from / to i. • Lemma 11.4. Let β be a branching and let C be a directed circuit such that no edge of C - Β is eligible relative to B. Then |C - B\ = 1. • Theorem 11.8. Let Η be a critical subgraph. Then there exists an optimum branching Β such that, for every directed circuit C in H, \C- B\ - 1. Proof. Let Β be an optimum branching that, among all optimum branchings, contains a maximum number of edges of the critical subgraph. Consider any edge eEHB. Let e be incident into vertex /, and let e' be the edge of Β incident into /. If e were eligible, then (B U e) - e' would also be an optimum branching, containing a larger number of edges

OPTIMUM BRANCHINGS

329

Figure 11.7. (a) A directed graph G. (6) A critical subgraph of G.

GRAPH ALGORITHMS

330

of Η than Β does; a contradiction. Thus no edge of Η - Β is eligible relative to B. So, by Lemma 11.4, for each circuit C in H, \C - B\ = 1. • Let C,, C , . . . , C be the directed circuits in H. Note that no two circuits of Η can have a common edge. In other words these circuits are edgedisjoint. For each C , let e° be an edge of minimum weight in C,. 2

k

(

Corollary 11.8.1. There exists an optimum branching Β such that: 1. \C,-B\ = 1, i = l , 2 , . . . , A . 2 . If no edge of Β - C, is incident into a vertex in C,, / = C ~B

1, 2 , . . . ,

(11.14)

=e. a

l

k, then

l

Proof. Let, among all optimum branchings satisfying item 1, β be a branching containing a minimum number of edges from the set {e\, e , •. •, e° }. We now show that Β satisfies item 2 . If not, suppose, for some i, e" ε Β, but no edge of Β - C, is incident into a vertex in C,. Let e= C,- B. Then (B - ef) U e is clearly an optimum branching that satisfies item 1 but has fewer edges than Β from the set {e\, e\,..., e° ). This is a contradiction. • 2

k

k

This result is very crucial in the development of Edmonds' algorithm. It suggests that we can restrict our search for optimum branchings to those that satisfy (11.14). Now we construct, from the given graph G, a simpler graph G'. We also show how to construct from an optimum branching of G' an optimum branching of G that satisfies (11.14). As before, let Η be the critical subgraph of G and let C,, C , . . . , C be the directed circuits in H. The graph G' is constructed by contracting all the edges in each C„ i = l, 2,...,k. In G', vertices of each circuit C, are represented by a single vertex a,, called a pseudo-vertex. The weights of the edges of G' are the same as those of G, except for the weights of the edges incident into the pseudo-vertices. These weights are modified as follows. Let e = (j, /') be an edge of G such that / is a vertex of some circuit C and ι is not in C . Then in G', e is incident into the pseudo-vertex a . Define e as the unique edge in C that is incident into vertex Then in G' the weight of e, denoted by w'(e), is given by 2

k

r

r

r

r

w\e) = w(e) - w(e) + w(e° ) . r

(11.15)

For example, consider the edge e incident into the directed circuit {e , e , e , e ) of the critical subgraph of the graph G of Fig. 11.7. Then e, = e , and the weight of e in G' is given by x

2

3

5

4

s

x

OPTIMUM BRANCHINGS

331

w'(e ) = w(e ) - w(e ) + w(e ) x

x

5

4

=5-6+5 = 4. Note that e is a minimum weight edge in the circuit {e , e , e , e ). Let Ε and Ε', respectively, denote the edge sets of G and G'. We now show how to construct from a branching fl' of G' a branching Β of G that satisfies (11.14) and vice versa. For any branching Β of G that satisfies (11.14) it is easy to see that 2

4

3

4

s

(11.16)

Β' = ΒΠΕ'

is a branching of G'. Further B' as defined is unique for a given B. Next consider a branching B' of G'. For each C,, let us define C[ as follows: 1. If the in-degree in B' of a pseudo-vertex a, is zero, then

c; = c , - ° . c

2. If the in-degree in Β' of α, is nonzero, and e is the edge of B' incident into a , then t

c; = c , - e . Then it is easy to see that (11.17)

B = B ' \ J C ; .

is a branching of G that satisfies (11.14). Further Β as defined is unique for a given B'. Thus we conclude that there is a one-to-one correspondence between the set of branchings of G that satisfy (11.14) and the set of branchings of G'. Furthermore, the weights of the corresponding branchings Β and B' satisfy k

k

w(B) - w(B') = Σ w(C,) - Σ w(e^) . 1=1

(11.18)

1=1

This property of Β and B' implies that if Β is an optimum branching of G that satisfies (11.14), then B' is an optimum branching of G' and vice versa. Thus we have proved the following theorem. Theorem 11.9. There exists a one-to-one correspondence between the set of

332

GRAPH ALGORITHMS

all optimum branchings in G that satisfy (11.14) and the set of all optimum branchings in G'. • Edmonds' algorithm for constructing an optimum branching is based on Theorem 11.9 and is as follows: Algorithm 11.8. Optimum Branching (Edmonds) 51. From the given graph G = G construct a sequence of graphs G , G,, G , . . . , G , where 1. G is the first graph in the sequence whose critical subgraph is acyclic and 2. G 1 :£ i =£ k, is obtained from G,_, by contracting the circuits in the critical subgraph //,_, of G,_, and altering the weights as in (11.15). 52. Since H is acyclic, it is an optimum branching in G . Let B = H . Construct the sequence B _ , B _ ,..., B , where 1. Β,, 0 ^ i s k - 1, is an optimum branching of G, and 2. B , for i s O , is constructed by expanding, as in (11.17), pseudovertices in B . • 0

2

0

k

k

n

k

k

k

1

k

2

k

k

0

t

l+l

As an example let G be the graph in Fig. 11.7a, and let H be the graph in Fig. 11.76. H is the critical subgraph of G„. After contracting the edges of the circuits in H and modifying the weights, we obtain the graph G, shown in Fig. 11.8a. The critical subgraph H of G, is shown in Fig. 11.86. //, is acyclic. So it is an optimum branching of G,. An optimum branching of G is obtained from H by expanding the pseudo-vertices a, and a (which correspond to the two directed circuits in H ), and it is shown in Fig. 11.8c. Tarjan [11.58] gives an 0(m log n) implementation of Edmonds' algorithm, where m is the number of edges and η is the number of vertices. See also Camerini, Fratta, and Maffioli [11.59]. Chu and Liu [11.60] and Bock [11.61] have also independently discovered Edmonds' algorithm. 0

0

0

0

l

0

l

2

0

11.5 PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

The optimal assignment and the timetable scheduling problems, the study of which involves the theory of matching, are discussed in this section. Obtaining an optimal assignment requires as a first step the construction of a perfect matching in an appropriate bipartite graph. With this in view, we first discuss an algorithm for constructing a perfect matching in a bipartite graph.

PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

333

Figure 11.8. (a) Graph G . (b) H a critical subgraph of G,. (c) An optimum branching of graph G of Fig. 11.7a. 1

lt

GRAPH ALGORITHMS

334

11.5.1

Perfect Matching

Consider the following personnel assignment problem in which η available workers are qualified for one or more of η available jobs, and we are interested to know whether we can assign jobs to all the workers, one job per worker, for which they are qualified. If we represent the workers by one set X={x , x ,...,x } of vertices and the jobs by the other set Y = {y , y , • • •, y„) of vertices of a bipartite graph G, in which x, is joined to y if and only if the worker x, is qualified for the job _y, then it is clear that the personnel assignment problem is to find whether the graph G has a perfect matching or not. One method of finding a solution for this problem would be to apply an algorithm to find a maximum matching. Such an algorithm will be discussed in Chapter 12. If the maximum matching consists of η edges, then it shows that the graph has a perfect matching, and the maximum matching is a perfect matching. The main drawback of this method is that if the graph does not have a perfect matching, then we will know this only at the end of the procedure. Now we discuss an algorithm that either finds a perfect matching of G or stops when it finds a subset S of X such that |Γ(5)| < |S|, where Γ(5) is the set of vertices adjacent to those in 5. Clearly, by Hall's theorem (Theorem 8.13) there exists no perfect matching in the latter case. The basic idea behind the algorithm is very simple. We start with an initial matching M. If Μ saturates all the vertices in X, then it is the one for which we are looking. Otherwise, we choose an unsaturated vertex u in X and systematically search for an augmenting path Ρ starting from u. This search would result in a tree called a Hungarian tree of the graph. At any stage, if we find an augmenting path, we perform the augmentation and get the new matching, which saturates one more vertex in X, and proceed as before. If such a path does not exist, then we would have obtained a set S C X, violating the necessary and sufficient condition for the existence of a perfect matching. The details of this search procedure are discussed below. Let Μ be a matching in G, and let u be an unsaturated vertex in X. A tree Η in G is called an M-alternating tree rooted at u if: t

x

2

n

2

t

;

1. u belongs to the vertex set of H. 2. For every vertex ν of H, the unique path from u to i; in Η is an Λί-alternating path (i.e., an alternating path relative to M). Let us denote by S the subset of vertices of X and by Τ the subset of vertices of Y that occur in H. The alternating tree is grown as follows. Initially, Η consists of only the vertex u. It is then grown in such a way that at any stage, there are two possibilities.

PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

Μ

335

(6)

Figure 11.9. Examples of alternating trees. 1. All vertices of Η except u are saturated (e.g., see Fig. 11.9a). 2. Η contains an unsaturated vertex different from u (e.g., see Fig. 11.96), in which case we have an augmenting path and hence we get a new matching. In the first case either Γ(5) = Τ or Τ C Γ(5). la. Γ ( 5 ) = Γ . Since | 5 | = | Γ | + 1 in the tree H, we get in this case |T(S)| = |S| - 1, and so the set S does not satisfy the necessary and sufficient condition required by Hall's theorem. Hence we conclude that there exists no perfect matching in G. lb. Τ C Γ(5). So there exists a vertex y in Y that does not occur in Τ but that occurs in Γ(5). Let this vertex y be adjacent to vertex χ in S. If y is saturated with the vertex ζ as its mate, then we grow Η by adding the vertices y and ζ and the edges (x, y) and (y, z). We are then back to the first case. If y is unsaturated, we grow Η by adding the vertex y and the edge (*, y), resulting in the second case. The path from u to y in Η is an augmenting path relative to M. This method is presented in the following algorithm. Algorithm 11.9. Perfect Matching SI. Let G be a bipartite graph with bipartition (X, Y) and 1*1 = \Y\. Let M be the null matching, that is, M = 0 . Set /' = 0. 0

0

336

GRAPH ALGORITHMS

52. If all the vertices in X are saturated in the matching M, then HALT. (Λί, is a perfect matching in G.) Otherwise pick an unsaturated vertex Μ in A' and set S = {u} and Τ = 0 . 53. If Γ ( 5 ) = T, then HALT. (Now | Γ ( 5 ) | < | 5 | , and hence there is no perfect matching in G.) Otherwise select a vertex y from Γ(5) - T. 54. If y is not saturated in M„ go to S5. Otherwise set ζ = mate of y , S = 5 U {z} and T= TU{y}, and then go to S3. 55. (An augmenting path Ρ is found.) Set M = Μ, Θ Ρ and i = i + 1. Go to S2. • l+1

As an example, consider the bipartite graph G shown in Fig. 11.10a. In this graph the edges of an initial matching Μ are shown in dashed lines. The vertex x, is not saturated in M. The M-alternating tree rooted at x, is now developed. We terminate the growth of this tree as shown in Fig. 11.106 when we locate the augmenting path x,, y , x , y . We then augment Μ and obtain the new matching shown in Fig. 11.10c. Vertex x is not saturated in this new matching. So we proceed to develop the alternating tree rooted at x , with respect to the new matching. This tree terminates as shown in Fig. ll.lOd. Further growth of this tree is not possible since at this stage Γ(5) = Τ, where 5 = {χ,, x , x , x } and Τ = { > Ί ' y-ii y i ) - Hence the graph in Fig. 11.10a has no perfect matching. Sometimes we may be interested in finding perfect matchings having a specific property. Itai, Rodeh, and Tanimoto [11.62] discuss an algorithm applicable for a class of such problems. x

3

2

4

4

2

11.5.2

3

4

Optimal Assignment

Consider an assignment problem in which each worker is qualified for all the jobs. Here it is obvious that every worker can be assigned a job (of course, we assume, as before, that there are η workers and η jobs). In fact, any maximum matching performs this, and we have got n\ such matchings. A problem of interest in this case is to take into account the effectiveness of the workers in their various jobs, and then to make that assignment, which maximizes the total effectiveness of the workers. The problem of finding such an assignment is known as the optimal assignment problem. The bipartite graph for this problem is a complete one; that is, if X = {x,, x , . . . ,x„) represents the workers and Y = { y y , . . . , y } represents the jobs, then for all i and /', x, is adjacent to y Also, we assign a weight Wjj = w(x,, yj) to every edge (x , y ) , which represents the effectiveness of worker x, in job y (measured in some unit). Then the optimal assignment problem corresponds to finding a maximum weight perfect matching in this weighed graph. Such a matching is referred to as an optimal matching. We now discuss an 0(n ) algorithm due to Kuhn [11.63], and Munkres [11.64], for the optimal assignment problem. We follow the treatment given in Bondy and Murty [11.65]. 2

u

r

(

;

3

t

2

n

PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

Figure 11.10

337

338

GRAPH ALGORITHMS

A feasible vertex labeling is a real-valued function / on the set X U Y such that f(x) + / ( y ) ^

W(JC,

for all χ £ X and y e Y.

y)

f(x) is then called the label of the vertex x. For example, the following labeling is a feasible vertex labeling: /(JC) = max{ tv(jc, y)} , if JC £ X ; /(y) = 0 ,

ify£y.

From this it should be clear that there always exists a feasible vertex labeling irrespective of what the weights are. For a given feasible vertex labeling / , let E denote the set of all those edges (JC, y) of G such that /(JC) + / ( y ) = H>(JC, y). The spanning subgraph of G with the edge set E is called the equality subgraph corresponding to / . We denote this subgraph by G . The following theorem relating equality subgraphs and optimal matchings forms the basis of the Kuhn-Munkres algorithm. f

f

f

Theorem 11.10. Let / b e a feasible vertex labeling of a graph G = (V, E). If G contains a perfect matching Λί*, then Λί* is an optimal matching in G. f

Proof. Suppose that G contains a perfect matching Λί*. Since G is a spanning subgraph of G, Λί* is also a perfect matching in G. Let w(M*) denote the weight of Λί*, that is, f

f

νν(Λί*) = Σ

w(e).

Since each edge e £ Λί* belongs to the equality subgraph and the vertices of the edges of Λί* cover each vertex of G exactly once, we get w(M*)= Σ

w(e)

= Σ Μ

.

(11.19)

On the other hand, if Λί is any perfect matching in G, then w(M)=

Σ w(e)

(EM

=ΞΣ/(Ι>). uev

(11.20)

PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

339

Now combining (11.19) and (11.20), we see that w(M*)>w(M). Thus M* is an optimal matching in G.

•

In the Kuhn-Munkres algorithm, we first start with an arbitrary feasible vertex labeling / and find the corresponding G . We will choose an initial matching Μ in G and apply Algorithm 11.9. If a perfect matching is obtained in G , then, by Theorem 11.10, this matching is optimal. Otherwise Algorithm 11.9 terminates with a matching M' that is not perfect, giving an Μ'-alternating tree Η that contains no Μ'-augmenting path and that cannot be grown further in G . We then modify / to a feasible vertex labeling / ' with the property that both M' and Η are contained in G ., and Η can be extended in G ,. We make such a modification in the feasible vertex labeling whenever necessary, until a perfect matching is found in some equality subgraph. Details of the Kuhn-Munkres algorithm are presented next. f

f

f

f

f

f

Algorithm 11.10. Optimal Assignment (Kuhn and Munkres) 5 1 . G is the given complete bipartite graph with bipartition (AT, Y) and \X\ = \ Y\. W= [w ] is the given weight matrix. Set i = 0. 52. Start with an arbitrary feasible vertex labeling / in G. Find the equality subgraph G and then select an initial matching Λί,· in G . 53. If all the vertices in X are saturated in Λί,, then Λί, is a perfect matching, and hence by Theorem 11.10, it is an optimal matching. So HALT. Otherwise let u be an unsaturated vertex in X. Set S = {«} and T = 0. iy

f

f

54. Let T (S) be the set of vertices that are adjacent in G to the vertices in S. If T (S)DT, then go to S5. Otherwise (i.e., if V (S)=T) compute f

f

f

f

d, = min { /(*) + f(y)~

w(x, y)}

(11.21)

and get a new feasible vertex labeling / ' given by \f(v)-d , = \f(v) + d , [/(f), f

f'(v)

f

if i ; E S ; if vET; otherwise.

(11.22)

(Note that d > 0 and T (S) = T.) Replace / by / ' and G by G .. 55. Select a vertex y from T (S) - T. If y is not saturated in M,, go to S6. f

f

f

f

f

340

GRAPH ALGORITHMS

Otherwise set ζ = mate of y in M,,, S - S U {2} and Γ = Τ U {y}, and then go to S4. S6. (An augmenting path Ρ is found.) Set M = Μ , Φ Ρ and / = i + 1. Go to S3. • i+l

To illustrate the Kuhn-Munkres algorithm consider a complete bipartite graph G having the following weight matrix W= [w ]: tJ

4 4 1 3 3 2 2 1 W= 5 4 4 3 1 1 2 2J An initial feasible vertex labeling / of G may be chosen as follows: f( )

= 4,

Xl

f(x ) = 3,

f(x ) = 5,

2

3

f(x ) = 2; 4

The equality subgraph G^-is shown in Fig. 11.lie. Applying Algorithm 11.9, we find that G has no perfect matching because for the set S = {JC, , x , x ), T = r(5) = { y y } . Using (11.21) we compute f

2

1 ;

2

d = l. f

Figure 11.11

3

PERFECT MATCHING, OPTIMAL ASSIGNMENT, AND TIMETABLE SCHEDULING

341

The following new labeling/' is then obtained using (11.22): /'(*,) = 3 , f'(y )

= u

l

f'(x ) / ' ( >

/'(*,) = 4 ,

= 2,

2

2

) =

i .

/'(>'3) = o ,

/'(x ) = 2; 4

/'(y ) = 4

o .

The equality subgraph G . is shown in Fig. 11.116. Using Algorithm 11.9 on G ., we obtain the perfect matching Μ consisting of the edges (x , y ), (x , y ), ( J C , y ), and (x , y ) . This matching is an optimal matching. Megiddo and Tamir [11.66] discuss an 0(n log n) algorithm for a class of weighted matching problems that arise in certain applications relating to scheduling and optimal assignments. More recent results on the perfect matching and the optimal assignment problems can be found in Karp [11.67], Derigs [11.68], Avis [11.69], Avis and Lai [11.70], and Grigoriadis and Kalantari [11.71]. More discussions of matching algorithms and related references can be found in Chapter 12. f

f

2

l

x

11.5.3

3

3

4

2

4

Timetable Scheduling

In a school there are ρ teachers x , x ,... ,x and q classes y , y ,..., y. Given that teacher x, is required to teach class y for ρ periods, we would like to schedule a timetable having the minimum possible number of periods. This is a special case of what is known as the timetable scheduling problem. Suppose that we construct a bipartite graph G = (X, Y) in which the vertices in X represent the teachers and those in Y represent classes, and vertex x, G X is connected to vertex y G Y by p parallel edges. Since in any one period each teacher can teach at most one class and each class can be taught by at most one teacher, it follows that a teaching schedule for one period corresponds to a matching in G, and conversely each matching corresponds to a possible assignment of teachers to classes for one period. Thus the timetable scheduling problem is to partition the edges of G into as few matchings as possible. By Corollary 8.22.1, the minimum number of matchings in any partition of the edge set of a bipartite graph G is equal to the maximum degree in G. The proof of this theorem also suggests the following procedure for determining a partition having the smallest number of matchings: x

2

p

x

l

s

2

q

η

tj

Step 1. Let G be the given bipartite graph. Set i = 0 and G = G. Step 2. Construct a matching M of G, that saturates all the maximum degree vertices in G,. Step 3. Remove M, from G,. Let G denote the resulting graph. If G has no edges, then M , A / , , . . . ,M, is a required partition of the edge set of G. Otherwise set i = i + 1 and go to step 1. 0

t

1 + 1

0

( + 1

342

GRAPH ALGORITHMS

Clearly, the complexity of this procedure depends on the complexity of implementing step 2, which requires finding a matching that saturates all the maximum degree vertices in a bipartite graph G = (X, Y). Such a matching may be found as follows. (See proof of Theorem 8.21.) Let X denote the set of maximum degree vertices in X, and let Y denote the set of maximum degree vertices in Y. Let G denote the subgraph of G formed by the edges incident on the vertices in X . Similarly G is the subgraph on the edges incident on the vertices in Y . By Theorem 8.22 there is a matching M that saturates all the vertices in X„. The matching M is a maximum matching in G„. Similarly in G there is a maximum matching M that saturates all the vertices in Y . Following the procedure used in the proof of Theorem 8.21, we can find from M and M a matching Λί that saturates the vertices in X and Y . Then Λί is a required matching saturating all the maximum degree vertices in G. The complexity of finding Λί and M is the same as the complexity of finding a maximum matching that, as we shall see in Chapter 12, is 0(n ), where η is the number of vertices in the bipartite graph. It is easy to see that the complexity of constructing Μ from M and Λί,, is 0(n ). Thus the overall complexity of implementing step 2 is 0 ( n ) . This bound can be improved since all that we need in step 2 is a matching that saturates all the maximum degree vertices. Cole and Hopcroft [11.72] present such an algorithm of complexity 0(m log n), where m is the number of edges in the bipartite graph. An earlier algorithm for this problem due to Gabow and Kariv [11.73] had complexity 0(min {n log n, m log «}). Since step 2 will be repeated Δ times where Δ is the maximum degree in G and Δ == n, it follows that the complexity of constructing the required timetable is 0(mn log n). A more general problem follows. Let us assume that only a limited number of classrooms are available. The question is: How many periods are now needed to schedule a complete timetable? Suppose that / lessons are to be given and that they have to be scheduled in a p-period timetable. This timetable would require an average of \l/p] lessons to be given per period. So it is clear that at least \l/p] rooms will be needed in some one period. Interestingly one can always arrange / lessons in a p-period timetable so that at most \l/p] rooms are occupied in any one period. See Exercises 8.24 and 8.25. A discussion of the general form of the timetable scheduling problem and references related to this may be found in Even, Itai, and Shamir [11.74]. a

b

a

a

b

b

a

a

b

b

b

a

a

β

b

b

b

5

2

a

2 5

2

11.6

2

THE CHINESE POSTMAN PROBLEM

A postman picks up mail at the post office, delivers it along a set of streets, and returns to the post office. Of course, he must cover every street at least once, in either direction. The question is: What route would enable the

THE CHINESE POSTMAN PROBLEM

343

postman to walk the shortest distance possible? This problem known as the Chinese postman problem was first proposed by the Chinese mathematician Kwan [11.75]. If G denotes the weighted connected graph representing the streets and their lengths, then the Chinese postman problem is simply that of finding a minimum weight closed walk that traverses every edge of G at least once. We shall refer to such a minimum weight closed walk as an optimal Chinese postman tour. Any closed walk traversing every edge at least once will be referred to as a postman tour. If G is Eulerian, then clearly any Euler trail of G is an optimal postman tour. Suppose that G is not Eulerian. Then we can easily see that every postman tour of G corresponds to an Euler trail in the (Eulerian) graph G* that has the same vertex set as G and that has as many copies of an edge (i, / ) as the number of times it appears in the walk. Conversely, if G* is an Eulerian graph constructed by adding to G appropriate numbers of additional copies of edges, then each Euler trail of G* will correspond to a postman tour of G. We can consider G* as consisting of two types of edges: the original edges (the edges of G) and the pseudo-edges (the parallel edges asked to G to make it Eulerian). Our objective, therefore, is to obtain a G* such that the sum of the weights of its pseudo-edges is as small as possible. Note here that the weight of a pseudo-edge is the same as that of the corresponding edge. In order to gain an insight into the structure of G*, consider a vertex υ whose degree in G is odd. Clearly in G* an odd number of pseudo-edges must be incident on v. Let (u, v) be one such pseudo-edge. If u is of even degree in G, then there must be a pseudo-edge (w, u) incident on w, for otherwise the degree of u in G* will be odd. Continuing this argument, we can see that in G* there is a path of pseudo-edges that starts at υ and ends at a vertex whose degree in G is odd. Thus, if G has 2k, k^l odd degree vertices, then we can group these vertices into k distinct pairs (v ,v ). (υ ,ν ),... , ( i > * - i > 2k) in G* there is a path of pseudo-edges between the vertices of each pair. These observations suggest the following procedure to construct G* from G. Pick any two vertices, say u and υ, whose degrees in G are odd. Find an u-v path in G and add to G pseudo-edges along this path. Clearly, in the resulting graph G' both u and υ will have even degrees. Repeat this procedure, picking any two odd degree vertices in G'. This procedure will terminate when we have obtained an Eulerian graph G* in which all vertices have even degrees. It should be easy to see that the sum W* of the weights of pseudo-edges of the graph G* (constructed as above) is equal to the sum of the weights of the paths chosen to add the pseudo-edges. So, in order to minimize W* it is necessary that the paths selected must be the shortest ones. Also, our choice of pairs of odd-degree vertices influences the value of W*. In other words to l

v

3

Α

2

s u c n

t n a t

2

GRAPH ALGORITHMS

344

minimize W* we should group the odd-degree vertices into k pairs (w (l>

3)

V ), A

. . . ,(v k-U 2

2k)

V

s

u

c

h

t

h

a

v ),

lt

2

t

k i= 1

is as small as possible. Recall that d(v v ) denotes the distance between u, and v, in G. We can easily verify that the minimum value of W* and the corresponding shortest paths can be obtained by solving a maximum weight perfect matching problem in a complete bipartite graph as described in S3 and S4 of the following algorithm due to Edmonds [11.76] and Edmonds and Johnson [11.77]! it

t

Algorithm 11.11 Optimal Chinese Postman Tour (Edmonds and Johnson)

50. Given a weighted connected graph G, an optimal Chinese postman tour is required. 51. Identify all the odd-degree vertices of G. Let these be u,, i > , . . . , u , k a 1. If there are no odd-degree vertices in G, set G* = G and go to S5. 52. Compute the shortest paths between all pairs of odd-degree vertices. Let d(v , v ) denote the distance between u, and v 53. Construct a complete bipartite graph G' with bipartition as follows: 2k

2

t

t

r

X

=

{jCj,

x , • • · , X k} ' 2

2

Y={yi,y ,---,y k), 2

2

>0 o. =

w(x

n

y ) = Μ - d(v„ v,) f

for i V ; ,

where Μ is a large number. 54. Find a maximum weight perfect matching in G'. For each (x,, y ) in this perfect matching, add pseudo-edges to G along a shortest v-v, path. Let G* denote the resulting Eulerian graph. 55. Construct an Euler trail of G*. This trail defines an optimal Chinese postman tour. HALT. • t

It is easy to show that no edge can appear in more than one of the shortest paths identified in S4 in Algorithm 11.11. This means that no edge will be traversed more than twice in the optimal Chinese postman tour. Note that S2 can be carried out by Algorithm 11.5 for the all pairs shortest path problem. Step S4 can be carried out by Algorithm 11.10 for the optimal assignment problem. To carry out S5 we need an efficient algorithm to construct an Euler trail in an Eulerian graph. An algorithm due to Fleury [11.78], which achieves this, is described below.

THE CHINESE POSTMAN PROBLEM

345

Algorithm 11.12. Eulerian Trail (Fleury) 50. Given an Eulerian graph G = (V, E), an Euler trail of G is required. 5 1 . Let i = 0 and select an arbitrary vertex v of G and define T : v . 52. Given that the trail Γ,: v , e , u,, e ,..., e„ υ, has been constructed, select an edge e from 0

0

x

0

0

2

l+i

e,}

Ε — {e , e ,..., x

2

subject to the following conditions: ( ) i + i ' incident on υ,. a

e

s

(b) Unless there is no other choice, e

is not a bridge of the graph

l+l

G, = G - { e , , e , . . . , e , } . 2

If no such edge e exists, then HALT. 53. Define 7", : v , e , i>,, e ,.. .,e , v , where e 54. Set I = I + 1 and go to S2. • l+1

+1

0

x

2

l

+ l

l + l

l + 1

=(υ,,

υ ). ι+ι

Theorem 11.11. If G is Eulerian, then Fleury's algorithm constructs an Eulerian trail of G. Proof. Let G = (V, E) be an Eulerian graph. Suppose that Fleury's algorithm starts with vertex υ of G and terminates with the trail 0

T :v ,e ,v ,e ,v ,...,e ,v . p

0

i

1

2

2

p

p

Note that whereas the vertices u,'s of T may not all be distinct, the edges e,'s are. We need to show that T is an Eulerian trail of G. For i = 1, 2,...,p, let p

p

Gj = G - {e^e^..

.,e,} .

First we show that v = v . Since the algorithm terminates in v , the degree of υ in G is zero. If υ Φυ , then the degrees of v and v in T will be odd because they are the terminal vertices of T . This would then imply that the degree of v in G is odd since the degree of a vertex in G is equal to the sum of its degrees in G and T . A contradiction because G is Eulerian and every vertex in G has even degree. Hence v = v and T is a closed trail. Furthermore, this means that every vertex v, has even degree in T and so every vertex of G must also have even degree. We next show that T contains every edge of G. Suppose that this is not true. Let S denote the set of vertices of positive degree in G . Clearly S is nonempty and v e 5, where 5 = V — S. Let v be the last vertex of Τ that p

0

p

ρ

p

0

p

0

p

p

p

p

p

p

0

p

p

p

p

p

p

k

346

GRAPH ALGORITHMS

belongs to S. Since T terminates in υ Ε S, it follows that the edge k+i ~ ( k> k+i) * ' y edge of the cut (S,S) in G . In other words e is a bridge of G . For / = k, k + 1 , . . . , p, let G\ denote the subgraph of G, induced by the vertex set S. Since none of the edges e , e , . . . , e have both their end vertices in 5, it follows that G' = G' = · · • = G' . Also note that every vertex in G' has even degree because it is the subgraph of G induced by vertices of positive degree. Thus by Theorem 3.1, G' is Eulerian. Now consider any edge e Φ e of G incident on v . It follows from S2 of the algorithm that e is also a bridge of G , for otherwise e would not have been picked at this step. Thus e is a bridge of G' and also of G'. But G' has no bridges because, as we have seen, it is Eulerian and every edge of an Eulerian graph lies in a circuit of the graph (Corollary 3.1.1). Thus T is a closed trail and contains every edge of G. In other words Fleury's algorithm constructs an Eulerian trail of G. • p

e

v

s t r i e

v

ρ

o n

k

k+1

k

k+l

k

k + r

p

k+l

p

p

p

p

k+x

k

k

k

k+l

k

p

p

For a variant of the Chinese postman problem (namely, the rural Chinese postman problem) and its application in communication network protocol testing see Aho, Dahbura, Lee and Uyar [11.79]. For some generalizations of this problem and related algorithms see Orloff [11.80], Papadimitriou [11.81], Lenstra and Rinooy-Kan [11.82], Frederickson [11.83], and Dror, Stern, and Trudeau [11.84]. 11.7

DEPTH-FIRST SEARCH

In this section we describe a systematic method for exploring a graph. This method known as depth-first search, in short, DFS, has proved very useful in the design of several efficient algorithms. Some of these algorithms are discussed in the remaining sections of this chapter. Our discussion here is based on [11.85]. 11.7.1

DFS of an Undirected Graph

We first describe DFS of an undirected graph. To start with, we assume that the graph under consideration is connected. If the graph is not connected, then DFS would be performed separately on each component of the graph. We also assume that there are no self-loops in the graph. DFS of an undirected graph G proceeds as follows: We choose any vertex, say v, in G and begin the search from v. The start vertex υ, called the root of the DFS, is now said to be visited. We then select an edge (v, w) incident on υ and traverse this edge to visit w. We also orient this edge from ν to w. The edge (v, w) is now said to be examined and is called a tree edge. The vertex ν is called the father of w, denoted as FATHER(w). In general, while we are at some vertex x, two possibilities arise:

DEPTH-FIRST SEARCH

347

1. If all the edges incident on χ have already been examined, then we return to the father of χ and continue the search from FATHER(x). The vertex χ is now said to be completely scanned. 2. If there exist some unexamined edges incident on x, then we select one such edge (JC, y) and orient it from χ to y. The edge (x, y) is now said to be examined. Two cases need to be considered now: Case 1

Case 2

If y has not been previously visited, then we traverse the edge (x, y), visit y, and continue the search from y. In this case (x, y) is a tree edge and JC = FATHER(.y). If y has been previously visited, then we proceed to select another unexamined edge incident on x. In this case the edge (x, y) is called a back edge.

During the DFS, whenever a vertex χ is visited for the first time, it is assigned a distinct integer D F N ( J C ) such that D F N ( x ) = i, if χ is the ith vertex to be visited during the search. D N F ( J C ) is called the depth-first number of jt. Clearly, depth-first numbers indicate the order in which the vertices are visited during DFS. DFS terminates when the search returns to the root and all the vertices have been visited. As we can see from the preceding description, DFS partitions the edges of G into tree edges and back edges. It is easy to show that the tree edges form a spanning tree of G. DFS also imposes directions on the edges of G. The resulting directed graph will be denoted by G. The tree edges with their directions imposed by the DFS will form a directed spanning tree of G. This directed spanning tree will be called the DFS tree. Note that DFS of a graph is not unique since the edges incident on a vertex may be chosen for examination in any arbitrary order. As an example, we have shown in Fig. 11.12 DFS of an undirected graph. In this figure tree edges are shown as continuous lines, and back edges are shown as dashed lines. Next to each vertex we have shown its depth-first number. We have also shown in the figure the list of edges incident on each vertex v. This list for a vertex ν is called the adjacency list of v, and it gives the order in which the edges incident on ν are chosen for examination. We now present a formal description of the DFS algorithm. In this description the graph under consideration is not assumed to be connected. The array MARK used in the algorithm has one entry for each vertex. To begin with we set MARK(u) = 0 for every vertex υ in the graph, thereby indicating that no vertex has yet been visited. Whenever a vertex is visited, we set the corresponding entry in the MARK array equal to 1. The arrays D F N and FATHER are as defined before. TREE and BACK are two sets storing, respectively, the tree edges and the back edges as they are generated.

348

GRAPH ALGORITHMS

1

6

Vertex 1 2 3 4 5 6 7 8 9 10 11

Adjacency List (1,2), (1,3), (1,4) (2,1), (2,3), (2,8), (2,9), (2,10), (2,11) (3,1), (3,2), (3,4), (3,5), (3,6), (3,7) (4,1), (4,3), (4,5), (4,6) (5,3), (5,4) (6,3), (6,4) (7.3), (7,8) (8,2), (8,7) (9,2), (9,10), (9,11) (10,2), (10,9) (11,2), (11,9)

Figure 11.12. DFS of an undirected graph. Algorithm 11.13. DFS of an Undirected Graph 51. G is the given graph. Set TREE = 0 , BACK = 0 , and i = l. For every vertex υ in G, set FATHER(i>) = 0 and MARK(u) = 0. 52. (DFS of a component of G begins.) Select any vertex, say vertex r, with MARK(r) = 0. Set DFN(r) = i, MARK(r) = 1, and υ = r. (The vertex r is called the root of the component under consideration.) 53. If all the edges incident on υ have already been labeled "examined," then go to SS. (v is now completely scanned.) Otherwise select an edge (v, w) that is not yet labeled "examined" and go to S4. 54. Orient the edge (υ, w) from ν to w and label it "examined." Do the following and then go to S3.

DEPTH-FIRST SEARCH

1. If M A R K ( H - ) = 0, I = I

349

set

+ 1,

D F N ( M > ) = i,

T R E E = TREE U {(v, w)}, MARK(w) = 1, FATHER(w) = v, ν — w. 2. If MARK(w) = l, set BACK = BACK U {(v, w)}. 55. If FATHER(u) Φ0 (i.e., ν is not the root of the component under consideration), then set υ = FATHER(u) and go to S3. Otherwise go to S6. 56. If, for every vertex x, MARK(x) = 1, then go to S7; otherwise set i = / + l and go to S2. 57. (DFS is completed.) HALT. • Let Γ be a DFS tree of a connected undirected graph. As we mentioned before, Γ is a directed spanning tree of G. For further discussions, we need to introduce some terminology. If there is a directed path in Τ from a vertex υ to a vertex w, then υ is called an ancestor of w, and w is called a descendant of v. Furthermore, if υ Φ w, υ is called a proper ancestor of w, and w is called a proper descendant of v. If (v, w) is a directed edge in T, then ν is called the father of w, and w is called a son of v. Note that a vertex may have more than one son. A vertex ν and all its descendants form a subtree of Τ with vertex ν as the root of this subtree. Two vertices ν and w are related if one of them is a descendant of the other. Otherwise υ and w are unrelated. If υ and w are unrelated and DFN(i;), and v are descendants of s and s , respectively (see Fig. 11.13). Let Γ, and T denote the subtrees of Τ rooted at s, and s , respectively. Assume without loss of generality that D F N ( i j ) < D F N ( j ) . It is then clear from the DFS algorithm that vertices in 7Λ, are visited only after the vertex s, is completely scanned. Further, scanning of s, is completed only after all the vertices in Γ, are scanned completely. So there cannot exist an edge connecting u, and v . For if such an edge existed, it would have been visited before the scanning of Sj is completed. Thus we have the following. 2

2

2

2

x

2

2

2

2

2

350

GRAPH ALGORITHMS

Figure 11.13

»,

Theorem 11.12. If (v, w) is an edge in a connected undirected graph G, then in any DFS tree of G either ν is a descendant of w or vice versa. • The absence of cross edges in an undirected graph is an important property that forms the basis of an algorithm to be discussed in the next section for determining the biconnected components of a graph. 11.7.2

DFS of a Directed Graph

DFS of a directed graph is essentially similar to that of an undirected graph. The main difference is that in the case of a directed graph an edge is traversed only along its orientation. As a result of this constraint, edges in a directed graph G are partitioned into four categories (and not two as in the undirected case) by a DFS of G. An unexamined edge (v, w) encountered while at the vertex υ would be classified as follows. Case 1

w has not yet been visited.

In this case (v, w) is a tree edge. Case 2

w has already been visited.

a. If w is a descendant of ν in the DFS forest (i.e., the subgraph of tree edges), then (v, w) is called a forward edge.

DEPTH-FIRST SEARCH

351

b. If w is an ancestor of υ in the DFS forest, then (v, w) is called a back edge. c. If ν and w are not related in the DFS forest and D F N ( H ' ) DFN(u). The proof for this is along the same lines as that for Theorem 11.12. A few useful observations are now in order: 1. An edge (v, w), with DFN(w) > DFN(u), is either a tree edge or a forward edge. During the DFS it is easy to distinguish between a tree edge and a forward edge because a tree edge always leads to a new vertex. 2. An edge (v, w) with DFN(tv) < DFN(u) is either a back edge or a cross edge. Such an edge (v, w) is a back edge if and only if w is not completely scanned when the edge is encountered while examining the edges incident out of v. 3. DFS forest, the subgraph of tree edges, may not be connected even if the directed graph under consideration is connected. The first vertex to be visited in each component of the DFS forest will be called the root of the corresponding component. A description of the DFS algorithm for a directed graph is presented next. In this algorithm we use a new array SCAN that has one entry for each vertex in the graph. To begin with we set SCAN(u) = 0 for every vertex υ, thereby indicating that none of the vertices is completely scanned. Whenever a vertex is completely scanned, the corresponding entry in the SCAN array is set to 1. As we pointed out earlier, when we encounter an edge (v,w) with D F N ( H > ) ), we shall classify it as a back edge if SCAN(»v) = 0; otherwise (v,w) is a cross edge. We also use two arrays, FORWARD and CROSS, that store respectively, forward and cross edges. Algorithm 11.14. DFS of a Directed Graph 51.

52. 53.

G is the given directed graph with no self-loops. Set T R E E = 0 , FORWARD = 0 , BACK = 0 , CROSS = 0 , and i = l. For every vertex υ in G, set MARK(u) = 0, FATHER(u) = 0, and SCAN(u) = 0. (DFS with a new root begins.) Select any vertex, say vertex r, with MARK(r) = 0. Set DFN(r) = t, MARK(r) = 1, and υ = r. If all the edges incident out of ν have already been labeled "examined," set SCAN(i>) = 1, and then go to S5. (v is now completely scanned.) Otherwise select an edge (v, w) that is not yet labeled "examined' and go to S4.

352

54.

GRAPH ALGORITHMS

Label the edge (v, w) "examined," do the following, and then go to S3. 1. If MARK(vv) = 0, then set j = i + 1, D F N ( H - ) = i,

TREE = TREE U {(i>, w)}, MARK(w) = 1, FATHER(w) = v, υ = w. 2. Otherwise set FORWARD = FORWARD U {(υ, w)} if DFN(tv) > DFN(u), BACK = BACK U {(i>, w)} if DFN(w) < DFN(u), and SCAN(M>)

= 0;

CROSS = CROSS U {(v, w)}, otherwise. 55. If FATHER(u) Φ0 (i.e., ν is not a root), then set ν = FATHER(u) and go to S3. Otherwise go to S6. 56. If for every vertex x, MARK(x) = 1, then go to S7; otherwise set / = ι + 1 and go to S2. 57. (DFS is completed.) HALT. • As an example, DFS of a directed graph is shown in Fig. 11.14a. Next to each vertex we have shown its depth-first number. The tree edges are shown as continuous lines, and the other edges are shown as dashed lines. The DFS forest is shown separately in Fig. 11.14ft. We pointed out earlier that the DFS forest of a directed graph may not be connected, even if the graph is connected. This can also be seen from Fig. 11.14b. This leads us to the problem of discovering sufficient conditions for a DFS forest to be connected. In the following we prove that the DFS forest of a strongly connected graph is connected. In fact, we shall be establishing a more general result. Let Τ denote a DFS forest of a directed graph G = (V, E). Let G, = (V , Ej), with |Vj| 2 : 2 , be a strongly connected component of G. Consider any two vertices υ and ic in 6 , . Assume without loss of generality that D F N ( U ) < D F N ( M ' ) . Since Gj is strongly connected, there exists a directed path Ρ in G, from ν to w. Let χ be the vertex on Ρ with the lowest depth-first number and let T be the subtree of Γ rooted at x. Note that cross edges and back edges are the only edges that lead out of the subtree T . Since these edges lead to vertices having lower depth-first numbers than D F N ( J C ) , it follows that once path Ρ reaches a vertex in T , then all the subsequent vertices on Ρ will also be in T . In particular, w also lies in T . So it is a descendant of x. Since D F N ( * ) ^ D F N ( u ) < D F N ( w ) , it follows from the DFS algorithm that υ is also in T . Thus any two vertices ν and w in G, have a common ancestor that is also in G,. f

x

x

x

x

x

x

DEPTH-FIRST SEARCH

353

8

Figure 11.14. (a) DFS of a directed graph, (b) DFS forest of graph in (a). We may conclude from this that all the vertices of G, have a common ancestor r, that is also in G,. It may now be seen that among all the common ancestors in Τ of vertices in G,, vertex r, has the highest depth-first number. Further, it is easy to show that if υ is a vertex in G„ then any vertex on the tree path from r, to υ will also be in G . So the subgraph of Γ induced by V is connected. Thus we have the following. (

t

Theorem 11.13. Let G, = (V), E ) be a strongly connected component of a directed graph G = (V, E). If Γ is a DFS forest of G, then the subgraph of Τ induced by V, is connected. • t

354

GRAPH ALGORITHMS

Following is an immediate corollary of Theorem 11.13. Corollary 11.13.1. The DFS forest of a strongly connected graph is connected. • It is easy to show that the DFS algorithms 11.13 and 11.14 are both of complexity 0(n + m), where η is the number of vertices and m is the number of edges in a graph. 11.8

BICONNECTIVITY AND STRONG CONNECTIVITY

In this section we discuss algorithms due to Tarjan [11.85] and Hopcroft and Tarjan [11.86] for determining the biconnected components and the strongly connected components of a graph. These algorithms are based on DFS. We begin our discussion with the biconnectivity algorithm. 11.8.1

Biconnectivity

We may recall (Chapter 8) that a biconnected graph is a connected graph with no cut-vertices. A maximal biconnected subgraph of a graph is called a biconnected component of the graph. A crucial step in the development of the biconnectivity algorithm is the determination of a simple criterion that can be used to identify cut-vertices as we perform a DFS. Such a criterion is given in the following two lemmas. Let G = (V, E) be a connected undirected graph. Let Τ be a DFS tree of G with vertex r as the root. Then we have the following. +

Lemma 11.5. Vertex υ Φ r is a cut-vertex of G if and only if for some son s of ν there is no back edge between any descendant in Τ of s (including itself) and a proper ancestor of v. Proof. Let G' be the graph that results after removing vertex ν from G. By definition, υ is a cut-vertex of G if and only if G' is not connected. Let s,, s ,..., s be the sons of ν in T. For each i, 1 < i s Jfc, let V, denote the set of descendants of s, (including itself), and let G be the subgraph of G' induced on V). Further let V" -V - U * V „ where V = V- {v}, and let G" be the subgraph induced on V". Note that all the proper ancestors of υ are in V". Clearly, G , G ,...,G and G" are all subgraphs of G, which together contain all the vertices of G'. We can easily show that all these subgraphs are connected. Further, by Theorem 11.12 there are no edges connecting vertices belonging to different G,'s. So it follows that G' will be connected if 2

k

(

=1

x

2

k

'Note that a biconnected component is the same as a block defined in Section 1.7.

BICONNECTIVITY AND STRONG CONNECTIVITY

355

and only if for every i, 1 < ι < A:, there exists an edge (a, b) between a vertex a £ V, and a vertex b £ V". Such an edge (a,b) will necessarily be a back edge, and b will be a proper ancestor of v. We may therefore conclude that G' will be connected if and only if for every son s, of ν there exists a back edge between some descendant of s, (including itself) and a proper ancestor of v. The proof of the lemma is now immediate. • Lemma 11.6. The root vertex r is a cut-vertex of G if and only if it has more than one son. Proof. Proof in this case follows along the same line as that for Lemma 11.5. • In the following we refer to the vertices of G by their depth-first numbers. To embed into the DFS procedure the criterion given in Lemmas 11.5 and 11.6, we now define, for each vertex υ of G, L O W ( D ) = min({u} U {*v| there exists a back edge (x, w) such that JC is a descendant of υ, and w is a proper ancestor of ν in

T}). (11.23)

Using the LOW values defined, we can restate the criterion given in Lemma 11.5 as in the following theorem. Theorem 11.14. Vertex ν Φ r is a cut-vertex of G if and only if υ has a son * such that LOW(s) > v. • Noting that LOW(u) is equal to the lowest numbered vertex that can be reached from ν by a directed path containing at most one back edge, we can rewrite (11.23) as Low(u) = min({i>} U {LOW(i)|i is a son of v} U {vv|(u, w) is a back edge}). This equivalent definition of LOW(u) suggests the following steps for computing LOW(y): 1. When ν is visited for the first time during DFS, set LOW(u) equal to the depth-first number of v. 2. When a back edge (u, w) incident on υ is examined, set LOW(u) to the minimum of its current value and the depth-first number of w. 3. When the DFS returns to ν after completely scanning a son s of υ, set LOW(u) equal to the minimum of its current value and LOW(s).

356

GRAPH ALGORITHMS

Figure 11.15. G,, G , G , G , G —biconnected components of a graph. 2

3

4

5

Note that for any vertex v, computation of LOW(u) ends when the scanning of υ is completed. We next consider the question of identifying the edges belonging to a biconnected component. For this purpose we use an array STACK. To begin with STACK is empty. As edges are examined, they are added to the top of STACK. Suppose DFS returns to a vertex υ after completely scanning a son s of v. At this point computation of LOW(s) will have been completed. Suppose it is now found that L O W ( s ) s v. Then, by Theorem 11.14, ν is a cut-vertex. Further, if s is the first vertex with this property, then we can easily see that the edge (v, s) along with the edges incident on s, and its descendants will form a biconnected component. These edges are exactly those that lie on top of STACK up to and including (u, s). They are now removed from STACK. From this point on the algorithm behaves in exactly the same way as it would on the graph G', which is obtained by removing from G the edges of the biconnected component that has just been identified. For example, a DFS tree of a connected graph may be as in Fig. 11.15, where G,, G ,..., G are the biconnected components in the order in which they are identified. A description of the biconnectivity algorithm now follows. This algorithm is essentially the same as Algorithm 11.13, with the inclusion of appropriate steps for computing LOW(y) and identifying the cut-vertices and the edges belonging to the different biconnected components. Note that in this algorithm the root vertex r is treated as a cut-vertex, even if it is not one, for the purpose of identifying the biconnected component containing r. 2

5

Algorithm 11.15. Biconnectivity 51. G is the given connected graph. For every vertex ν in G, set FATHER(t>) = 0 and MARK(u) = 0. Set ι = 1 and STACK = 0 . 52. Select any vertex, say vertex r, with MARK(r) = 0. Set DFN(r) = 0, LOW(r) = i, MARK(r) = 1, ν = r.

BICONNECTIVITY AND STRONG CONNECTIVITY

357

53. If all the edges incident on υ have already been labeled "examined," then go to S5. Otherwise select an edge (v, w) that is not yet labeled "examined." Label this edge "examined," add it to the top of STACK, and go to S4. 54. Do the following and then go to S3: 1. If MARK(tv) = 0, set i = i + 1, DFN(w) = /', LOW(w) = i, FATHER(w) = v, MARK(vv) = 1. ν — w. 2.

If M A R K ( H O = 1,

set

LOW(i>) = min{LOW(u), D F N ( H > ) } . 55. If FATHER(u) Φ 0, go to S6; otherwise go to S8. 56. If LOW(u)>DFN(FATHER(i;)), then remove all the edges from the top of STACK up to and including the edge (FATHER(u), υ). (A biconnected component has been found.) 57. Set LOW(FATHER(u)) = min{LOW(i;), LOW(FATHER(y))}, υ = FATHER(u) and go to S3. 58. (All biconnected components have been found.) HALT. • See Fig. 11.16 for an illustration of this algorithm.

Figure 11.16. Illustration of Algorithm 11.15. LOW values are shown in parentheses. Biconnected components are {e , e , e , e , e }, {e,}, and {e , 2

4

(2)

5 (2)

3

4

s

6

7

358

GRAPH ALGORITHMS

11.8.2

Strong Connectivity

Recall that a graph is strongly connected if for every two vertices ν and w there exists in G a directed path from υ to w and a directed path from w to v; further a maximal strongly connected subgraph of a graph G is called a strongly connected component of the graph. Consider a directed graph G = (V, E). Let G =(V , E ), G = (V , E ) , . . . , G = (V*, £*) be the strongly connected components of G. Let Γ be a DFS forest of G and 7Ί, T ,...,T be the induced subgraphs of Τ on the vertex sets V , V ,...,V , respectively. We know from Theorem 11.13 that T , T ,...,T are connected. Let r,, l < / < & , be the root of T,. Assume that if i<j, then DFS terminates at vertex r, earlier than at r . Then we can see that for each i < ;', either r, is to the left of r or r, is a descendant of r in T. Further G„ 1 < i < £, would consist of those vertices that are descendants of r, but are in none of G,, G , . . . , G,_,. The first step in the development of the strong connectivity algorithm is the determination of a simple criterion that can be used to identify the roots as we perform a DFS. The following observations will be useful in deriving such a criterion. These observations are all direct consequences of the fact that there exist no directed circuits in the graph obtained by contracting all the edges in each one of the sets E , E ,...,E (see Section 5.1). X

2

2

2

x

X

k

2

x

x

2

2

k

k

k

y

;

j

2

x

2

k

1. There is no back edge of the type (υ, w) with υ Ε V, and w E V i Φ j . In other words all the back edges that leave vertices in V also end on vertices in Vj. 2. There is no cross edge of the type (v, w) with υ E V , w Ε V, ί Φ], and r is an ancestor of r,. Thus for each cross edge (v, w) one of the following two is true: a. υ Ε V, and wE V for some i and / with i Φ j and r to the left of r . b. For some i, vEV and wEV jy

t

t

j

j

)

l

(

r

Assuming that the vertices of G are named by their DFS numbers, we define for each ν in G, LOWLINK(y) = min({u} U {w\ there is a cross edge or a back edge from a descendant of ν to w, and w is in the same strongly connected component as v}). Suppose vEV Then it follows from the above definition that LOWLINK(u) is the lowest numbered vertex in V, that can be reached from υ by a directed path that contains at most one back edge or one cross edge. From the observations that we have just made it follows that all the edges of such a directed path will necessarily be in G,. As an immediate consequence we get r

BICONNECTIVITY AND STRONG CONNECTIVITY

LOWLINK(r,) = r,

for all 1 < ι < k .

359 (11.24)

Suppose υ £ V and ν ¥= r,. Then there exists a directed path Ρ in G, from υ to r,. Such a directed path Ρ should necessarily contain a back edge or a cross edge because r, < y, and only cross edges and back edges lead to lower numbered vertices. In other words Ρ contains a vertex w < v. So for ν Φ r,, we get t

LOWLINK(u) < υ .

(11.25)

Combining (11.24) and (11.25) we get the following theorem, which characterizes the roots of the strongly connected components of a directed graph. Theorem 11.15. A vertex ν is the root of a strongly connected component of a directed graph G if and only if LOWLINK(t;) = v. • The following steps can be used to compute LOWLINK(u) as we perform a DFS. 1. On visiting υ for the first time, set LOWLINK(u) equal to the DFS number of v. 2. If a back edge (u, w) is examined, then set LOWLINK(u) equal to the minimum of its current value and the DFS number of w. 3. If a cross edge (u, w) with w in the same strongly connected component as υ is explored, set LOWLINK(u) equal to the minimum of its current value and the DFS number of w. 4. When the search returns to υ after completely scanning a son s of v, set LOWLINK(u) to the minimum of its current value and LOWLINK(s). To implement step 3 we need a test to check whether w is in the same strongly connected component as v. For this purpose we use an array STACK1 to which vertices of G are added in the order in which they are visited during the DFS. STACK1 is also used to determine the vertices belonging to a strongly connected component. Let ν be the first vertex during DFS for which it is found that LOWLINK(u) = v. Then by Theorem 11.15, υ is a root and in fact it is r . At this point the vertices on top of STACK1 up to and including υ are precisely those that belong to G j . Thus G, can easily be identified. These vertices are now removed from STACK1. From this point on the algorithm behaves in exactly the same way as it would on the graph G', which is obtained by removing from G the vertices of G,. As regards the implementation of step 3 in LOWLINK computation, let υ £ V , and let (υ, w) be a cross edge encountered while examining the edges t

t

GRAPH ALGORITHMS

360

incident on v. Suppose w is not in the same strongly connected component as v. Then it would belong to a strongly connected component G whose root r is to the left of r, (observation 2a p. 358). The vertices of such a component would already have been identified, and so they would no longer be on STACK1. Thus w will be in the same strongly connected component as υ if and only w is on STACK1. A description of the strong connectivity algorithm now follows. This is the same as Algorithm 11.14 with the inclusion of appropriate steps for computing LOWLINK values and for identifying the vertices of the different strongly connected components. We use in this algorithm an array POINT. To begin with POINT(u) = 0 for every vertex v. This indicates that no vertex is on the array STACK1. ΡΟΙΝΤ(υ) is set to 1 when υ is added to STACK1, and it is set to zero when ν is removed from STACK1. y

;

Algorithm 11.16. Strong Connectivity 51.

G is the given directed graph. For every vertex υ in G, set MARK(u) = 0, FATHER(u) = 0, and POINT(u) = 0. Set i = 1 and STACK1 = 0 . 52. Select any vertex, say vertex r, with MARK(r) = 0. Set DFN(r) = i, LOWLINK(r) = i, MARK(r) = 1. Add r to STACK1, set POINT(r) = 1 and υ = r. 53. If all the edges incident on υ have already been labeled "examined," then go to S5. Otherwise select an edge (υ, w) that is not yet labeled "examined." Label it "examined" and go to S4. 54. Do the following and then go to S3: 1. If MARK(w) = 0, set i = i + 1, D F N ( H O = i, L O W L I N K ( H - ) = i, FATHER(H-) =

υ,

M A R K ( M > ) = 1.

Add w to STACK1, and set P O I N T ( H O = 1 and ν = w. 2. If MARK(w) = l, DFN(w)
REDUCTIBILITY OF A PROGRAM GRAPH

361

1 ID

9 (7)

10(9)

Figure 11.17. Illustration of Algorithm 11.16. LOWLINK values are shown in parentheses. Strongly connected components are {3, 4, 5}, {6, 7, 8, 9, 10}, {2}, and {1, 11, 12, 13}. 56. If FATHER(u) = 0, go to S7; otherwise set 1. LOWLINK(FATHER(u)) = min {LOWLINK( FATHER( v)), LOWLINK(i;)}, 2. υ = F A T H E R ^ ) , and then go to S3. 57. If for every vertex x, MARK(x) = 1, then go to S8; otherwise set i = i + 1 and go to S2. 58. (All strongly connected components have been identified.) HALT. • See Fig. 11.17 for an illustration of this algorithm.

11.9

REDUCIBILITY OF A PROGRAM GRAPH

A program graph is a directed graph G with a distinguished vertex s such that there is a directed path from s to every other vertex of G. In other words every vertex in G is reachable from s. The vertex s is called the start vertex of G. We assume that there are no parallel edges in a program graph. This assumption will involve no loss of generality as far as our discussions in this section are concerned. The flow of control in a computer program can be modeled by a program graph in which each vertex represents a block of instructions that can be executed sequentially. Such a representation of computer programs has

GRAPH ALGORITHMS

362

proved very useful in the study of several questions relating to what is known as the code optimization problem. For many of the code optimization methods to work, the program graph must have a special property called reducibility. See [11.87] through [11.95]. Reducibility of a program graph G is defined in terms of the following two transformations on G: 5,: S:

Delete self-loops (u, υ) in G. If (v, w) is the only edge incident into w, and w Φ s, delete vertex w. For every edge (w, x) in G add a new edge (v, x) if (v, x) is not already in G. (This transformation is called collapsing vertex w into vertex v.)

2

For example collapsing vertex 5 into vertex 4 in the program graph of Fig. 11.18a results in the graph shown in Fig. 11.186. A program graph is reducible if it can be transformed into a graph consisting only of vertex s by repeated applications of the transformations S, and S . For example, the graph in Fig. 11.18a is reducible. It can be verified that this graph can be reduced by collapsing the vertices in the order 5, 8, 4, 3, 10, 9, 7, 6, 2. Cocke [11.88] and Allen [11.93] were the original formulators of the notion of reducibility, and their definition is in terms of a technique called interval analysis. The definition given above is due to Hecht and Ullman [11.96], and it is equivalent to that of Cocke and Allen. If a graph G is reducible, then it can be shown [11.96] that any graph G' obtained from G by one or more applications of the transformations 5, and S is also reducible. Thus the order of applying transformations does not matter in a test for reducibility. Further, some interesting classes of programs such as "go-to-less" programs give rise to graphs that are necessarily reducible [11.96], and most programs may be modeled by a reducible graph using a process of "node splitting" [11.97]. Suppose we wish to test the reducibility of a graph G. This may be done by first deleting self-loops using transformation 5, and then counting the number of edges incident into each vertex. Next we may find a vertex w with only one edge (v, w) incident into it and apply transformation S , collapsing w into v. We may then repeat this process until.we reduce the graph entirely or discover that it is not reducible. Clearly each application of S requires 0(n) time, where η is the number of vertices in G and reduces the number of vertices by 1. Thus the complexity of this algorithm is 0(n ). Hopcroft and Ullman [11.98] have improved this to 0(m log m), where m is the number of edges in G. Tarjan [11.99] and [11.100] has subsequently given an algorithm that compares favorably with that of Hopcroft and Ullman. Hecht and Ullman [11.96] and [11.101] have given several useful structur2

2

2

2

2

REDUCIBIUTY OF A PROGRAM GRAPH

6

9

7

363

10

(/>)

Figure 11.18. (a) A reducible program graph G. HIGHPT1 values are given in parentheses, (b) Graph obtained after collapsing in G vertex 5 into vertex 4.

al characterizations of program graphs. One of these is given in the following Theorem. Theorem 11.16. Let G be a program graph with start vertex s. G is reducible if and only if there do not exist distinct vertices ν Φ s and w Φ s,

GRAPH ALGORITHMS

364

OS

Figure 11.19. A basic nonreducible graph.

directed paths from s to ν and P from s to w, and a directed circuit C containing υ and w, such that C has no edges and only one vertex in common with each of P, and P (see Fig. 11.19). • 2

2

Proof of Theorem 11.16 may be found in [11.96] and [11.102]. We now proceed to discuss Tarjan's algorithm for testing the reducibility of a program graph. This algorithm uses DFS and is based on a characterization of reducible program graphs that we shall prove using Theorem 11.16. Our discussion here is based on [11.99]. Let G be a program graph with start vertex s. Let Γ be a DFS tree of G with s as the root. Henceforth we refer to vertices by their depth-first numbers. Theorem 11.17. G is reducible if and only if G contains no directed path Ρ from s to some vertex ν such that υ is a proper ancestor in Τ of some other vertex on P. Proof. Suppose G is not reducible. Then there exist vertices υ and w, directed paths P, and P , and circuit C that satisfy the condition in Theorem 11.16. Assume that ν < w. Let C, be the part of C from υ to w. Then C, contains some common ancestor « of υ and w. The directed path consisting of P followed by the part of C from w to u satisfies the condition in the theorem. Conversely, suppose there exists a directed path Ρ that satisfies the condition in the theorem. Let then υ be the first vertex on P, which is a proper ancestor of some earlier vertex on P. Suppose w is the first vertex on P, which is a descendant of v. Let P, be the part of Τ from s to v, and let P be the part of Ρ from s to w. Also let C be the directed circuit consisting of the part of Ρ from w to ν followed by the path of tree edges from υ to w. Then we can see that υ, w, P , P , and C satisfy the condition in Theorem 11.16. So G is not reducible. • 2

2

2

t

2

REDUCTIBILITY OF A PROGRAM GRAPH

365

For any vertex v, let HIGHPTl(u) be the highest numbered proper ancestor of ν such that there is a directed path Ρ from ν to HIGHPTl(i>) and Ρ includes no proper ancestors of υ except HIGHPTi(u). We define HIGHPTl(u) = 0 if there is no directed path from υ to a proper ancestor of υ. As an example, in Fig. 11.18a we have indicated in parentheses the HIGHPT1 values of the corresponding vertices. Note that in HIGHPTl(u) calculation we may ignore forward edges since if Ρ is a directed path from ν to w and Ρ contains no ancestors of ν except ν and w, we may substitute for each forward edge in Ρ a path of tree edges or a part of it and still have a directed path from υ to w that contains no ancestors of υ except ν and w. Tarjan's algorithm is based on the following characterization of program graphs. Theorem 11.18. G is reducible if and only if there is no vertex υ with an edge ( M , V) incident into υ such that w < H I G H P T l ( u ) , where w is the highest numbered common ancestor of w and v. Proof. Suppose G is not reducible. Then by Theorem 11.17 there is a directed path Ρ from s to υ with υ a proper ancestor of some other vertex on P. Choose Ρ as short as possible. Let »v be the first vertex on Ρ that is a descendant of v. Then all the vertices except υ that follow w on Ρ are descendants of υ in T. In other words the part of Ρ from w to υ includes no proper ancestors of w except v. So HIGHPTl(>v) 5 : v. Thus w, H I G H P T I ( H ' ) , and the edge of Ρ incident into w satisfy the condition in the theorem. Conversely, suppose the condition in the theorem holds. Then the edge (u, v) is not a back edge because in such a case the highest numbered common ancestor w of u and υ is equal to v, and so w = υ > HIGHPTl(u). Thus («, v) is either a forward edge or a cross edge. Let P, be a directed path from υ to HIGHPTl(u), which passes through no proper ancestors of ν except HIGHPTl(u). Then the directed path consisting of the tree edges from s to u followed by the edge (u, v) followed by P, satisfies the condition in Theorem 11.17. So G is not reducible. • Now it should be clear that testing reducibility of a graph G using Theorem 11.18 involves the following main steps: 1. Perform a DFS of G with s as the root. 2. Calculate HIGHPTl(u) for each vertex υ in G. 3. For cross edges check the condition in Theorem 11.18 during the HIGHPT1 calculation. 4. For forward edges check the condition in Theorem 11.18 after the HIGHPT1 calculation.

366

GRAPH ALGORITHMS

ν

ZOO

Figure 11.20

<

'6 2

u

Note that, as we observed earlier, forward edges may be ignored during the HIGHPT1 calculation. Further as we observed in the proof of Theorem 11.18, back edges need not be tested for the condition in Theorem 11.18. To calculate HIGHPT1 values, we first order the back edges (u, v) by the number of v. Then we process the back edges in order, from highest to lowest v. Initially all vertices are unlabeled. To process the back edge (u, v), we proceed up the tree path from u to v, labeling each currently unlabeled vertex with v. (We do not label υ itself.) If a vertex w gets labeled, we examine all cross edges incident into w. If (z, w) is such a cross edge (see Fig. 11.20), we proceed up the tree path from ζ to v, labeling each unlabeled vertex with v. If ζ is not a descendant of v, then G is not reducible by Theorem 11.18, and the calculation stops. We continue labeling until we run out of cross edges incident into just labeled vertices; then we process the next back edge. When all the back edges are processed, the labels give the HIGHPT1 values of the vertices. Each unlabeled vertex has HIGHPT1 equal to zero. We now describe Tarjan's algorithm for testing reducibility. In this algorithm we use η queues, called buckets, one for each vertex. The bucket BUCKET(w) corresponding to vertex w contains the list of back edges (w, w) incident into vertex w. While processing back edge (w, w), we need to keep track of certain vertices from which u can be reached by directed paths. The set CHECK is used for this purpose. Algorithm 11.17. Program Graph Reducibility (Tarjan)

SI.

Perform a DFS of the given η-vertex program graph G. Denote vertices by their DFS numbers. Order the back edges (M, V) of G by the number of v.

REDUCIBILITY OF A PROGRAM GRAPH

52.

53. 54.

55. 56. 57. 58. 59. 510. 511. 512. 513. 514. 515. 516. 517.

367

For i = l, 2 , . . . , , i set HIGHPTl(i) = 0, BUCKET(i) = the empty list. Add each back edge (u, w) to B U C K E T ( H > ) . Set w = η - 1. Test if BUCKET(w) is empty. If yes, go to S6; otherwise go to S7. Set w = w - 1. If w < 1, go to S16; otherwise go to S5. (Processing of a new back edge begins.) Delete a back edge (x, w) from BUCKET(w) and set CHECK = {x}. Test if CHECK is empty. If yes (processing of a back edge is over), go to S5; otherwise go to S9. Delete u from CHECK. Test if Μ is a descendant of w. If yes, go to S l l ; otherwise go to S17. Test if u = w. If yes, go to S8; otherwise go to S12. Test if HIGHPTl(u) = 0. If yes, go to S13; otherwise go to S15. Set HIGHPTl(u) = w. For each cross edge (u, w) add ν to CHECK. Set w = F A T H E R ( w ) . G O to S l l . If Μ £ HIGHPTl(u) for each forward edge (u, v) (the graph is reducible), then HALT. Otherwise go to S17. HALT. The graph is not reducible. •

Note that S10 in Algorithm 11.17 requires that we be able to determine whether a vertex w is a descendant of another vertex u. Let ND(u) be the number of descendants of vertex u in T. Then we can show that w is a descendant of u if and only if u s w < u + ND(w). (See Exercise 11.8). We can calculate ND(u) during the DFS in a straightforward fashion. The efficiency of Algorithm 11.17 depends crucially on the efficiency of the HIGHPT1 calculation. To make the HIGHPT1 calculation efficient, in Algorithm 11.17 we need to avoid examining vertices that have already been labeled. Tarjan suggests a procedure to achieve this. The following observation forms the basis of this procedure: Suppose at S12 we are examining a vertex u for labeling. At this stage let u' be the highest unlabeled proper ancestor of u. This means that all the proper ancestors of u except Μ', which lie between u' and u in T, have already been labeled. Thus u' is the next vertex to be examined. Furthermore when u is labeled, then all the vertices for which u is the highest unlabeled proper ancestor will have u' as their highest unlabeled proper ancestor. To implement the method that follows from this observation, we shall use sets numbered 1 to n. A vertex w Φ1 will be in the set numbered v, that is, SET(y), if υ is the highest numbered unlabeled proper ancestor of w. Since

368

GRAPH ALGORITHMS

vertex 1 never gets labeled, each vertex is always in a set. Initially a vertex is in the set whose number is its father in T. Thus initially add i to SET(FATHER(/)) for i = 2, 3 , . . . , n. To carry out S15, we find the number u' of the set containing u and let that be the new u. Further when u becomes labeled, we shall combine the sets numbered u and u' to form a new set numbered «'. Thus u' becomes the highest numbered unlabeled proper ancestor of all the vertices in the old SET(u). It can now be seen that replacing S12 through S15 in Algorithm 11.17 by the following sequence of steps will implement the method described. Of course, SETs are initialized as explained earlier. S12'. Set u' = the number of the set containing u. S13'. Test if HIGHPTl(u) = 0. If yes, go to S14'; otherwise go to S16'. S14\ Set 1. HIGHPTl(u) = w; and 2. SET(u') = SET(u)USET(u'). SIS'. For each cross edge (v, u) add ν to CHECK. S16'. Set u = u'. Go to S l l . It can be verified that the modified HIGHPT1 calculations require 0(n) set unions, 0(m + n) executions of step S12', and 0{m + n) time exclusive of set operations. If we use the algorithm described in Fischer [11.103] and Hopcroft and Ullman [11.104] for performing disjoint set unions and for performing S12', then it follows from the analysis given in Tarjan [11.105] that the reducibility algorithm has complexity 0(ma(m, n)), where a(m, n) is a very slowly growing function that is related to a functional inverse of Ackermann's function A(p, q), and it is defined as follows: a(m, n) - min{z s l| A(z, 4[m/n]) > log

2

n).

The definition of Ackermann's function is 2q, 0, A(p, q) = w, {A(p-\,A{p,q

1)),

p = 0; o = 0and p ^ l ρ > 1 and q = 1 : ρ > 1 and q ^ 2 ,

(11.26)

Note that Ackermann's function is a very rapidly growing function. It is easy to see that A(3,4) is a very large number, and it can be shown that a(m, η) ^ 3 if m Φ 0 and log η < A(3,4). The algorithm for set operations mentioned is also described in Aho, Hopcroft, and Ullman [11.106] and Horowitz and Sahni [11.107]. Algorithm 11.17 is nonconstructive, that is, it does not give us the order 2

REDUCIBILITY OF A PROGRAM GRAPH

369

in which vertices have to be collapsed to reduce a reducible graph. However, as we shall see now, this information can be easily obtained as this algorithm progresses. During DFS let us assign to the vertices numbers called SNUMBERs from η to 1 in the order in which scanning at a vertex is completed. In fact this is the same as the order in which the corresponding entries of the array SCAN in Algorithm 11.14 are set to 1. The following can be easily verified: 1. 2. 3. 4.

If (v, w) If ( v , w) If ( v , w) If ( v , w)

is is is is

a a a a

tree edge, then S N U M B E R ( U ) < S N U M B E R ( H < ) . cross edge, then SNUMBER(u) < S N U M B E R t » . back edge, then SNUMBER(iv) > SNUMBER(iv). forward edge, then SNUMBER(tv) < S N U M B E R i » .

As an example, in Fig. 11.21 we have shown in parentheses the SNUMBERs of the corresponding vertices of the graph in Fig. 11.18a. Suppose we apply the reducibility algorithm, and each time we label a vertex υ let us associate with it a pair (HIGHPTl(u), SNUMBER(tv)). When the algorithm is finished, let us order the vertices so that a vertex labeled (x y ) appears before a vertex labeled (x , y ) if and only if x > x or x = x and y )). u

x

(10)6

2

x

x

2

2

7(9)

x

9(7)

2

2

10 (β)

Figure 11.21. Graph of Fig. 11.18a with SNUMBER values in parentheses.

GRAPH ALGORITHMS

370

Suppose v , v , v ,... is a reduction order for a reducible program graph G. Let Γ be a DFS tree of G with vertex s as the root. Using Theorem 11.18 and the properties of SNUMBERs outlined earlier, it is easy to show that the tree edge (M, v ) is the only edge incident into v . Suppose we collapse v into u and let G' be the resulting reducible graph. Also let 7" be the tree obtained from Τ by contracting the edge (u, v ). Then clearly 7" is a DFS tree of G'. Further: a

b

c

a

a

a

a

1. A cross edge of G corresponds to nothing or to a cross edge or to a forward edge of G'. 2. A forward edge of G corresponds either to nothing or to a forward edge of G'. 3. A back edge of G corresponds either to nothing or to a back edge of G'. It can now be verified that the relative HIGHPT1 values of the vertices in G' will be the same as those in G. This is also true for the SNUMBERs. Thus v , v ,... will be a reduction order for G'. Repeating these arguments, we get the following theorem. b

c

Theorem 11.19. If a program graph G is reducible, then we may collapse the vertices of G in the reduction order using the transformation S (interspersed with applications of 5,). •

2

For example, it can be verified from HIGHPT1 values given in Fig. 11.18a and the SNUMBER values given in Fig. 11.21 that the sequence 4, 5, 3, 8, 10, 9, 7, 6, 2 is a reduction order for the graph in Fig. 11.18a. Closely related to the problem of testing reducibility, is the problem of testing whether a graph is series-parallel (see Exercise 7.12 for the definition of series-parallel graphs). For a characterization of series-parallel graphs using DFS and a recognition algorithm for series-parallel graphs see Shinoda, Chen, Yasuda, Kajitani, and Mayeda [11.108], and Valdes, Tarjan, and Lawler [11.109]. 11.10

Sf-NUMBERING OF A GRAPH

In this section we present yet another application of DFS—computing an sf-numbering of a graph. This numbering is necessary for the planarity testing algorithm to be discussed in the following section. Given an η-vertex biconnected graph G = (V, E) and an edge (s, t) of G, a numbering of the vertices of G is called an si-numbering of G if the following conditions are satisfied, where g(v) denotes the corresponding st-number of vertex v:

sf-NUMBERING OF A GRAPH

371

1. For all υ 6 V, 1 < g(v) =£ n, and for u Φ v, g(u) Φ g(v). 2. g{s) = l. 3. *(*) = «· 4. For υ £ V— {s, t} there are adjacent vertices u and w such that g(u)
Figure 11.22. sr-graph G.

GRAPH ALGORITHMS

372

Path Finding Algorithm (Applied from Vertex v)

51. Pick a "new" edge incident on v. i. If (v, w) is a back edge ( D F N ( w ) < D F N ( u ) ) , then mark e "old" and HALT. Note: The path consists of the single edge e. ii. If (v, w) is a tree edge ( D F N ( w ) > D F N ( i ; ) ) , then do the following: Starting from υ traverse the directed path that defined L O W ^ ) and mark all edges and vertices on this path "old." HALT. Note: The path here starts with the tree edge (v, t) and ends in the vertex u such that D F N ( M ) = LOW(»v). This path has exactly one back edge. iii. If (w, v) is a back edge ( D F N ( w ) > DFN(u)) do the following: Starting from ν traverse the edge (w, v) backward and continue backward along tree edges until an "old" vertex is encountered. Mark all the edges and vertices on this path "old" and HALT. Note: The path in this case is directed into v. 52. If all the edges incident on υ are "old" HALT. Note: The path produced is empty. The following facts hold true after each application of the path finding algorithm. Note that the algorithm is always applied from an "old" vertex. 1. All ancestors of an "old" vertex are old too. This is true before the first application of the algorithm since t is the only ancestor of s and it is "old." This property remains true after any one of the applicable steps of the algorithm. 2. When the algorithm is applied from an "old" vertex v, it either produces an empty path or it produces a path that starts at v, passes through "new" vertices and edges and ends at another "old" vertex. This is obvious when (v, w) or (w, v) is a back edge [cases (i) and (iii) of Sl]. This is also true when (υ, w) is a tree edge because in the biconnected graph G the vertex u defining LOW(w) is an ancestor of υ and therefore u is "old." Even and Tarjan's fi-numbering algorithm presented next uses a stack STACK that initially contains only t and s with s on top of t. In this description we do not explicitly include the details of DFS. We also assume that to start with t, s and the edge (r, s) are "old." Algorithm 11.18. sr-Numbering (Even and Tarjan) Sl.

Set i = 1.

PLANARITY TESTING

373

52. Let ν be the top vertex on STACK. Remove ν from STACK. If υ = r, then set g(v) = i and HALT. 53. (υΦ t). Apply the path finding algorithm from v. If the path is empty, then set g(v) = i, i = / + 1 and go to S2. 54. (The path is not empty.) Let the path be P: v, u ,..., u , w. Place u , u u , , υ on STACK in this order (note that υ comes on top of STACK) and go to S2. • 2

k

k

k

Theorem 11.20. Algorithm 11.18 computes an sr-numbering for every biconnected graph G = (V, E). Proof. The following facts about the algorithm are easy to verify: 1. No vertex appears in more than one place on STACK at any time. 2. Once a vertex ν is placed on STACK, no vertex under ν receives a number until υ does. 3. A vertex is permanently removed from STACK only after all edges incident on υ become "old." We now show that each vertex υ is placed on STACK before / is removed. Clearly this is true for υ = s because initially t and s are placed on STACK with s on top of t. Consider any vertex ν Φ s, t. Since G is biconnected, there exists a directed path Ρ of tree edges from s to v. Let P: s, u , u ,... ,u = v. Let m be the first index such that u is not placed on STACK. Since u _ is placed on STACK, t can be removed only after « _ , is removed (fact 2), and u _ is removed only after all edges incident on « _ , are "old" (fact 3). So u must be placed on STACK before t is removed. We need to show that the numbers assigned to the vertices are indeed si-numbers. Since each vertex is placed on STACK and eventually removed, every vertex ν gets a number g(v). Clearly all numbers assigned are distinct. Also g(s) = 1 and g(r) = η because s is the first vertex and t is the last vertex to be removed. Every time a vertex ν Φ s, t is placed on STACK, there is an adjacent vertex placed above ν and an adjacent vertex placed below v. By fact 2 the one above gets a lower number and the one below gets a higher number. • x

2

m

k

m

m

x

m

m

l

m

See Erbert [11.112] for another si-numbering algorithm.

11.11

PLANARITY TESTING

In this section we discuss the problem of testing the planarity of a graph. In recent years this and related problems on planar graphs have assumed great

GRAPH ALGORITHMS

374

importance in view of their practical applications. See, for example, Wimer, Koren, and Cederbaum [11.113]. The most successful approach so far for testing the planarity of a graph has been attempts to construct a representation of a planar embedding of the graph. If such a representation can be obtained, then the graph is planar; if not, the graph is nonplanar. All the planarity testing algorithms based on this idea can be grouped into two categories: path addition algorithms and vertex addition algorithms. Path addition algorithms first find a cycle in the graph. When this cycle is removed, the remaining edges of the graph would form several connected components. Each of these components is then embedded in the plane along with the original cycle and the embeddings of the component are combined, if possible, to give an embedding of the entire graph. In [11.114] Hopcroft and Tarjan proposed a linear time algorithm for testing planarity of a graph. This algorithm finds a cycle in the graph using depth-first search. This cycle is then embedded in the plane, thereby dividing the plane into two regions—one inside the cycle and the other outside the cycle. The connected components of the graph, obtained after removing the cycle, are then successively embedded either in the inside region or in the outside region. During the embedding of any component, if necessary, all the components that are already embedded in the inside region may be moved to the outside region, and all those already embedded in the outside region may be moved to the inside region. All these rearrangements are done in an efficient manner without actually drawing the embedding, so that the algorithm has 0{n) time bound. Expositions of Hopcroft and Tarjan's algorithm may be found in Melhorn [11.14], Reingold, Nievergelt, and Deo [11.115], and Even [11.116]. Earlier, Demoucron, Malgrange, and Pertuiset [11.117] had given an algorithm similar to Hopcroft and Tarjan's that avoids the rearrangement of already embedded components by choosing the components, for embedding at each stage, in an appropriate manner. Rubin [11.118] developed an 0(n ) space and 0(n ) time implementation of this algorithm and showed that, for all practical purposes, his implementation behaves as an O(n) algorithm. 2

2

Vertex addition algorithms use an alternate approach to embed a graph in the plane. These algorithms start with a single embedded vertex and add all the edges incident on that vertex. The other end vertices of these edges are not embedded. They then embed an unembedded vertex and add all edges incident on it in the same way. This process of embedding is continued until the entire graph is constructed. For these algorithms to work correctly, the vertices must be embedded in a special order. In [11.110] Lempel, Even, and Cederbaum proposed a vertex addition algorithm. In this algorithm, at any time during the embedding process, the subgraph embedded up to that time is represented by certain formulas that are then manipulated, by applying certain transformations, to check whether the next vertex can be embedded. The best implementation of Lempel, Even, and Cederbaum's algorithm was reported by Booth and Lueker

PLANARITY TESTING

375

[11.119]. They developed a data structure called a Ρβ-tree to represent the permutations of a set in which elements of certain subsets of the given set are required to occur consecutively, and presented efficient algorithms to manipulate the Ρβ-tree. Using PQ-trees, Booth and Lueker developed an 0(n) time implementation of Lempel, Even, and Cederbaum's algorithm. We now proceed to discuss the essential features of Lempel, Even, and Cederbaum's algorithm to test the planarity of a graph. Hereafter we refer to this algorithm as the LEC algorithm. Let G = (V, E) be a simple biconnected graph with η = |V| vertices and m = | £ | edges. The LEC algorithm first renumbers the vertices of G as 1, 2 , . . . , η using an si-numbering. The vertices of G are thereafter referred to by their si-numbers, and they are processed in that order for embedding. The si-numbering is essential for the algorithm to work correctly. The graph G, with its vertices labeled according to an si-numbering, is called an st-graph. Clearly the edges ( 1 , 2 ) , (η - 1, n), and ( 1 , n) are present in any si-graph G. Furthermore, if each edge is oriented from its lower numbered vertex to its higher numbered vertex, then G may be viewed as a directed graph. The following observations follow easily from the definition of si-numbering. Observation 1. In G the in-degree of vertex 1 is zero, the out-degree of vertex η is zero, and for every other vertex D , 2 s p < N - 1 , the in-degree and out-degree are nonzero. Observation 2. For any vertex v, 2 S D < « , there exists a path in G from vertex 1 to υ such that all the internal vertices on the path are less than v. These observations may be verified for the si-graph G shown in Fig. 11.22. For any si-graph G, let G , 1 ^ k s n, denote the subgraph of G induced by the vertex set V = { 1 , 2 , . . . , k}. Thus G consists of all those edges of G whose end vertices are both in V . Now we define the graph B as follows: G is a subgraph of B . In addition to G , B contains all the edges of G that emanate from vertices of V and enter, in G, vertices of V- V . These edges are called virtual edges and the vertices they enter in V- V are called virtual vertices. These vertices are labeled as their counterparts in G but are kept separate. Thus, in B there may be several virtual vertices with the same label, each with exactly one entering edge. For example, Fig. 11.23 shows the graph B of the si-graph shown in Fig. 11.22. If the si-graph G is planar, then there exists a planar embedding G of G. Note that G contains a planar embedding G of G , l < A : < n . Using observation 1, the following lemma can be proved. k

k

k

k

k

k

k

k

k

k

k

k

k

9

k

k

Lemma 11.7. If G is a planar embedding of G contained in a planar embedding G of an si-graph G, then all the edges and vertices of G - G are drawn on one region of G . • k

k

k

k

376

GRAPH ALGORITHMS

Figure 11.23. Graph B9. Thus we can assume, without loss of generality, that if G is a planar graph, then there exists a planar embedding of Bk in which all the virtual edges are drawn in the outside region. Since the edge (1, n) is a virtual edge in every Bk, I s k^n1, it follows that vertex 1 is on the outside region of every Gk. Thus we can draw the graph Bk in the following form. Vertex 1 is drawn at the bottom level. All the virtual vertices appear at the highest level on the horizontal line. The remaining vertices of Gk are drawn in such a way that vertices with higher labels are drawn higher. Such a realization of Bk is called a bush form of Bk. For example, a bush form of the graph B9 is shown in Fig. 11.24. Since Bk and its bush form are isomorphic, hereafter we shall refer to the bush form of Bk also by Bk. Note that in the bush form Bk, the virtual vertices are labeled k + 1 or higher, and the si-numbering ensures that there exists at least one virtual vertex labeled k + 1. Lempel, Even, and Cederbaum proved that if G is planar, then there exists a bush form of Bk in which all the virtual vertices with labels k + 1 appear next to each other on the horizontal line. Let B'k be such a bush form isomorphic to Bk. For example, the bush form B'9 corresponding to B9 is shown in Fig. 11.25. If for a given Bk a corresponding B'k exists, then the bush form Bk+1 can be constructed from B'k as follows. Merge all the virtual vertices labeled k + 1 into one vertex and pull it down from the horizontal line. Add all the edges of G that emanate from vertex k + 1 as virtual edges. Now vertex A: + 1 is considered embedded. Thus, if for each Bk, 1 < λ < η - 2 , the corresponding B'k exists, then we can construct the bush forms B2, Bi,...,Bn_l starting with Bx. Note that

PLANARITY TESTING

©©©©©©©

Figure 11.24. Bush form B9

Figure 11.25. Bush form B9.

377

378

GRAPH ALGORITHMS

B' _ = B _ , and applying this procedure to B„_ will give a planar embedding of G. n

n

l

l

Algorithm 11.19. Planarity Testing (Lempel, Even and Cederbaum)

50. Given a graph G, test if G is planar. 51. Find an sf-numbering of G; renumber the vertices according to the s/-numbering and obtain the si-graph (bush form B, consists of the vertex 1 and all the edges in G incident out of vertex 1 as virtual edges). 52. Set k = l. 53. (B' is a bush form isomorphic to bush form B in which all the virtual vertices labeled k + 1 appear consecutively.) If B' does not exist, HALT (G is not planar). 54. Construct B from B' (B is constructed from B' by merging all the virtual vertices labeled k + 1 into a single vertex k + 1 and adding all the edges in G incident out of vertex k + 1 as virtual edges). 55. If k = η - 1, HALT. (The graph is planar.) Otherwise set k = k + 1 and go to S3. • k

k

k

k+l

k

k+i

k

The crucial step in the LEC algorithm is the construction of B' from B for every l < t < n - 2 . Such bush forms would exist if the given graph G is planar. We now state two lemmas that form the basis of an algorithm for constructing B' from B . The term subbush used in the following discussions is defined first. Let υ be a cut vertex of a bush form B . If V denotes the vertex set of a component of B - v, then the subgraph of B induced by the vertex set V U {v} is called a subbush of B relative to v. k

k

k

k

k

k

k

k

Lemma 11.8. Let υ be a cut vertex of B . If υ > 1, then exactly one subbush of B , relative to υ, contains vertices lower than v. • k

k

Lemma 11.9. Let Η be a maximal biconnected component of B and >Ί > >2> · · · ' y be the vertices of Η that are also end vertices of the edges of B - H. In every bush form isomorphic to B , y y ,..., y, are on the outside window (the circuit forming the boundary of the infinite region) of Η and in the same order, except that the orientation may be reversed. • k

q

k

k

lf

2

The proof of the lemmas use observation 2 and Lemma 11.7, and may be found in [11.116]. From Lemma 11.9 it follows that a bush form isomorphic to B can be obtained by flipping a subbush. Lemma 11.8 implies that a cut vertex υ of B is the lowest vertex in each of the subbushes, except the one that contains vertex 1. These subbushes have the same structure as a bush form, except k

k

FURTHER READING

379

that their lowest vertex is υ rather than 1. If there are ρ such subbushes of B relative to v, then these subbushes can be permuted around ν in any of the ρ! permutations to obtain a bush form isomorphic to B . Also each of these subbushes can be flipped over. These transformations, namely, permutation and flipping, maintain the bush form. In fact Lempel, Even, and Cederbaum proved the following (See Even [11.116]).

k

k

Theorem 11.21. If ΒI and B are bush forms of the same B , then there exists a sequence of permutations and flippings that transform B into B\ such that in B and B the virtual vertices appear in the same order. • 2

k

k

l

k

2

3

k

k

The above theorem implies that each bush form B can be transformed into a bush form B' in which all the virtual vertices labeled k + 1 appear next to each other. For example, the bush form B shown in Fig. 11.25 is obtained from the bush form B of Fig. 11.24 by flipping the subbush containing the set of vertices {1, 3, 4, 9} and permuting the subbushes around the cut-vertex 1. Now the problem is to find, from among all possible permutations and flippings, an appropriate sequence that will transform B into B' . Moreover, one would like to do these transformations efficiently, without drawing the actual bush forms. As we stated earlier, Booth and Lueker showed that these transformations can be implemented efficiently using Ρβ-trees. PQ-trees can be used to represent the information pertaining to bush forms as well as to obtain the bush form B from k

k

g

9

k

k

k+X

For more recent results on planar graphs and the planarity testing problem see Nishizeki and Chiba [11.120] and de Fraysseix and Rosenstiehl [11.121]. The planar layout problem and some related results are discussed in Otten and Van Wijk [11.122], Rosenstiehl and Tarjan [11.123], Chiba, Nishizeki, Abe and Ozawa [11.124], and Jayakumar, Thulasiraman, and Swamy [11.125]. For algorithms for the maximal planarization problem see Chiba, Nishioka, and Shirakawa [11.126], Ozawa and Takahashi [11.127], Jayakumar, Thulasiraman, and Swamy [11.128], and Marek-Sadowska [11.129]. Eades and Tamassia [11.130] have given an extensive annotated bibliography of algorithms for planar drawings with various properties and related applications.

11.12

FURTHER READING

In the past two decades a large number of graph algorithms have been reported in the literature and certain new research areas (motivated by recent technological developments) have emerged. References to some of these may be found in Section 12.12 of this book. See Lengauer [11.131] for an extensive coverage of applications of combinatorial algorithms in integrated circuit layout design.

380

11.13

GRAPH ALGORITHMS

EXERCISES

11.1

A transitive reduction of a directed graph G = (V, E) is defined to be any graph G' = ( V , £ ' ) with as few edges as possible such that the transitive closure of G' is equal to the transitive closure of G. Design an algorithm to find a transitive reduction of a directed graph. How is this algorithm related to the transitive closure algorithm? (Aho, Garey, and Ullman [11.132].)

11.2

Prove the validity of the following modification of Dijkstra's algorithm for finding the shortest paths between a specified vertex s and all other vertices in a weighted directed graph G, which has no directed circuits of negative lengths. In the following, Ε denotes the edge set of G, and w(u, v) denotes the length of the directed edge (", v). 51. Set σ(ί) = 0, and set σ{μ) = °° for u Φ s. 52. Set S = {s}. 53. If 5 = Φ, HALT; otherwise select u* such that u* Ε S and °-(w*) = min {σ(κ)} . 54. For each ν such that (u*, υ) Ε Ε, set σ(υ) = ππη{σ(ι>), a(u*) + w(u*, υ)}. If this process decreases σ(ν), set 5 = S U {v}. 55. Set S = 5 - {u*}, and go to S3. (Edmonds and Karp [11.31].)

11.3

Prove the validity of the following algorithm due to Dantzig [11.37] for finding the shortest paths between all pairs of vertices in a weighted graph that has no directed circuits of negative length. Here W (i, j) is the distance from i to / where 1 s j , j < k and no vertices higher than k are used on the path. w(i, ;') = W\i, j) is the length of the edge (/, ;') and w(i, j) = <» if no such edge exists. Also w(i, i) = 0 for all i. 5 1 . Set k = 2. 52. For 1 < j < k do k

W (i, k) = min M i , k), W ~\i, k

k

j) + w(j, k)}

W (k, i) = min {w(k, i), w(k j) + W '\j, k

53. For 1

k

i)} .

< i , j
k

k

EXERCISES

381

S4. If k = n, STOP; otherwise set k = k + 1 and go to S2. Also show how negative-length circuits can be detected while carrying out this algorithm. 11.4

Using topological sorting, design an algorithm to find the shortest path from a vertex s to all the other vertices in an acyclic weighted directed graph.

11.5

Suppose the shortest path from ι to j is not unique. Which path is chosen by Floyd's algorithm (Algorithm 11.5)?

11.6

An optimum branching may not be an optimum spanning directed tree. Show how Edmonds' algorithm can be modified to find an optimum spanning directed tree.

11.7

Using DFS, design algorithms for the following problems: (a) Topological sorting of the vertices of a graph. (b) Finding bridges of a graph. (c) Finding a spanning forest. (d) Finding a set of fundamental circuits and a set of fundamental cutsets. (e) Testing whether a graph is bipartite. (f) Testing whether a subset of edges is a cut of a graph.

11.8

Consider a graph G and let Γ be a DFS spanning forest of G. Let ND(y) be the number of descendants of υ (including itself). Prove that a vertex w is a descendant of υ if and only if DFN(u) < DFN(w) < DFN(u) + ND(u).

11.9

Modify the DFS algorithm to include computation of ND(tv).

11.10

Let Γ be a DFS tree of a connected graph G. Let G be a complete subgraph of G. Show that all the vertices of G lie on one directed path in T. Let Γ be a DFS tree of a directed graph G. If C is a directed circuit in G and υ is the vertex on C with minimum depth-first number, show that ν is an ancestor in Τ of every vertex on C. s

s

11.11

11.12

Breadth-first search (BFS) explores a connected graph G as follows: 51. To begin with no vertices of G are labeled. 52. Select vertex s and label it with 0. 53. Set i = 0. 54. Let S be the set of all unlabeled vertices adjacent to at least one vertex labeled i. 55. If S is empty HALT. Otherwise label all the vertices in S with i + l.

382

GRAPH ALGORITHMS

S6. Set i = i + l and go to S4. Show that the edges traversed while labeling the vertices form a spanning tree of G. 11.13 Show how the BFS algorithm can be used to compute the distance from a vertex s to all the vertices in a connected graph G. (By distance between vertices u and υ here we mean the length of a u-v path having the smallest number of edges.) 11.14

Show that the set of back edges of a reducible program graph is unique, that is, all DFS spanning forests have the same set of back edges (Hecht and Ullman [11.101]).

11.15

Given a biconnected graph G, design a DFS-based algorithm to extract a minimal biconnected subgraph of G.

11.16

Prove the following theorem known as the planar separator theorem ([11.133], [11.134] and [11.135]). Let G = (V, E) be a planar graph with a nonnegative real number w(v), called weight of v, associated with each vertex v. Let W= Σ w(v) be the total weight of G. Then there is a partition A, S, Β of V such that 1) W(A) = Z w(v)^2l iW, W(fl)<2/3W. 2) | 5 | < 4 Vn 3) S separates A from B, that is, Ε Π (A x Β) = Φ. 4) Partition A, S, Β can be constructed in time 0(n). (Note: See also [11.14] for a proof of this and several related results.) υεν

/

u&A

11.14

REFERENCES

11.1

D. Gries, Compiler Construction for Digital Computers, Wiley, New York, 1971. S. Warshall, "A Theorem on Boolean Matrices," J. ACM, Vol. 9, 11-12 (1962). H. S. Warren, "A Modification of Warshall's Algorithm for the Transitive Closure of Binary Relations," Comm. ACM, Vol. 18, 218-220 (1975). V. L. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. Faradzev, "On Economical Construction of the Transitive Closure of a Directed Graph," Soviet Math. Dokl., Vol. 11, 1209-1210 (1970). V. Strassen, "Gaussian Elimination is Not Optimal," Numerische Math., Vol. 13, 354-356 (1969). M. J. Fischer and A. R. Meyer, "Boolean Matrix Multiplication and Transitive Closure," Conf. Record, IEEE 12th Annual Symp. on Switching and Automata Theory, 1971, pp. 129-131. Μ. E. Furman, "Application of a Method of Fast Multiplication of Matrices

11.2 11.3 11.4 11.5 11.6 11.7

REFERENCES

11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 11.22 11.23 11.24 11.25 11.26 11.27 11.28

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in the Problem of Finding the Transitive Closure of a Graph," Soviet Math. Dokl., Vol. 11, 1252 (1970). I. Munro, "Efficient Determination of the Transitive Closure of a Directed Graph," Information Processing Lett., Vol. 1, 56-58 (1971). P. E. O'Neil and E. J. O'Neil, "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure," Inform, and Control, Vol. 22, 132-138 (1973). J. Eve and R. Kurki-Suonio, "On Computing the Transitive Closure of a Relation," Acta Informatica, Vol. 8, 303-314 (1977). P. Purdom, "A Transitive Closure Algorithm," BIT, Vol. 10, 76-94 (1970). C P . Schnorr, "An Algorithm for Transitive Closure with Linear Expected Time," SIAMJ. Computing, Vol. 7, 127-133 (1978). Μ. M. Syslo and J. Dzikiewicz, "Computational Experience with Some Transitive Closure Algorithms," Computing, Vol. 15, 33-39 (1975). K. Melhorn, Graph Algorithms and NP-Completeness, Springer-Verlag, Berlin, 1984. L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., 1962. E. W. Dijkstra, "A Note on Two Problems in Connexion with Graphs," Numerische Math., Vol. 1, 269-271 (1959). L. R. Ford, Jr., "Network Flow Theory," Paper P-923, RAND Corp., Santa Monica, Calif., 1956. R. E. Bellman, "On a Routing Problem," Quart. Appl. Math., Vol. 16, 87-90 (1958). E. F. Moore, "The Shortest Path Through a Maze," Proc. Intl. Symp. Theory of Switching, Part II, University Press, Cambridge, Mass., 1957, pp. 285-292. S. E. Dreyfus, "An Appraisal of Some Shortest-Path Algorithms," Operations Research, Vol. 17, 395-412 (1969). T. C. Hu, "A Decomposition Algorithm for Shortest Paths in a Network," Operations Research, Vol. 16, 91-102 (1968). H. Frank and I. T. Frisch, Communication, Transmission and Transportation Networks, Addison-Wesley, Reading, Mass., 1971. N. Christofides, Graph Theory: An Algorithmic Approach, Academic Press, New York, 1975. E. L. Lawler, Combinatorial Optimization: Networks and Matroids Holt, Rinehart and Winston, New York, 1976. M. Gordon and M. Minoux, Graphs and Algorithms, (trans, by S. Vajda), Wiley, New York, 1984. E. Minieka, Optimization Algorithms for Networks and Graphs, Marcel Dekker, New York, 1978. P. M. Spira and A. Pan, "On Finding and Updating Spanning Trees and Shortest Paths," SIAM J. Comput., Vol. 4, 375-380 (1975). D. B. Johnson, "Efficient Algorithms for Shortest Paths in Sparse Networks," J. ACM, Vol. 24, 1-13 (1977).

384

GRAPH ALGORITHMS

11.29

R. A. Wagner, "A Shortest Path Algorithm for Edge-Sparse Graphs," /. ACM, Vol. 23, 50-57 (1976). D. B. Johnson, "A Note on Dijkstra's Shortest Path Algorithm," /. ACM, Vol. 20, 385-388 (1973). J. Edmonds and R. M. Karp, "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems," J. ACM, Vol. 19, 248-264 (1972). M. L. Fredman, "New Bounds on the Complexity of the Shortest Path Problem," SIAM J. Comp., Vol. 5, 83-89 (1976). M. A. Comeau and K. Thulasiraman, "Structure of the Submarking Reachability Problem and Network Programing," IEEE Trans. Circuits and Systems, Vol. 35, 89-100 (1988). Y. Liao and C. K. Wong, "An Algorithm to Compact VLSI Symbolic Layout with Mixed Constraints," IEEE Trans. CAD. Circuits and Systems, Vol. 2, 62-69 (1983). T. Lengauer, "On the Solution of Inequality Systems Relevant to IC Layout," J. Algorithms, Vol. 5, 408-421 (1984). R. W. Floyd, "Algorithm 97: Shortest Path," Comm. ACM, Vol. 5, 345 (1962). G. B. Dantzig, "All Shortest Routes in a Graph," in Theory of Graphs, Gordon and Breach, New York, 1967, pp. 91-92. Y. Tabourier, "AH Shortest Distances in a Graph: An Improvement to Dantzig's Inductive Algorithm," Discrete Math., Vol. 4, 83-87 (1973). J. Y. Yen, "Finding the Lengths of All Shortest Paths in N-Node, NonNegative Distance Complete Networks Using N /2 Additions and Ν Comparisons," J. ACM, Vol. 19, 423-424 (1972). T. A. Williams and G. P. White, "A Note on Yen's Algorithm for Finding the Length of All Shortest Paths in N-Node Non-Negative Distance Networks," /. ACM, Vol. 20, 389-390 (1973). A. R. Pierce, "Bibliography on Algorithms for Shortest Path, Shortest Spanning Tree and Related Circuit Routing Problems (1956-1974)," Networks, Vol. 5, 129-149 (1975). N. Deo and C. Y. Pang, "Shortest Path Algorithms-Taxonomy and Annotation," Networks, Vol. 14, 275-323 (1984). R. E. Tarjan, "Data Structures and Network Algorithms," CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 44, Society for Industrial Applied Mathematics, Philadelphia, 1983. B. Carre, Graphs and Networks, Clarendon Press, Oxford, 1979. R. E. Tarjan, "A Unified Approach to Path Problems," /. ACM., Vol. 28, 577-593 (1981). G. N. Frederickson, "Fast Algorithms for Shortest Paths in Planar Graphs with Applications," SIAM J. Comp., Vol. 16, 1004-1022 (1987). A. Moffat and T. Takoka, "An All-Pairs Shortest Path Algorithm with Expected Time 0(n log n)," SIAMJ. Comp., Vol. 16, 1023-1031 (1987). J. B. Kruskal, Jr., "On the Shortest Spanning Subtree of a Graph and the Travelling Salesman Problem," Proc. Am. Math. Soc, Vol. 7, 48-50 (1956).

11.30 11.31 11.32 11.33 11.34 11.35 11.36 11.37 11.38 11.39

3

11.40 11.41 11.42 11.43 11.44 11.45 11.46 11.47

2

11.48

3

REFERENCES

11.49 11.50 11.51 11.52 11.53 11.54 11.55 11.56 11.57 11.58 11.59 11.60 11.61 11.62 11.63 11.64 11.65 11.66 11.67 11.68 11.69 11.70

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R. C. Prim, "Shortest Connection Networks and Some Generalizations," Bell Sys. Tech. J., Vol. 36, 1389-1401 (1957). A. Kerschenbaum and R. Van Slyke, "Computing Minimum Spanning Trees Efficiently," Proc. 25th Ann. Conf. of the ACM, 1972, pp. 518-527. A. C. Yao, "An 0(\E\ log log |V|) Algorithm for Finding Minimum Spanning Trees," Information Processing Lett., Vol. 4, 21-23 (1975). D. Cheriton and R. E. Tarjan, "Finding Minimum Spanning Trees," SIAM J. Comput., Vol. 5, 724-742 (1976). R. E. Tarjan, "Sensitivity Analysis of Minimum Spanning Trees and Shortest Path Trees," Information Processing Lett., Vol. 14, 30-33 (1982). C. H. Papadimitriou and M. Yannakakis, "The Complexity of Restricted Minimum Spanning Tree Problems," J. ACM, Vol. 29, 285-309 (1982). R. L. Graham and P. Hall, "On the History of the Minimum Spanning Tree Problem," Mimeographed, Bell Laboratories, Murray Hill, N.J., 1982. J. Edmonds, "Optimum Branchings," J. Res. Nat. Bur. Std., Vol. 71B, 233-240 (1967). R. M. Karp, "A Simple Derivation of Edmonds' Algorithm for Optimum Branchings," Networks, Vol. 1, 265-272 (1972). R. E. Tarjan, "Finding Optimum Branchings," Networks, Vol. 7, 25-35 (1977). P. M. Camerini, L. Fratta, and F. Maffioli, "A Note on Finding Optimum Branchings," Networks, Vol. 9, 309-312 (1979). Y. Chu and T. Liu, "On the Shortest Arborescence of a Directed Graph," Scientia Sinica [Peking], Vol. 4, 1396-1400, (1965); Math. Rev., Vol. 33, #1245 (D. W. Walkup). F. C. Bock, "An Algorithm to Construct a Minimum Directed Spanning Tree in a Directed Network," in Developments in Operations Research, B. Avi-Itzak, Ed., Gordon and Breach, New York, 1971, pp. 29-44. A. Itai, M. Rodeh, and S. L. Tanimoto, "Some Matching Problems for Bipartite Graphs," J. ACM, Vol. 25, 517-525 (1978). H. W. Kuhn, "The Hungarian Method for the Assignment Problem," Naval Res. Legist. Quart., Vol. 2, 83-97 (1955). J. Munkres, "Algorithms for the Assignment and Transportation Problems," J. SIAM, Vol. 5, 32-38 (1957). J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, 1976. N. Megiddo and A. Tamir, "An 0(N log N) Algorithm for a Class of Matching Problems," SIAM J. Comput., Vol. 7, 154-157 (1978). R. M. Karp, "An Algorithm to Solve the m x η Assignment Problem in Expected Time (mn log «)," Networks, Vol. 10, 143-152 (1980). V. Derigs, "A Shortest Augmenting Path Method for Solving Minimal Perfect Matching Problems," Networks, Vol. 11, 379-390 (1981). D. Avis, "A Survey of Heuristics for the Weighted Matching Problem," Networks, Vol. 13, 475-493 (1983). D. Avis and C. W. Lai, "The Probabilistic Analysis of a Heuristic for the Assignment Problem," SIAM J. Comp., Vol. 17, 732-741 (1988).

386

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11.71

Μ. D. Grigoriadis and B. Kalantri, "A New Class of Heuristic Algorithms for Weighted Perfect Matching," J. ACM, Vol. 35 , 769-776 (1988). R. Cole and J. Hopcroft, "On Edge Coloring Bipartite Graphs," SIAMJ. Comp., Vol. 11, 540-546 (1982). Η. N. Gabow and O. Kariv, "Algorithm for Edge Coloring Bipartite Graphs and Multigraphs," SIAMJ. Comp., Vol. 11, 117-129 (1982). S. Even, A. Itai, and A. Shamir, "On the Complexity of Time-Table and Multicommodity Flow Problems," SIAM J. Comput., Vol. 5, 691-703 (1976). Meo-Ko Kwan, "Graphic Programming Using Odd and Even Points," Chinese Math., Vol. 1, 273-277 (1962). J. Edmonds, "The Chinese Postman Problem," Operations Research, Vol. 13, Suppl. 1, 373 (1965). J. Edmonds and E. L. Johnson, "Matching, Euler Tours and the Chinese Postman," Math. Programming, Vol. 5, 88-124 (1973). R. J. Wilson, Introduction to Graph Theory, Oliver and Boyd, Edinburgh, 1972. Α. V. Aho, A. T. Dahbura, D. Lee, and M. U. Uyar, "An Optimization Technique for Protocol Conformance Test Generation Based on UIO Sequences and Rural Chinese Postman Tours," in Protocol Specification, Testing, and Verification, VIII, S. Aggarwal and Κ. K. Sabnani, Eds., Elsevier North-Holland, New York, 1988, pp. 75-86. C. S. Orloff, "A Fundamental Problem in Vehicle Routing," Networks, Vol. 4, 35-64 (1974). C. H. Papadimitriou, "On the Complexity of Edge Traversing," J. ACM, Vol. 23, 544-554 (1976). J. Lenstra and A. Rinooy-Kan, "On General Routing Problems," Networks, Vol. 6, 273-280 (1976). G. N. Frederickson, "Approximation Algorithms for Some Postman Problems," J. ACM, Vol. 26, 538-554 (1979). M. Dror, H. Stern, and P. Trudeau, "Postman Tour on a Graph with Precedence Relation on Arcs," Networks, Vol. 17, 283-294 (1987). R. E. Tarjan, "Depth-First Search and Linear Graph Algorithms," SIAM J. Comput., Vol. 1, 146-160 (1972). J. Hopcroft and R. Tarjan, "Efficient Algorithms for Graph Manipulation," Comm. ACM, Vol. 16, 372-378 (1973). Α. V. Aho, J. E. Hopcroft, and J. D. Ullman, "On Finding the Least Common Ancestors in Trees," SIAM J. Comput., Vol. 5, 115-132 (1976). J. Cocke, "Global Common Subexpression Elimination," SIGPLAN Notices, Vol. 5, 20-24 (1970). J. D. Ullman, "Fast Algorithms for the Elimination of Common Subexpressions," Acta Informatica, Vol. 2, 191-213 (1973). K. Kennedy, "A Global Flow Analysis Algorithm," Int. J. Computer Math., Vol. 3, 5-16 (1971). M. Schaefer, A Mathematical Theory of Global Program Optimization, Prentice-Hall, Englewood Cliffs, N.J., 1973.

11.72 11.73 11.74 11.75 11.76 11.77 11.78 11.79

11.80 11.81 11.82 11.83 11.84 11.85 11.86 11.87 11.88 11.89 11.90 11.91

REFERENCES

11.92 11.93 11.94 11.95 11.96 11.97 11.98 11.99 11.100 11.101 11.102 11.103 11.104 11.105 11.106 11.107 11.108

11.109 11.110

387

Α. V. Aho and J. D. Ullman, The Theory of Parsing, Translation and Compiling, Vol. II, Compiling, Prentice-Hall, Englewood Cliffs, N.J., 1973. F. E. Allen, "Control Flow Analysis," SIGPLAN Notices, Vol. 5, 1-19 (1970). F. E. Allen, Program Optimization, Annual Review in Automatic Programming, Vol. 5, Pergamon, New York, 1969. M. S. Hecht, Flow Analysis of Computer Programs, Elsevier, New York, 1977. M. S. Hecht and J. D. Ullman, "Flow Graph Reducibility," SIAM J. Comput., Vol. 1, 188-202 (1972). J. Cocke and R. E. Miller, "Some Analysis Techniques for Optimizing Computer Programs," Proc. 2nd Int. Conf. on System Sciences, Honolulu, Hawaii, 1969. J. E. Hopcroft and J. D. Ullman, "An η log η Algorithm for Detecting Reducible Graphs," Proc. 6th Annual Princeton Conf. on Information Sciences and Systems, Princeton, N.J., 1972, pp. 119-122. R. E. Tarjan, "Testing Flow Graph Reducibility," Proc. 5th Annual ACM Symp. on Theory of Computing, 1973, pp. 96-107. R. E. Tarjan, "Testing Flow Graph Reducibility," /. Comput. Sys. Sci., Vol. 9, 355-365 (1974). M. S. Hecht and J. D. Ullman, "Characterizations of Reducible Flow Graphs," J. ACM, Vol. 21, 367-375 (1974). J. M. Adams, J. M. Phelan, and R. H. Stark, "A Note on the HechtUllman Characterization of Non-Reducible Flow Graphs," SIAM J. Cornput., Vol. 3, 222-223 (1974). M. Fischer, "Efficiency of Equivalence Algorithms," in Complexity of Computer Computations, R. E. Miller and J. W. Thatcher, Eds., Plenum Press, New York, 1972, pp. 153-168. J. E. Hopcroft and J. D. Ullman, "Set Merging Algorithms," SIAM J. Comput., Vol. 2, 294-303 (1973). R. E. Tarjan, "On the Efficiency of a Good but Not Linear Set Union Algorithm," /. ACM, Vol. 22, 215-225 (1975). Α. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974. E. Horowitz and S. Sahni, Fundamentals of Data Structures, Computer Science Press, Potomac, Md., 1976. S. Shinoda, W-K. Chen, T. Yasuda, Y. Kajitani, and W. Mayeda, "A Necessary and Sufficient Condition for any Tree of a Connected Graph to be a DFS-Tree of One of Its 2-Isomorphic Graphs," IEEE ISCAS, New Orleans, 2841-2844 (1990). J. Valdes, R. E. Tarjan, and E. L. Lawler, "The Recognition of Series Parallel Digraphs," SIAM J. Comp., Vol. 11, 298-313 (1982). A. Lempel, S. Even, and I. Cederbaum, "An Algorithm for Planarity Testing of Graphs," Theory of Graphs, International Symp., Rome, July 1966, P. Rosenstiehl, Ed., Gordon and Breach, New York, 1967, pp. 215-232.

388

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11.111 S. Even and R. E. Tarjan, "Computing an si-Numbering," Th. Comp. Sci., Vol. 2, 339-344 (1976). 11.112 J. Erbert, "sf-Ordering the Vertices of Biconnected Graphs," Computing, Vol. 30, 19-33 (1983). 11.113 S. Wimer, J. Koren, and I, Cederbaum, "Floor Plans, Planar Graphs and Layouts," IEEE Trans. Circuits and Systems, Vol. CAS-35, 267-278 (1988). 11.114 J. Hopcroft and R. E. Tarjan, "Efficient Planarity Testing," /. ACM, Vol. 21, 549-568 (1974). 11.115 Ε. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, N.J., 1977. 11.116 S. Even, Graph Algorithms, Computer Science Press, Potomac, Md., 1979. 11.117 G. Demoucron, Y. Malgrange, and R. Pertuiset, "Graphes Planaires: Reconnaissance et Construction de Representation Planaires Topologiques," Rev. Francaise de Rech. Operationelle, Vol. 8, 33-47 (1984). 11.118 F. Rubin, "An Improved Algorithm For Testing the Planarity of a Graph," IEEE Trans. Computers, Vol. C-24, 113-121 (1975). 11.119 K. S. Booth and G. S. Lueker, "Testing for the Consecutive Ones Property, Interval Graphs and Graph Planarity Testing Using PQ-tree Algorithms," J. Comp. and Sys. Sciences, Vol. 13, 335-379 (1976). 11.120 T. Nishizeki and N. Chiba, "Planar Graphs: Theory and Algorithms," Annals of Discrete Mathematics, Vol. 32, North-Holland, Amsterdam, 1988. 11.121 H. de Fraysseix and P. Rosenstiehl, "A Depth-First Search Characterization of Planarity," Annals of Discrete Mathematics, Vol. 13, 75-80 (1982). 11.122 R. H. J. M. Otten and J. G. Van Wijk, "Graph Representation in Interactive Layout Design," Proc. IEEE Intl. Symp. Circuits and Systems, 1978, pp. 914-918. 11.123 P. Rosenstiehl and R. E. Tarjan, "A Unified Approach to Visibility Representation of Planar Graphs," Discrete and Computational Geometry, Vol. 1, Springer-Verlag, Berlin, 1986, pp. 343-353. 11.124 N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa, "A Linear Algorithm for Embedding Planar Graphs Using PQ-trees," J. Comp. and Syst. Sciences, Vol. 30, 54-76 (1985). 11.125 R. Jayakumar, K. Thulasiraman, and Μ. N. S. Swamy, "Planar Embedding: Linear Time Algorithms for Vertex Placement and Edge Ordering," IEEE Trans. Circuits and Systems, Vol. 35, 334-344 (1988). 11.126 N. Chiba, I. Nishioka, and I. Shirakawa, "An Algorithm of Maximal Planarisation of Graphs," Proc. IEEE Intl. Symp. Circuits and Systems, 1979, pp. 649-652. 11.127 T. Ozawa and H. Takahashi, "A Graph-Planarization Algorithm and Its Applications," Graph Theory and Algorithms, Lecture Notes in Computer Science, Vol. 108, 95-107 (1981). 11.128 R. Jayakumar, K. Thulasiraman, and Μ. N. S. Swamy, "0(n ) Algorithms for Graph Planarisation," IEEE Trans. CAD. Integrated Circuits and Systems, Vol. 8, 257-267 (1988). 2

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11.129 Μ. Marek-Sadowska, "Planarisation Algorithms for Integrated Circuits Engineering," Proc. IEEE Intl. Symp. on Circuits and Systems, New York, 1978, pp. 919-923. 11.130 P. Eades and R. Tamassia, "Algorithms for Drawing Graphs," An Annotated Bibliography, Technical Report No. CS-89-09, Dept. Comp. Science, Brown University, 1989. 11.131 T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, Wiley, New York, 1990. 11.132 Α. V. Aho, M. R. Garey, and J. D. Ullman, "The Transitive Reduction of a Directed Graph," SIAM J. Computing, Vol. 1, 131-137 (1972). 11.133 R. Lipton and R. E. Tarjan, "A Separtor Theorem for Planar Graphs," Proc. of the Conf. on Theoretical Computer Science, Waterloo, Canada, 1-10 (1977). 11.134 R. Lipton and R. E. Tarjan, "Applications of a Planar Separator Theorem," Proc. 18th Annual Symp. Foundations of Computer Science Conf., 1977, pp. 162-170. 11.135 J. D. Ullman, Computational Aspects of VLSI, Computer Science Press, Rockville, Md., 1984.

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

CHAPTER 12

FLOWS IN NETWORKS

Network optimization, a branch of operations research, refers to the class of optimization problems defined on graphs or networks. These problems include the minimum cost flow problem, the maximum flow problem, covering problems, and such problems as the optimal assignment problems and the Chinese postman problem considered in the previous chapter. While important by themselves, they also arise as subproblems while studying larger and more complex problems. For instance, as we have seen in the previous chapter, shortest path and matching algorithms serve as building blocks in the algorithm for the Chinese postman problem. In this chapter we study, in sufficient detail, the maximum flow problem. While the theoretical richness of this problem has an unsurpassing beauty, it is also practically very important. It provides the main link between graph theory and operations research. The celebrated max-flow min-cut theorem of Ford and Fulkerson is equivalent to some of the celebrated graph theorems such as Hall's marriage theorem and Menger's theorems on connectivities. This equivalence has helped develop efficient matching and connectivity algorithms. The chapter is organized as follows. After stating the maximum flow problem in Section 12.1, we develop in Section 12.2 the maximum flow minimum cut theorem. We then proceed to develop two fundamentally different approaches to the maximum flow problem. In Sections 12.3 to 12.6, Ford and Fulkerson's labeling algorithm and several refinements of this algorithm are discussed. In Section 12.7 we develop the recent work of Goldberg and Tarjan, which represents a significant departure from earlier works based on Ford and Fulkerson's algorithm. In Section 12.8 we study the complexity of the maximum flow problem in the case of a special class of 390

THE MAXIMUM FLOW PROBLEM

391

networks called 0-1 networks. The next two sections show how network flow techniques can be used to develop algorithms for matching and connectivity problems. We conclude with a brief introduction to the theory of NP-completeness.

12.1

THE MAXIMUM FLOW PROBLEM

A transport network TV is a connected directed graph that has no self-loops and that satisfies the following conditions: 1. There is only one vertex with zero in-degree; this vertex is called the source and is denoted by s. 2. There is only one vertex with zero out-degree; this vertex is called the sink and is denoted by t. 3. Each directed edge e = (/, / ) in Ν is associated with a nonnegative real number called the capacity of the edge; it is denoted by c(e) or c(i, ;'). If there is no edge e directed from ι to ;', then we define c(e) = 0. A transport network represents a model for the transportation of a commodity from its production center to its market through communication routes. The capacity of an edge may then be considered as representing the maximum rate at which a commodity can be transported along the edge. A flow f in a transport network Ν is an assignment of a nonnegative real number f(e) = f(i, j) to each edge e = (i, j) such that the following conditions are satisfied: 1. / ( j , / ) < c(i, / ) for every edge (i, j) in TV. 2. Σ . „ , / ( / , / ) = Σ j / ( / , ι) for all i Φ s, t. βΙΙ

(12.1) (12.2)

The value f(e) of the flow in edge e may be considered as the rate at which material is transported along e under the flow/. Condition (12.1), called the capacity constraint, requires that the rate of flow along an edge cannot exceed the capacity of the edge. Condition (12.2), called the conservation condition, requires that for each vertex i, except the source and the sink, the rate at which material is transported into i is equal to the rate at which it is transported out of i. As an example, a transport network TV with a flow/is shown in Fig. 12.1. In this figure next to each edge e we have shown the capacity c(e) and the flow /(e) in that order. The value of a flow /, denoted by v a l ( / ) , is defined as val(/)=

Σ/(*,/)· all

i

(12.3)

392

FLOWS IN NETWORKS

2. 1

4,4

/

/ \

2, 1 Λ .3,2

6,4

1

N.

5,2,

'

10,2

2,2\ 3,2

\

d Figure 12.1. A transport network. '3

c

Soon we prove using the conservation condition that val(/)=

Σ/(*,/)= Σ/(/,')· all

ι

(12.4)

all /

This would only confirm the intuitively obvious fact that, because of the conservation condition, the total amount of material transported out of the source is equal to the total amount transported into the sink. A flow / * in a transport network Ν is said to be maximum if there is no flow fin Ν such that val(/) > val(/*). 12.2

MAXIMUM FLOW MINIMUM CUT THEOREM

A cut (S, S) in a transport network Ν is said to separate the source s and the sink / if s Ε S and t Ε S. Such a cut will be referred to as an s-t cut. The capacity c(K) = c(5, 5) of a cut Κ = {S, S) is denned as c(K) = Σ Φ, j).

(12.5)

ies ,es

Note that the capacities of the edges directed from S to Sjdo not contribute to the capacity of the cut Κ = (S, S)._We denote by f(S, S) the sum of the flows in the edges directed from 5 to 5. The quantity f(S, S) is defined in a similar way. _ For example, consider the cut Κ = (S, S) in the transport network shown in Fig. 12.1, where S = {a, b, c,s)

and

S = { d , t) .

The edges directed from 5 to S are e , e , and e . Hence 3

4

g

MAXIMUM FLOW MINIMUM CUT THEOREM

393

c(S, S) = c(e ) + c(e ) + c(e ) = 1 1 , 3

4

8

/(S,S)=/(e ) + /(e ) + / ( ) = 8, 3

4

e 8

f(S, 5) = / ( e ) = 2 . 9

Theorem 12.1. For any flow/and any s-t cut (S, S) in a transport network N, val(/) = / ( S , S ) - / ( S , S ) .

(12.6)

Proof. From the definitions of a flow and the value of a flow, we have V

a-

V

-\

λ · ·

·\

i

v

a

l

Z / 0 , / ) - Z / ( ; , ) = (o,

( / ) >

if*

all /

all ;

>•

=

·*;

ifies-{ }.

i

s

'

1

'

Summing these equations over all the vertices in S, we get Σ Σ /(Λ / ) - Σ Σ /(/, 0 = v a l ( / ) . ι i e i all y

(12.7)

i 6 S all

In this equation both / ( / , j) and - / ( / , / ) , for i £ S and / £ 5, appear exactly once on the left-hand side and get canceled out. So (12.7) simplifies to

Σ Σ/('·,/)-Σ Σ/κ/,ο

= νΗΐ(/).

Thus / ( S , 5 ) - / ( 5 , S ) = val(/).

•

Note that (12.3) is a special case of (12.6). Corollary 12.1.1. For any flow / a n d any s-t cut Κ = (S, S) in a transport network N, val(/)
(12.8)

Proof. Since every / ( i , ;') is nonnegative, we get from (12.6) v a l ( / ) = / ( S , S ) - / ( S , S)
•

An edge (i, / ) is said to be f-saturated if / ( / , / ) = c(i, ;') and f-unsaturated otherwise; it is f-positive if / ( / , / ) > 0, and f-zero if / ( i , / ) = 0.

394

FLOWS IN NETWORKS

Note that equality in (12.8) holds if and only if f(S, S) = 0 and f(S, S) = c(S, 5). Injrther words val(/) = c(S, 5) if and only if_all the edges directed from S to S are /-saturated and those directed from S to S are /-zero. An s-t cut Κ in a transport network Ν is a minimum cut if there is no s-t cut K' in Ν such that c(/C) < c(K). Corollary 12.1.2. Let / be a flow and Κ be an s-t cut such that v a l ( / ) = c(K). Then / is a maximum flow and Κ is a minimum s-f cut. Proof. Let / * be a maximum flow and K* be a minimum s-t cut. By Corollary 12.1.1, val(/*)
e,

e

2

Ο

Ο

Ο ··· Ο

S = U

Μ,

U

0

2

«,_,

e

k

Ο ··· Ο

Ο

II,

U

k

= V

be a path in Ν from the source to some vertex v. Note that Ρ need not be a directed path. Edge e, of Ρ is called a forward edge of Ρ if it is oriented from u,_, to u,. Otherwise it will be called a backward edge of P. For each edge e, in P, let /(«,.),

if e, is a forward edge ; if e, is a backward edge .

\

· )

With the path Ρ we associate a nonnegative number e(P) defined by e(P) = m i n { , ( P ) } . C

(12.10)

Note that e(P) > 0. We call a path f-unsaturated if all the forward edges of the path are /-unsaturated and all the backward edges of the path are /-positive. An s-t path Ρ is called an f-augmenting path if Ρ is /-unsaturated. From

MAXIMUM FLOW MINIMUM CUT THEOREM

'3

395

d

Figure 12.2 (12.9) and (12.10) it follows that for an s-t path P, e(P) > 0 if and only if P is /-augmenting. Given an s-t path P in the network N, we can define a new flow / as follows:

I

f(e) + e(P), f(e) - e(P), /(e),

if e is a forward edge of P ; if e is a backward edge of P ; otherwise .

We can easily see that val(/) = val(/) + e(P). Thus val(/) > val(/) if and only if P is /-augmenting. In other words a flow/ is not maximum if there exists an /-augmenting path. As an example, consider the network N of Fig. 12.1. Let flow / be as shown in this figure. In the path P consisting of e2, es. e-,, and e8, the edges e2, eg, and e-, are forward edges, and e5 is the only backward edge. With respect to the flow /, e2(P) = 1, e5(P) = 1, e7(P) = 1, and e8(P) = 2. Hence e(P) = min{e2(P), e 5 (P), e,(P), e8(P)} = 1. Since e(P) > 0, P is/-augmenting. The revised flow /based on the path P is as shown in Fig. 12.2. Note that/is obtained by increasing the flows in all the forward edges of P by e(P) and decreasing the flows in the backward edges of P by e(P). The flows in the remaining edges remain unaltered. Theorem 12.2. A flow / in a transport network N is maximum if and only if there is no /-augmenting path. Proof. Necessity If there is an /-augmenting path P in N, then clearly / is not a maximum flow because the revised flow / based on P has a larger value than /·

396

FLOWS IN NETWORKS

Sufficiency Suppose that Ν contains no/-augmenting path. Let S denote the set of all vertices in Ν that can be reached from the source by /-unsaturated paths. Clearly, sES. Further, tES because there is no /-augmenting path in N. _ Now we show that val(/) = c(S, S), Jhereby proving (by Corollary 12.1.2) that / i s a maximum flow and (S, S) is a minimum cut. Consider a directed edge (i>, w) with υ Ε S and wES. Since vES, there is an /-unsaturated s-υ path Q. The edge (υ, w) must be /-saturated, for otherwise Q can be extended to yield an /-unsaturated s-w path, and this is not possible since w Ε S. Similarly we can show that every edge_(u, w) directed from S to 5 should be /-zero. Thus / ( S , 5) = c(S, S) and f(S, S) = 0. Hence val( / ) = / ( S , S) - / ( S , S) = c(S, 5 ) . It now follows from Corollary 12.1.2 t h a t / i s a maximum flow and (S,S) a minimum cut. •

is

In the course of the proof of Theorem 12.2 we have also established the following well-known result due to Ford and Fulkerson [12.1] and Elias, Feinstein and Shannon [12.2]. Theorem 12.3 (Max-Flow Min-Cut Theorem). In a transport network the value of a maximum flow is equal to the capacity of a minimum cut. • The max-flow min-cut theorem can be used to prove several results of combinatorial significance. We discuss two of these results in Sections 12.9 and 12.10. For others, Ford and Fulkerson [12.3] and Berge [12.4] may be consulted.

12.3

FORD-FULKERSON LABELING ALGORITHM

We now discuss an algorithm also due to Ford and Fulkerson [12.1] for determining a maximum flow in a transport network. This algorithm, based on Theorem 12.2, consists of two phases. Given a flow /. In the first phase we use a labeling procedure to check whether an /-augmenting path exists. If such a path does not exist, then by Theorem 12.2 the present flow/is maximum. Otherwise we proceed to the second phase in which, using the labels generated in the first phase, we determine an /-augmenting path Ρ and obtain the revised flow / based on P. We then repeat phase 1 with the new flow /. Note that v a l ( / ) > v a l ( / ) . In the first phase a label of the form (d , Δ„) is assigned to a vertex v. The first symbol d in the label indicates the vertex from which υ receives its label. It would also indicate the direction of labeling—forward or backward. v

v

FORD-FULKERSON LABELING ALGORITHM

397

Soon it will become clear that when a vertex υ gets labeled, it means that there exists an /-unsaturated s-υ path Ρ and that for this path P, e(P) = Δ„. The first phase begins by labeling the source s as ( - , °°); here the value of d is irrelevant. The labeling of the vertices then proceeds according to the following rules: Suppose a vertex u is labeled and a vertex υ is unlabeled. Let e be an edge connecting u and v. s

Forward Labeling If e = («, v), then forward labeling of υ from « along e is possible if c ( e ) > / ( e ) . If such a labeling is done, then ν gets the label ( M , Δ„), where +

A„ = m i n { A „ , c ( e ) - / ( e ) } . Backward Labeling If e = (v, u), then backward labeling of υ from u along e is possible if /(e) > 0. If such a labeling is done, then ν gets the label ( ι Γ , Δ„), where Δ„ = min{A , /(e)} . u

In the first phase a vertex gets labeled at most once. This phase terminates when either (1) the vertex ί receives a label or (2) with t unlabeled, no more vertices can be labeled. If t receives a label in the first phase, then it follows from the rules for labeling that there exists an /-augmenting path Ρ and that e(P) = Δ,. In the second phase the path Ρ is traced back using the d symbols, and the revised flow / based on Ρ is also determined. Phase 1 is then repeated using the new flow / . If phase 1 terminates without labeling t, then it means that there is no /-augmenting path, and hence the present flow / is maximum. A description of the Ford-Fulkerson's algorithm is now presented. v

Algorithm 12.1. Maximum Flow in a Transport Network (Ford and Fulkerson) 51. Select any flow / in the given transport network. We may select /(e) = 0 for every edge e in N. 52. (Phase 1 begins.) Label s as ( - , ) . 53. If there exists an unlabeled vertex that can be labeled through either forward labeling or backward labeling, then select one such vertex v, label it, and then go to S4. Otherwise go to S7. 54. If υ = t, then go to S5. (Phase 1 ends.) Otherwise go to S3. 55. (Phase 2 begins.) Let the label of υ be (
+

v

u

398

FLOWS IN NETWORKS

56. If u = s, erase all the labels (phase 2 ends) and go to S2. Otherwise set v = u and go to S5. 57. (The present flow / is maximum.) HALT. ■ We now illustrate this algorithm. Consider the transport network N shown in Fig. 12.3. In this figure, next to each edge e we have shown c(e) and /(e) in this order. We take as an initial flow /(e) = 0 for all the edges in N. Starting with the label ( - , °°) for the source s, we label (S3 of Algorithm 12.1) the vertices a, b, c, d, and t in this order. The resulting labels are a: (s+, 3), b: (a + ,3), c: (a+, 3), d: (c + ,2), t:

(b\2).

Phase 1 now terminates because vertex / has just been labeled. In phase 2 we determine an augmenting path P and the revised flow /, as follows (S5 and S6). The first symbol in the label of t is b+. This means that in the path P vertex b precedes /. The first symbol in the label of b indicates that a precedes b in P. Similarly we see that Ä precedes a in P. Thus

All the edges in P are forward edges. So to obtain the revised flow /, we

<«+.3> (Λ2Ι Figure 12.3. Network for illustrating Ford-Fulkerson's labeling algorithm.

FORD-FULKERSON LABELING ALGORITHM

399

increase the flows in all the edges of Ρ by Δ, = 2. The flow /, has a value equal to 2, and it is shown in Fig. 12.4a. We now erase the labels of all the vertices. Starting with / , we then obtain a new set of labels as shown in Fig. 12.4a. An augmenting path with respect to /, consists of the vertices s, c, d, t in this order. All the edges in this path Ρ are forward edges. The flows in these edges are increased by 2. The resulting flow f is shown in Fig. 12.4i>. 2

3,2 5, 0

3, 0

<<Λ

4,0

2)

6,0

2,0 c

d

4)

(c . +

(a)

4,2 1.C\

3,2

2,2

'5,0

3,0 \ l , 0

S

K°°) 4,2

2,2

, 1) a

4, 1

i 1, 1

3,2

. 2, 2

5,0

3,0 V '

S 4,3

2,2

1

6,3

(c)

Figure 12.4. Illustration of Ford-Fulkerson's labeling algorithm.

400

FLOWS IN NETWORKS

A new set of labels based on f is shown in Fig. 12.46. An augmenting path with respect to f consists of the vertices s, c, b, a, d, t. The edge connecting a and 6 is a backward edge in this path. All the other edges of this path are forward edges. We now increase the flows in the forward edges by 1 and decrease the flow in the backward edge by 1. The resulting flow / is shown in Fig. 12.4c. Starting with / we then proceed to label the vertices. This process terminates as in Fig. 12.4c without labeling vertex t. Thus there is no augmenting path relative to the flow / . So / is a maximum flow. Let 5 denote the set of labeled vertices in Fig. 12.4c. Thus S = {s, a, 6, c}. Then it is clear from the proof of Theorem 12.2 that the cut (S, S) is a minimum cut and val(/ ) = capacity of (S, S). 2

2

3

3

3

3

3

12.4 EDMONDS AND KARP MODIFICATION OF THE LABELING ALGORITHM

In the Ford-Fulkerson algorithm, which we described in the previous subsection, vertices can be labeled in any order. In other words selection of an augmenting path (when it exists) can be made in any arbitrary manner. We now illustrate, with an example, a problem to which this arbitrariness might lead. Consider the transport network TV shown in Fig. 12.5. Suppose we start the labeling algorithm with the zero flow and alternately use the paths P,: s, a, 6, t and P : s, 6, a, t as augmenting paths. At each step the flow value increases by exactly 1, and the maximum flow of 2M is achieved after 2M augmentation steps. Thus the number of computational steps used in this case is not bounded by a function of the number of vertices and the number of edges in N. This number is in fact a function of the capacity M, which can be arbitrarily large. 2

b

a

Figure 12.5

EDMONDS AND KARP MODIFICATION OF THE LABELING ALGORITHM

401

Furthermore, Ford and Fulkerson [12.3] showed that their algorithm may fail if the capacities are irrational numbers. They gave an example where the flow converges in infinitely many steps to one fourth of the value of a maximum flow. To avoid this problem, Edmonds and Karp [12.5] have suggested a refinement to the labeling algorithm: At each step augment the flow along a shortest path. Here by a shortest path we refer to a path having the smallest number of edges. It can be easily seen that a shortest augmenting path will be selected if in the labeling procedure we scan on a "first-labeled firstscanned" basis; that is, if vertex υ has been labeled before vertex u, then scan υ before u. Here to scan a labeled vertex υ means to label (whenever possible) all the unlabeled vertices adjacent to v. Edmonds and Karp have also shown that this refinement will guarantee that the number of computational steps required to implement the labeling algorithm will be independent of the capacities. We now proceed to develop this result. Consider a flow / in a transport network N. Let

Ρ:

Ο

Ο

Ο ···

S = Μ

Μ,

«

0

Ο

Ο···

Μ,-Ι

2

Ο u

Ο

k-\

k

U

=

t

be an augmenting path. Recall that _ f c(e,) - f(e,),

€

if e, is a forward edge ; 'f , is backward edge .

' I f( i). _

e

e

a

Further e(/») = min

{e,(P)}.

Thus e(F) = e,(P), for some i. The corresponding e, is then called a bottleneck. Assume that the labeling algorithm starts with an initial flow /„ and constructs a sequence of flows / , , f , / ,... . Note that when an edge e is a bottleneck in the forward direction in an augmenting path, then during the augmentation it gets saturated. Also, if e is a bottleneck in the backward direction, then during the augmentation the flow through e is reduced to zero. This observation leads to the following result. 2

3

Lemma 12.1. If k < ρ and e is a bottleneck in the forward (backward) direction both in the augmenting path that changes f to f and in the augmenting path that changes f to f , then there exists an /, k
p

l+1

p+l

k+i

402

FLOWS IN NETWORKS

Let λ'(«, V) denote the length of a shortest /-unsaturated path from u to v. Here, of course, the path need not be directed. Further note that an edge e is used as a forward edge in the path only if it is not saturated, that is, fi(e) 0 . Lemma 12.2. For every vertex υ and every k = 0, 1 , 2 . . . , X (s,v)<X \s,v),

(12.11)

X (v,t)<X \v,t).

(12.12)

k

k+

k

k+

Proof. We shall prove (12.11), and the proof of (12.12) will follow in a similar manner. Suppose there is no f -unsaturated path from s to v. Then λ* (ί, ν) is assumed to be infinite and (12.11) follows trivially. So assume that + 1

k+l

Ρ:

e,

Ο S = U

Ο

e

2

Ο

Μ,

0

U

e

3

Ο ··· Μ

2

Ο p-\

U 3

e

p

Ο "ρ

=

υ

is a shortest f -unsaturated path from s to v. Suppose that e, is used as a forward edge in P. Then f ^e )
k+

k

l

l

k

i

l

t

k

«,_,) + 1 ,

X (s,u,)sX {s, k

k

k

+ l

(12.13)

because a shortest f -unsaturated path from s to u,_, followed by e, in the forward direction is an f -unsaturated path from s to «,. Clearly, in the latter case k

k

Α*(ί,Μ _ ) = Α*(ϊ,ΐί ) + 1. ι

Ι

(12.14)

1

In either case (12.13) holds. Similarly we can show that (12.13) holds when e, is used as a backward edge in the path P. Now summing (12.13) for i = 1, 2 , . . . , ρ and noting that X (s, u ) = 0, we get k

0

A*(s, u )

Thus (12.11) follows.

=

X \s,v). k+

•

Lemma 12.3. If the "first-labeled first-scanned" principle is used and if k < I

EDMONDS AND KARP MODIFICATION OF THE LABELING ALGORITHM

403

and e is used as a forward (backward) edge in the augmenting path that changes f to f and as a backward (forward) edge in the augmenting path that changes /, to then k

k+]

X'(s, ή s A*(s, i) + 2 . Proof. Assume that e is directed from u to v. Then A * ( i , v) = \ (s, u) + 1 ,

(12.15)

k

because e is used as a forward edge while augmenting f . Further, k

A'(s, t) = \'(s, v) + l + \'(u, t),

(12.16)

because e is used as a backward edge while augmenting But A'(s, υ) a A * ( j , υ) and A ' ( M , t)>\ (u,

ή .

k

(12.17)

Now we get from (12.15), (12.16), and (12.17) λ'(ί, t) a

A*(J,

u) + \ (u, 0 + 2 k

A * ( i , f) + 2 .

=

•

Theorem 12.4 (Edmonds and Karp). If, in Ford-Fulkerson's labeling algorithm, each flow augmentation is done along a shortest augmenting path, then a maximum flow can be obtained after no more than m(n + 2 ) / 2 augmentations, where m is the number of edges and n is the number of vertices in the transport network. Proof. Consider any edge e directed from « t o u . Consider the sequence of flows / * , / * , . . . , where Λ, < k < ..., such that e is used as a forward edge while augmenting f and is a bottleneck. By Lemma 12.1 there exists a sequence l , l ,..., such that 2

k

x

2

*,
k < l < Jfc < · · • , 2

2

3

and e is used as a backward edge while augmenting / , . By Lemma 12.3 λ*·(Μ) + 2 ^ λ ' ' ( ί , 0

404

FLOWS IN NETWORKS

and A''(s,0 + 2 ^ A * ' ( s , ί ) · +1

So λ*·0, r) + 4 ( / - l ) < A * > ( i , t). Since \ '(s, k

r) < η - 1

and A*'(*,0 1. 2

we get 1+ 4(/-1)* — · Thus e can be used as a bottleneck in the forward direction at most (n +2)14 times. Similarly it can be used as a bottleneck in the backward direction at most (n + 2)14 times. Thus each edge can serve as a bottleneck at most (n + 2)/2 times. Therefore the total number of augmentations is at most m(n + 2)12. • In the proof of Theorem 12.4 no assumption has been made about the nature of the capacities, except that they are nonnegative. Thus it is clear from this theorem that if the "first-labeled first-scanned" principle is used, then Algorithm 12.1 will terminate after a finite number of augmentations for any real nonnegative capacities. Since an augmenting path can be found in O(m) steps, it follows from Theorem 12.4 that Algorithm 12.1 is of complexity O(m n). Zadeh [12.6] has characterized a class of networks for which 0(n ) augmentations are necessary when each augmentation is done along a shortest augmenting path. Thus the upper bound in Theorem 12.4 cannot be improved upon except for a linear scale factor. 2

3

12.5

DINIC MAXIMUM FLOW ALGORITHM

In this section we discuss an algorithm, due to Dinic [12.7], for the maximum flow problem. This algorithm, like Edmonds and Karp's, also

DINIC MAXIMUM FLOW ALGORITHM

405

performs augmentations along shortest paths, but achieves a complexity of 0(mn2) by finding in one application of the labeling procedure all possible shortest augmentations. An outline of Dinic's algorithm is as follows. The algorithm starts with the zero flow (/(e) = 0 for every edge e). At this point let lx be the length of a shortest augmenting path from s to t. The algorithm identifies all shortest augmenting paths of length /, and using these paths constructs a new flow /,. Let /2 denote the length of a shortest augmenting path in the network when the flow is /,. The algorithm would then identify all shortest augmenting paths of length l2, perform augmentations using these paths, and construct a new flow/2 with val(/ 2 )>val(/i). This procedure is repeated, constructing flows / 3 , / 4 , . . . , until we reach a flow fk for which no augmenting path is available (lk+1 = °°). At this point the algorithm would terminate with fk as a maximum flow. As we can see from the preceding, Dinic's algorithm involves repeated applications of two procedures: 1. Identifying shortest augmenting paths relative to each flow/,. 2. Performing augmentations. The first objective is achieved by constructing what is called a layered network relative toflow/ . The second is achieved by finding what is called a maximal flow in this layered network. These concepts are discussed next. Given a transport network N = (V, E) with a flow/. The residual network Nf = (V, Ef) relative to / is defined as follows. Nf has the same vertex set V as N. For each edge e = {u, v) in N, Nf has the following: 1. An edge e' = (u, v), if /(e) < c(e), with capacity cf(e') = c(e) - /(e). (Note: In this case e E £ is a candidate for use as a forward edge in an augmenting path relative to flow /. So such an edge e'E Ef is called a forward edge of ΝΛ.

Figure 12.6

406

FLOWS IN NETWORKS

2. An edge e' = (v, u), if / ( e ) > 0 , with capacity c (e') = /(e). (Note: In this case e £ Ε is a candidate for use as a backward edge in an augmenting path relative to flow /. So such an edge e' £ E is called a backward edge of ty.) f

f

Note that if 0 < / ( e ) < c(e), the edge (u, υ) as well as the edge (υ, u) will be present in N . Also note that the undirected path corresponding to each directed s-t path in is an /-augmenting path in N. As an example, for the transport network and the flow shown in Fig. 12.1, the corresponding residual network is shown in Fig. 12.6. The layered network with respect to flow / is defined by the following algorithm: f

Algorithm 12.2. Layered Network 51. 52. 53. 54.

Set V = {s} and ι = 0. Set T= [υ\υΦ V for and (u, v)EE for some uE V ). If T = 0, the present flow / is maximum. HALT. If t Ε Τ, then set / = i + 1, V, = {/} and E, = {(u, ν) Ε E \u £ V and vEV ). Include all the edges in E, in the layered network. HALT. 55. Set V , = T. Set E, = {(«, υ) Ε E \u Ε V and ν Ε V } and include the edges in E, in the layered network. 56. Set i = / + l and go to S2. 0

t

f

t

f

t

l+i

1+

f

1+ 1

(

The sets Vj's defined in Algorithm 12.2 are called layers of the network and / is referred to as the length of the layered network. The capacity of an edge e in the layered network is the same as its capacity c (e) in N . The algorithm described for constructing the layered network is essentially a breadth-first search (see Exercise 11.12 for a definition of breadth-first search) of N with the source s as the start vertex. Also the set Vj contains exactly those vertices that are at distance i (in N ) from s (see Exercise 11.13). Note that here distance from s to ν refers to the length of an s-v path that has the smallest number of edges. Since / £ VJ, the sink is at distance / from s. Thus each directed s-t path in the layered network corresponds to a shortest /-augmenting path. Also, every shortest /augmenting path corresponds to a directed s-t path in the layered network. Thus the layered network serves the important role of identifying all shortest /-augmenting paths. Though we have used the residual network N to describe construction of the layered network, in practice, this is done directly from Ν and the flow/. Since \E \ ^ 2 | £ | , and breadth-first search of a directed graph examines each edge exactly once, it follows that the complexity of constructing a layered network is 0(m), where m is the number of edges in TV. We next draw attention to the following important properties of a layered network: f

f

f

f

f

f

DINIC MAXIMUM FLOW ALGORITHM

1. 2. 3. 4.

407

There is no edge («, v) with u G V), vEVj and j> i + 1. There is no edge (u, v) with w G V„ vEVj and / ^ i. AH edges in £, are directed from V, to V,_,. The layered network is an acyclic directed network.

A flow g in a layered network is a maximal flow if for each directed s-t path P : s = v , u , , . . . , v, = t in the layered network there exists at least one edge e,. = (v _ , υ,) on this path such that g(e ) = c^e,). Note that a maximal flow may not be a maximum flow. However, we can see that once a maximal flow has been constructed, no unsaturated s-t path of length / will be available for further flow augmentation. For this reason a maximal flow is also called a blocking flow. Given a blocking flow g, a new flow / ' of value v a l ( / ' ) > v a l ( / ) can easily be constructed as follows: 0

i

(

l

1. If (u, v), with u Ε K,_, and ν Ε V is a forward edge, then / ' ( « , υ) = f(u, v) + g(u, v). 2. If (u, v), with u G Vj., and υ Ε V, is a backward edge, then / ' ( u , M ) = f(v, u)-g(u, v). 3. For all other edges (u, v) of N, / ' ( u , i>) = / ( u , u). n

In the following, by phase we refer to that part of Dinic's algorithm that starts with a flow / , constructs the layered network relative to / , finds a maximal flow g in it and augments the flow in the original network to a flow / ' . Dinic's algorithm may consist of several phases. We now show that the number of these phases is bounded by n, the number of vertices in TV. Let LN(j) denote the layered network constructed in phase i and /, denote the length of LN(i'). Lemma 12.4. Consider a directed s-t path P: s = v , y , , . . . , υ, = / in LN(/ + 1). Let V, denote the ith layer of LN(i). If every vertex of Ρ appears in LN(i), then for all a = 0, 1 , 2 , . . . , υ Ε V implies a > b. 0

α

b

Proof. The result is clearly true for a = 0. Suppose that it is true for v , υ , , . . . , v , and suppose that v EV . We need to show that a + 1 s: c. If not, then c > a + l. Since by induction v was in layer V with a s b, it follows that the edge e* = (v , v ) was not in LN(i) since edges in a layered network connect vertices in consecutive layers. Hence the flow / in e* would not have been altered by the augmenting procedure of phase i. But e* is in LN(i + 1). Since at the beginning of phase i + 1, e* was available for augmentation, and the flow in e* was not affected by phase i, e* must have been available for inclusion in phase i. This is a contradiction and the lemma follows. • 0

a

a+l

c

a

a

a+i

b

408

FLOWS IN NETWORKS

Theorem 12.5. l

>/,.

i+x

Proof. Consider a directed s-t path P: s = v , v ,..., 0

x

υ,

= t in LN(i + 1).

Suppose that every vertex of Ρ appears in LN(i).

Case 1

Since sink l = D , 6 V , we have, by the previous lemma, l >l If Ί+ι h the entire path Ρ is in LN(i) as well as in LN(i + 1), and so none of the edges of Ρ were saturated before as well as after the construction of a maximal flow in LN(i). This is not possible since a maximal flow in LN(i) saturates at least one edge in every s-t path. So > /,. t i

l

l+x

r

=

Suppose that not all of the vertices of Ρ appear in LN(/).

Case 2

Let e* = (υ , υ ) be the first edge of Ρ such that for some b, v G V but v is not in LN(i). Thus flow in e* was not affected by augmentations during phase i. Since e* was available for inclusion in LN(i + 1 ) at the beginning of phase (i + 1), it must have been available for inclusion in LN(i'), but it was not included. The only possibility is that v entered LN(i') in the layer υ,, which contains the sink, but this is a special layer and contains only t. (Note: υ Φ t because v is not in LN(/).) So b + 1 = /,. By the previous lemma a2b. Thus / > β + 1 > 6 + 1 = /„ completing the proof for this case. • α

α+χ

a

b

a+x

a+x

α+χ

a+x

ι + 1

Theorem 12.5 means that the length of the layered network increases from phase to phase. Since the length of a path is bounded by n, it follows that the number of phases in Dinic's algorithm is also bounded by n. Using a backtracking algorithm, Dinic has shown that a maximal flow can be obtained in 0(mn) time. Since a layered network can be constructed in 0(m) time, it follows that the complexity of a phase is 0(mn). Since there are at most η phases, we have the following. Theorem 12.6. Dinic's algorithm is of complexity 0(mn ). 2

•

12.6 MAXIMAL FLOW IN A LAYERED NETWORK: THE MPM ALGORITHM

In this section we present yet one more refinement of the maximum flow algorithm. This comes from a very simple, elegant, and ingenious algorithm of Malhotra, Pramodh-Kumar, and Maheswari [12.8] (MPM) to construct in 0(n ) time a maximal flow in a layered network. Consider a layered network with c (e) denoting the capacity of edge e in the network. The in-potential of a vertex υ is the sum of the capacities of all the edges directed into v. The out-potential of ν is the sum of the capacities 2

f

MAXIMAL FLOW IN A LAYERED NETWORK: THE MPM ALGORITHM

409

of all the edges directed out of v. The potential of ν is the smallest of these two. A vertex with minimum potential is called a reference. In the following algorithm a pass refers to the sequence of steps in which a reference vertex in the current network is identified, a flow of value equal to the potential of the reference vertex r is pushed form ί to ί through the current network and a new layered network is constructed (S2 to S4 in the algorithm.) Algorithm 12.3. Maximal Flow in a Layered Network (Malhotra, Pramodh-Kumar, and Maheswari)

50. Given a layered network, a maximal flow g is required. 5 1 . Set g(e) = 0 for every edge e in the network. Compute the potential of every vertex. Pick a vertex of the smallest potential, say P*. Let this vertex be denoted by r. 52. (Beginning of a pass. Push out P* units of flow from the reference r to the sink t.) Let r be in layer /. Push P* units of flow through the outgoing edges at r, saturating them one at a time so that at most one outgoing edge has flow added to it but remains unsaturated. (Note: For pushing the flow the outgoing edges may be selected in any arbitrary order.) Every time the flow in an edge is increased, decrease its capacity by the same amount. (The edge flows and the capacities after each push reflect the latest augmentation of flow in the layered network.) For each vertex υ' in layer (i +1) do the following. Let h(v') be the total flow pushed into υ' from the previous layer in the current pass. Now push h(v') units of flow through the outgoing edges at υ', saturating them one at a time so that at most one outgoing edge has flow added to it but remains unsaturated. Every time the flow in an edge is increased, decrease the edge capacity by the same amount. Repeat the above for every vertex in the layers i + 2, i + 3 , . . . , until the sink t is reached. (Note: The layers should be chosen in the increasing order of their numbers. We would never find a vertex of insufficient capacity to handle the flow pushed into it because the reference r is a vertex of smallest potential). 53. (Push backward P* units of flow from the reference r to the source s.) Push backward P* units of flow through the incoming edges at r, saturating them one at a time so that at most one incoming edge has flow added to it but remains unsaturated. Every time the flow in an edge is increased, decrease the edge capacity by the same amount. For each vertex u' in layer i - 1 do the following: Let fc(u') denote the total flow pushed out of u' to vertices in the next layer in the current pass. Now push backward k(u') units of flow through the incoming edges at u', saturating them one at a time so that at most one incoming edge has flow added to it but remains unsaturated.

FLOWS IN NETWORKS

410

Every time the flow in an edge is increased, decrease the edge capacity by the same amount. Repeat the above for every vertex in the layers i - 2, i - 3 , . . . , until the source s is reached. 54. (Prune.) Remove from the layered network all edges with zero capacities. Remove every vertex that has had all of its incoming edges or all of its outgoing edges removed. (Note: Vertex r will be deleted at this point.) Continue this pruning process until only vertices that have nonzero potentials remain. (End of the current pass.) If no such vertex is present. HALT. (A maximal flow has been found.) 55. Select a vertex of smallest potential in the pruned network. With r denoting this vertex and P* denoting its potential, go to S2 (in fact repeat S2 to S4). • Since at least one vertex is removed at each pruning stage, the MPM algorithm will terminate. Since at termination every s-t path in the layered network has at least one edge e with g(e) = c (e) the algorithm terminates with a maximal flow. If pushing flow along an edge saturates the edge, then the push is called a saturating push; otherwise it is a nonsaturating push. Clearly, the MPM algorithm performs at most m saturating pushes because once an edge gets saturated, no more flow is pushed through that edge. During each execution of S2 to S4, at most η nonsaturating pushes (one per vertex) are performed. Since these steps are repeated at most η times, the algorithm performs at most n nonsaturating pushes. Thus at most (m + n ) push operations are performed. Since all other operations can be implemented in 0(m) time, it follows that the MPM algorithm has complexity 0(n ). Thus we have the following complexity result for Dinic's algorithm when the MPM algorithm is used to find a maximal flow. f

2

2

2

Theorem 12.7. The complexity of the Dinic-MPM algorithm for the maximal flow problem is 0(n ) where η is the number of vertices in the transport network. • Prior to the MPM algorithm, Karzanov [12.9] and Cherkasky [12.10] had presented algorithms of complexities 0(n ) and 0(n m ), respectively. The other algorithms based on Dinic's method include Galil [12.11], Gaul and Naamad [12.12], Shiloach [12.13], Sleator [12.14], Sleator and Tarjan [12.15], Shiloach and Vishkin [12.16], Gabow [12.17], and Tarjan [12.18]. Some of these algorithms are fast on sparse graphs, while the others are fast on dense graphs. Algorithms due to Shiloach and Vishkin [12.16] and Tarjan [12.18] have both complexity Q(n ). Galil's algorithm [12.11] has 3

3

2

V2

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

411

complexity 0(n m ). Algorithms due to Galil and Naamad [12.12] and Sleator and Tarjan [12.15] have complexities 0(mn(log n) ) and 0(mn log «), respectively. Gabow's algorithm [12.17], based on scaling, is of complexity 0(mn log U), where U is the maximum of edge capacities assumed to be integers. Among the algorithms based on Dinic's method, the only parallel algorithm is that of Shiloach and Vishkin, which has a parallel running time of 0(/i log n) (improved later to 0(n )). 5l3

2li

2

2

12.7

2

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

Given a transport network Ν = (V, Ε) with η vertices and m edges, it is clear from our discussions thus far that an assignment / of nonnegative real numbers to the edges of Ν is a maximum flow if and only if / satisfies the following: 1. f(e)s c(e), for every e £ £ . 2. Σ„ /(u, υ) - Σ f(v, u) = 0, for all u<EV. 3. There is no s-t /-augmenting path. υ

All the algorithms (Sections 12.3 to 12.6) based on Ford-Fulkerson's work start with an assignment / t h a t satisfies (1) and (2) and terminate with an / satisfying (3), always maintaining (1) and (2) at each step of the algorithm. In contrast, we develop in this section, an algorithm that starts with an assignment / that satisfies (1) and (3) and terminates with a n / t h a t satisfies (2) always maintaining (1) and (3) at each step in the algorithm. This algorithm, due to Goldberg [12.19] and Goldberg and Tarjan [12.20], uses the idea of a preflow originally introduced by Karzanov. Given a transport network N = (V, E) with η vertices and m edges, a pseudo-flow f is an assignment of nonnegative real numbers to the edges of TV such that for ( / , / ) £ Ε .

f(i,j)^c(i,j) A pseudo-flow / is a preflow if e(y)

=

Σ f(u, υ) - Σ f{v, « ) ^ 0 U

for every ν Φ s, t.

u

And e(v) is refined to as excess at vertex v. Clearly a preflow / is a flow if e(v) = 0 for every νΦχ, t. Given a preflow / , let N = (V, E ) denote the residual network with respect to /. Recall that each edge (u, υ)Ε Ε induces an edge (u, υ) ε E , if / ( u , v) < C(M, V), and an edge (v, u) ε E if / ( u , υ) > 0. Edges of N are all called residual edges. In the former case (Μ, ν) ε E is called a forward edge f

f

f

f

f

f

412

FLOWS IN NETWORKS

and in the latter case (v, u)E E is a backward edge. As in Section 12.5, c (e) will denote the capacity of the residual edge e: f

f

c(e) - / ( e ) , /(e),

if e is a forward edge ; if e is a backward edge .

A valid labeling d of TV is an assignment of nonnegative integers to the vertices of Ν such that d(s) = n, d(t) = 0 and for every residual edge (v, w), d(v) £ d(w) + 1. It can be shown that if d(v) < n, then d(v) is a lower bound on the distance from ν to t in N , and if d(v) a n, then d{y) - η is a lower bound on the distance from υ to s in N . A vertex υ is active \i v^s, t, and e(i>) > 0. The maximum flow algorithm due to Goldberg [12.19] and Goldberg and Tarjan [12.20] starts with the preflow / that is equal to the capacity c(e) on every edge e directed away from s, and zero on all other edges, and with some initial valid labeling. The simplest choice of a valid initial labeling is d(s) = n, and d(v) = 0 for all υ Φ s. The algorithm then repetitively performs two basic operations, push and relabel, as described in the following: f

f

Push (v, w) Applicability, ν is active, (v, w) Ε E and d(v) = d{w) + 1. Action. Set δ = min{e(u), c {v, w)} and do the following: 1. Increase f{v, w) by δ if (v, w) is a forward edge; otherwise decrease f(w, υ) by δ. 2. Decrease e{v) by δ and increase 8{w) by δ. (Note: δ > 0 because both e(v) and c (v, w) are positive.) • f

f

f

Relabel (v) Applicability, ν is active, and for every (v, w) Ε E , d(v) s d(w). Action. Set d{v) = m i n , {d(w) + 1}. • f

(0iM

)e£

A push from ν to w is a saturating push if c (v, w) = 0 after the push; otherwise it is a nonsaturating push. Note that relabeling of υ sets the label of ν to the largest value allowed by the valid labeling constraints. Also, immediately after relabeling, a push operation will be applicable along at least one edge (v,w)E E. f

f

Lemma 12.5. Given a preflow / and a valid labeling d, the following hold true: 1. If ν is any active vertex, then either a push or a relabel operation is applicable to v. 2. The labeling d remains valid after each basic operation (push or relabel). 3. / remains a preflow after each basic operation.

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

413

Proof 1. By definition d(v) < d(w) + 1 for every (i>, w) e E . If the push operation is not applicable to υ, then d(v) < d(w) + 1 for every residual edge (v, w). (Note that at least one residual edge (υ, w) will be there because υ is active.) Since the labels are integers, d(v) ^ d(w) for all these residual edges, and so a relabel operation is applicable to v. 2. As we pointed out before, a valid labeling d remains valid after a relabel operation. Consider then a push operation along an edge (u, w)E E . This may add edge (w, v) to N and may delete (υ, w) from Nf. Since d{w) = d(v) - 1, we have d{w) ^ d(v), and so addition of (w, v) to Nf does not affect the valid labeling constraint. Deletion of (i>, w) from Nf removes the corresponding constraint and so leaves the labeling valid. 3. A relabel operation does not modify the preflow /. Consider then a push operation along the residual edge (u, w). This decreases e(v) by δ and increases e(w) by the same amount. But 0 < δ < e(v). So both e(v) and e(w) remain nonnegative after the push operation and the result follows. • f

f

f

Next we present the following generic maximum flow algorithm due to Goldberg and Tarjan.

Algorithm 12.4. Generic Maximum Flow Algorithm (Goldberg and Tarjan)

50. Given a transport network N = (V, E), a maximum flow in Ν is required. 5 1 . (Initialize preflow / and valid labeling d) Set (i) f(s, u) = c(s, u), for every (s, u) e E. (ii) f(v, w) = 0, for every (u, w)EE with υ Φ s. (iii) d(s) = n, and d{v) = 0 for all υ Φ s. 52. If there is no active vertex, HALT, ( / i s a maximum flow.) 53. Select an active vertex and apply a basic operation. Go to S2. • Note that in Algorithm 12.4 the basic operations can be applied in any order. Lemma 12.5 guarantees that the algorithm maintains the invariants that d is a valid labeling and / is a preflow. Thus this lemma also guarantees that S3 of the Algorithm 12.4 will be executed as long as there is an active vertex. So, at termination the preflow / i s a flow. Now we need to prove that the algorithm terminates in finite time, and that at termination, the flow is indeed a maximum flow. First we take up the question of finiteness of the algorithm.

FLOWS IN NETWORKS

414

Lemma 12.6. If / is a preflow and υ is an active vertex, then, in N , the source s is reachable from v. f

Proof. Let S denote the set of vertices reachable from ν by directed paths in N . Suppose that s0S. It is clear that there is no edge ( / , e E with i £_S and / G 5. This means that for every (/,;') ε Ε with i ε S and / ε S, f(i, j) = c(i, / ) , and that for everyji, / ) ε Ε with i ε S and ; ε S, f(i, j) = 0. In other words f(S, S) > 0 and f(S, S) = 0. So f

f

2e(i)=f(S,

5)-/(S,5)<0

Since / is a preflow,

2e(i)>0. So E , e ( i ) = 0. Since vCS and > 0 , it follows that e(J)<0 i ε 5, contradicting that / is a preflow. • es

for some

It is easy to see that the relabel operation applied to a vertex υ increases d(v). So for any vertex v, the label d(v) never decreases. In fact we show in the following lemma that the labels stay finite during an execution of the algorithm. Lemma 12.7. At any time during the execution of the generic maximum flow algorithm, d(v)s2n - 1 for all vEV. Proof. Clearly the lemma is trivial for s and t. Since the algorithm changes the labels of active vertices only, the result will follow if we show that for any active vertex υ, d(v)^2n -1. By Lemma 12.6, there exists in Tv^ a directed path Ρ from υ to s. Let Ρ: ν = υ , υ , υ ,..., ν, = s. Since each edge (ν,, v ) on this path is a residual edge and the labeling is a valid one, it follows that d(v,)^d(v ) + 1 . Therefore, we have d(u) = d(i> ) — d(v ) + l
χ

2

i+x

l+x

0

l

Since the relabel operation applies only to vertices v¥=s, t and for all ν ε V, d(v) never decreases and remains nonnegative during an execution of the algorithm, we have the following bound on the total number of relabel operations performed by the generic maximum flow algorithm. Lemma 12.8. The number of relabel operations performed by the generic maximum flow algorithm is at most 2n - 1 per vertex and at most (2n - 1) (n - 2) < 2 n overall. • 2

In the following two lemmas we derive bounds on the total numbers of saturating and nonsaturating push operations performed by the generic maximum flow algorithm.

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

415

Lemma 12.9. The total number of saturating push operations performed by the generic maximum flow algorithm is at most 2mn. Proof. Consider an edge (i>, w) Ε Ε. Suppose a saturating push occurs along the edge (v, w) Ε E . In order for another push to occur along this edge, the algorithm must first push flow along the edge (w, υ) E E . This cannot happen until d(w) increases by at least 2. Thus between any two saturating pushes along the edge (v, w)E E , d(w) must increase by at least 2. Similarly d(v) must increase by at least 2 between any two saturating pushes along (w, ν) E E . Since, d(v) + d(w) > 1 when the first push occurs, and by Lemma 12.8, d(v) + d{w) < An - 3 when the last push occurs, it follows that the total number of saturating pushes corresponding to each (v, w) Ε Ε is at most 2n - 1 so that the total number of such pushes over all edges in TV is at most (2n - l ) m < 2nm. • f

f

f

f

Lemma 12.10. The total number of nonsaturating push operations performed by the generic maximum flow algorithm is at most 4mn . 2

Proof. Let Φ=

Σ

d(v).

^active

First note that Φ is zero immediately after initialization; Φ is always nonnegative, and Φ is zero at termination. So the total decrease in Φ is equal to the total increase in Φ. Now we have the following important observations: 1. A saturating push operation may initiate a relabel operation and so may cause Φ to increase by at most 2n - 1. So, by Lemma 12.9, the total increase in Φ due to saturating push operations is at most 2mn (2n-l). 2. By Lemma 12.8, the total increase in Φ due to all the relabel operations is at most (2M - 1) (n - 2). 3. A nonsaturating push operation along (v, w) E E causes Φ to decrease by at least one because this makes ν inactive, and d(w) = d(v) - 1. (Note: When υ becomes inactive, w may become active because of the push. So in such a case the decrease may be just equal to 1). f

Thus from observations 1 and 2 the total increase in Φ is at most (2n - 1) (n - 2) + 2mn (2n - 1) s 4mn . So the total decrease in Φ is at most 4mn . The lemma follows, since by observation 3 the total number of nonsaturating push operations is no more than the total decrease in Φ. • 2

Thus we have proved the following.

2

416

FLOWS IN NETWORKS

Theorem 12.8. The generic maximum flow algorithm terminates after 0(mn ) basic operations. • 2

Having proved that the algorithm terminates in finite time, we next prove that at termination the preflow is a maximum flow. We do this by showing that at no time during the execution of the algorithm, N contains a directed s-t path. f

Lemma 12.11. Given a preflow / and a valid labeling d, in N there is no directed s-t path. f

Proof. Assume the contrary and let P: s = v , υ , , . . . , υ, = t be a directed s-t path in N . Since the labeling d is a valid one and each edge (v„ v ) on Ρ is a residual edge, it follows that ^ d(v ) + 1, for all O s f < / . So we have d(s) = d(v )^d(t) +l
f

i+x

l+i

0

Theorem 12.9. The generic maximum flow algorithm terminates with a maximum flow. Proof. At termination, e(v) = 0 for all v¥=s, t. So the algorithm terminates with a flow / . But N has no directed s-t path. In other words there is no /-augmenting path in N. So, by Theorem 12.2, / is a maximum flow. • f

The running time of the generic maximum flow algorithm depends on the order in which the active vertices are selected for performing the basic operations. In any case it is clear that a polynomial time implementation of this algorithm is possible. We now proceed to study refined implementations of this algorithm and study their time complexities. In these implementations we associate with each vertex ν a list L(y) of all the edges incident on ν—the edges directed into as well as those directed out of v. Thus the edge (v, w) will appear in both the lists L(v) and L{w). To begin with, the first edge in L(v) is called the current edge of v. Every time we select a vertex ν for a push operation the current edge is the one that will be chosen for consideration. Note that the edge (v, w) Ε Ε induces a residual edge (υ, w)EE if f(v, w)< c(v, w), and an edge (w, υ) Ε Ε induces a residual edge (v, w)EE if f{w, v)>0. Also recall that the applicability of a push operation at vertex ν is checked with respect to the residual edges directed out of v. To describe our first refined algorithm we need to describe a new operation push/relabel (v). This operation combines the basic operations— push and relabel—using the lists L(u)'s. When applied to an active vertex v, push/relabel (v) selects the current edge of υ and checks if the corresponding residual edge (υ, w)—called the current residual edge—would permit a push operation along (v, w). If the push operation is applicable, then the f

f

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

417

push/relabel (v) operation is applied to push the excess out of υ along the current residual edge. If the push operation is not applicable, then push/ relabel (v) advances the current edge to the next edge in L(v). If the current edge is the last edge in L(v), no advancing will be possible. In such a case the first edge in L(v) will be made the current edge and the relabel operation will be applied to v. Now we have a formal description of push/relabel (v). Push /relabel (u) Applicability, υ is active. Action. Let e = (v, w) be the current edge of v. 51. If the push operation is applicable along the current edge, then apply push(i>, w) and HALT. 52. If the push operation is not applicable along the current edge and if the current edge is not the last edge in L(v), then make the next edge in L(v) the current edge and HALT. 53. If the push operation is not applicable along the current edge and if the current edge is the last edge in L(v), then make the first edge in L(v) the current edge of v, apply relabel(u) and HALT. • The following lemma shows that the push/relabel operation uses the relabel operation correctly. Lemma 12.12. The push /relabel operation does a relabeling only when the relabeling operation is applicable. Proof. Clearly, the push/relabel operation applies the relabeling operation at a vertex υ only when υ is active. Consider now a residual edge (υ, w) just before relabeling. Let e be the corresponding edge in L(v). Assume that e is directed from υ to w. That is, e = (v, w). Case 1 Suppose that when e was the current edge, a push operation was not applicable along (u, w). Thus at that time one of the following was true. (i) (v, w) was a residual edge and d(v) < d(w) + 1. (ii) (υ, w) was not a residual edge (Note: (v, w)0Ef). If (i) is true, then at that time d(v) ^ d(w), and this inequality remains true at the time of relabeling because d(w) never decreases, and d(v) has not changed since e was the current edge. If (ii) is true, then a push operation along (w, v) must have been performed to make c (v, w)>0 (i.e., to make (v, w) a residual edge). This would not be possible until d{w)> d(v). Thus in either case d[v) ^ d(w) at the time relabeling is done. f

Case 2 Suppose that when e was the current edge, a push operation was applicable along (υ, w). This means that at that time (v, w) was a residual

418

FLOWS IN NETWORKS

edge and d{v) = d(w) + 1. Now note that when push/relabel (u) chose the edge next to e in L(u), one of the following two would have occurred in an earlier step: (i) d(w) increased and a push along (v, w) was no longer applicable. (ii) The residual edge (υ, w) got saturated and it was no longer a residual edge. If (i) is true, at that time d(v) s d(w), and the inequality remains true at the time of relabel because d(v) has not changed during the scan of L(v) and d(w) never decreases. If (ii) is true, then a later push along edge (w, v) must have occurred making (v, w) a residual edge just before relabeling. When such a push occurred d(w) = d(v) + l, and so d(v)^d(w). Again this inequality remains true at the time of relabel. Thus just before the push/relabel (υ) does a relabeling, d(v)^d(w) for every residual edge (v, w) and so a relabeling is applicable to v. The proof in the case e is directed from w to υ follows in a similar manner. • Suppose we define S3' as follows: S3'. Select an active vertex υ and push/relabel (u). Then replacing, in Algorithm 12.4, S3 by S3', we get a refined algorithm whose complexity is derived next. Theorem 12.10. The refined algorithm runs in 0(mn) time plus 0 ( 1 ) time per nonsaturating push, for a total of 0(mn ) time. 2

Proof. Consider a vertex v^s, t. Recall that deg(u) denotes the degree of ν (number of edges incident on v). By Lemma 12.7, each vertex will be labeled at most 2n - 1 times. Before a relabeling is done by the push/relabel (u) operation, one full scan of the edges in the list is required. Relabeling itself requires another full scan of the list L(v). Also, one full scan of L(v) will be done after the last relabeling of v. Thus at most 4n - 1 scans of L(v) will be performed by the algorithm. But each scan of L(v) has complexity 0(deg(i>)). Thus the total time spent by push/relabel operations selecting υ is 0{n deg(u)) plus 0(1) per push. Summing over all the vertices and using Lemmas 12.9 and 12.10 we get the desired result (Note: Σ deg(u) = m). • υ

A further refinement of the basic operations is possible. This involves repeated applications of the push/relabel operation to an active vertex until e(v) = 0 or d(v) increases. This operation is called the discharge operation. Suppose we maintain the active vertices in a list Q such that vertices are added at one end and removed from the other end. This will ensure that the active vertices added to Q are processed on a "first-in-first-out" (FIFO) basis. For this reason Q is called a FIFO queue.

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

419

Maintaining the active vertices in a FIFO queue and applying the discharge operations until Q is empty, we get what is called the FIFO algorithm. A formal description of this algorithm follows. Algorithm 12.5. FIFO Maximum Flow Algorithm (Goldberg and Tarjan) 50. Given a transport network TV with source s and sink t, a maximum flow in Ν is required. 51. (Initialize the preflow/, valid labeling d, and the FIFO queue Q). Set c(u. υ), 0,

if u = s ; otherwise.

Add to Q every vertex υ adjacent to s. 52. If Q = 0 , HALT (a maximum flow has been found). 53. Remove vertex υ from the front of Q. 54. Apply push/relabel (v). If a vertex w becomes active during this push/relabel, add w to the rear of Q. 55. If e{v) > 0 and d(v) has not increased, go to S4. 56. If e{v) > 0 and d(v) has increased, add υ to the rear of Q. 57. Go to S2. • Note that the discharge operation consists of repeated applications of S4 and S5. Also, note that in Algorithm 12.5, once an active vertex is chosen, one could go on applying push/relabel until e(v) = 0, instead of putting ν in the Q (step S6) if it is still active and is relabeled. However, this will not change the complexity. To analyze the FIFO algorithm, we need the concept of a pass. The first pass consists of applying the discharge operations on the vertices added to Q during initialization. Pass (/ + 1), ι s 1 consists of applying discharge operations on vertices that are added to the queue during pass i. Lemma 12.13. The FIFO maximum flow algorithm makes at most 4 n passes.

2

Proof. Let d = {d(v)\v is active}. Consider a pass and let ν denote an active vertex with d(v) = d at the beginning of the pass. We now have the following observations: m a x

m a x

1. Suppose that no label changes during the pass. This means that every active vertex considered during the pass successfully pushed out its excess. Some of these vertices may become active during later pushes

420

FLOWS IN NETWORKS

into them during the same pass. But no further push into ν is possible during this pass because it has the highest label. So this vertex remains inactive even at the end of the pass. Thus d decreases during the pass. 2. d does not change during the pass. This means that vertex υ becomes inactive during the pass and some other vertex changes its label by at least one to restore the value of d . 3. d increases. Clearly some vertex must increase its label by at least as much as d increases. max

m a x

m a x

m a x

m a x

From observations (2) and (3) and Lemma 12.8 it follows that the number of passes during which d stays the same or increases is at most 2n . Since d = 0 at the start of the algorithm and d =Ϊ 0 throughout the algorithm, it follows that the number of passes during which d decreases is also at most 2n . So the total number of passes made by the FIFO algorithm is at most An . • m a x

2

m a x

max

m a x

2

2

Theorem 12.11. The FIFO maximum flow algorithm is of complexity 0 ( n ) . 3

Proof. It is clear that during a pass each vertex v^s, t makes at most one nonsaturating push per each vertex. So by the previous lemma the total number of nonsaturating pushes during the entire algorithm is 0 ( n ) . This combined with Theorem 12.10 gives the desired complexity result. • 3

Besides the FIFO strategy for maintaining and selecting active vertices for processing, a few other strategies have also been considered. The highest label preflow push algorithm selects an active vertex that is farthest from the sink. In other words a vertex υ with d(v) = d is selected. The LIFO (last-in-first-out) push algorithm maintains all active vertices in a stack and selects according to the last-in-first-out rule. We next analyze the highest label preflow push algorithm and establish that its complexity is 0(n Vm). We follow Cheriyan and Maheswari [12.21]. A push along residual edge (v, w) is a nonzeroing push if this is the first push along (υ, w) after a relabel of v. A phase consists of all pushes that occur between two consecutive relabel steps. The length /, is the difference between d at the start of the phase and d at the end of the phase. Thus Σ, /, is the total change in d over the entire algorithm. As in the proof of Lemma 12.13, it can be shown that Σ, /, = 0(n ). In order to determine the number of nonsaturating pushes performed by the highest label preflow push algorithm, the computation is divided into phases. Clearly at any vertex there is at most one nonsaturating push per phase. Since there are 0(n ) phases, the complexity of this algorithm is 0 ( n ) . But we can do better than this. During a phase, each nonsaturating push leaves behind a current residual edge. We can partition these (nonsaturating pushes) edges into what are m a x

2

m a x

m a x

max

2

2

3

PREFLOW PUSH ALGORITHM: GOLDBERG AND TARJAN

421

called trajectories. A trajectory from χ to y is a directed path of current residual edges such that (i) no edge on this path corresponds to a saturating push or a nonzeroing push and that (ii) the most recent push into χ is either a saturating or nonzeroing push. This means that we can associate each trajectory with a unique nonzeroing or saturating push. Also it is important to note that all the nonsaturating pushes performed by the highest label preflow push algorithm are contained in the trajectories. Though the partition of nonsaturating pushes into trajectories can be done as the algorithm progresses, the partition may not be unique. Furthermore, a trajectory may span several phases, that is, it may contain nonsaturating pushes from different phases. Let us now consider the current residual edges left behind by the nonsaturating pushes performed during a phase. These edges constitute a directed forest Τ since each vertex has exactly one out-going current residual edge. To compute the total number of nonsaturating pushes we now proceed as follows. A trajectory is short if its length is less than n/Vm; otherwise it is long. A push is a short trajectory push if it is contained in a short trajectory; otherwise it is a long trajectory push. Clearly, over the entire algorithm there are 0(n Vrh) short trajectory pushes, since each trajectory is associated with a unique nonzeroing or saturating push, and over the entire algorithm the total number of saturating pushes and nonzeroing pushes is 0{mn). The long trajectory pushes can be accounted as follows. Consider now the forest Τ of current residual edges left behind during a phase, say phase i. Since the trajectories that intersect with the edges of Τ (i.e., those trajectories that contain pushes performed during phase i) are edge-disjoint, there are at most v m such long trajectories. Thus the contribution of phase i to the number of long trajectory pushes is v m / , . Summing this over i and recalling that Σ, /, = 0(n ), we see that there are at most nVm long trajectory pushes performed over the entire algorithm. Summarizing, the nonsaturating pushes performed by the highest label preflow push algorithm is 0(n Vm). Maintaining all active vertices in a list-based bucket data structure, a highest label vertex can be picked in constant time. Thus we have the following result due to Cheriyan and Maheswari [12.21]. 2

2

2

Theorem 12.12. The highest label preflow push algorithm has complexity 0(n Vm). Μ 2

Cheriyan and Maheswari [12.21] have also shown that the LIFO algorithm has complexity 0(mn ). They have defined what is called a maximal excess preflow push algorithm and established its complexity as 0(mn ). They have also demonstrated that the time bounds obtained for all these algorithms—the FIFO, the highest label, the LIFO, and the maximal excess 2

2

422

FLOWS IN NETWORKS

algorithms—are tight. Using the dynamic tree structure [12.15], Goldberg [12.19] and Goldberg and Tarjan [12.20] improved the time bound of the preflow push algorithm to 0(mn log n /m). Subsequently, Ahuja, Orlin, and Tarjan [12.22] obtained an 0(mn log(n/m(log U) + 2)) algorithm using scaling and the dynamic tree data structure. (Recall that U refers to the maximum of edge capacities assumed to be integers.) This algorithm, which is an improved version of an earlier algorithm by Ahuja and Orlin [12.23], modifies the push(v, w) procedure (see page 412) so that δ can take on any value between 0 and D where D = min{e(u), c (v, w)}. In [12.24] Tuncel provides an alternate proof of the complexity of the highest label preflow push algorithm (Theorem 12.12). Interestingly, he shows that this complexity result would hold even if the algorithm were to send flow over arbitrary admissible edges, not just the current edge. In [12.25] Cheriyan and Hagerup have presented a randomized algorithm with expected time complexity of 0(min{mn log n, mn + n (log n) }). as improved later by Tarjan. In [12.26] Alon gives a derandomization of this algorithm, which results in an algorithm of complexity 0(min{mn log n, mn + n l o g n}). Cheriyan, Hagerup, and Melhorn present in [12.27] an "incremental" algorithm that runs in 0(/j /log n) time. What is significant is that they show that the generic algorithm (Algorithm 12.4) can be made to work "incrementally," that is, it starts with just the edges leaving the source s and as the execution progresses more edges are added gradually to the execution subnetwork. Goldberg [12.19], Goldberg and Tarjan [12.20], and Cheriyan and Maheswari [12.21] also discuss parallel as well as distributed implementations of the preflow push algorithm. Note that the FIFO algorithm lends itself to natural distributed and parallel implementations. 2

u2

f

2

2

8/3

3

12.8

MAXIMUM FLOW IN 0 - 1 NETWORKS

A transport network is a 0-1 network if the capacity c(e) = 1 for every edge e in the network. A 0-1 network is of type 1 if it has no parallel edges, and is of type 2 if, for every vertex υ in the network, either d (v) = 1 or d (v) = 1. These networks arise when network flow techniques are used to solve combinatorial problems. (See Sections 12.9 and 12.10.) In view of these applications, we study, in this section, the complexity of constructing a maximum flow in a 0-1 network. Naturally, one would expect in these cases better complexity results than in the case of general networks. This is indeed true, as we shall see soon. Our discussion here follows the work of Even and Tarjan [12.28]. Thus we analyze Dinic's algorithm for its behavior when applied on 0-1 networks. It is easy to see that any residual network derived from a 0-1 network is also a 0-1 network. Also, a maximal flow in a 0-1 layered network can be found in 0(m) time, where m is the number of edges in the network. This is m

out

MAXIMUM FLOW IN 0-1 NETWORKS

423

because all the pushes involved in the application of the MPM algorithm are of the saturating type. Thus, Dinic's algorithm is of complexity 0(mri) in the case of 0-1 networks. However, we can get better results by improving the bound on the number of phases (or layered networks) needed. Consider now a 0-1 network Ν = (V, Ε) with η vertices and m edges. For a flow / in TV, let N = (V, E ) denote the corresponding residual network. Recall that c (e) denotes the capacity of the edge e in N . Also, each edge e = (u, v) in TV with /(e) = 0 corresponds to a unique edge e' = (u, v), in N with c (e') = 1. If /(e) = 1, then e corresponds to the edge e' = (v, u) with c (e') = 1. In the following we shall denote by F the value of a maximum flow in Ν and by F' the value of a maximum flow in N . Also, F will denote the value of flow /. The following two lemmas are crucial for the development of the results of this section. f

f

f

f

f

f

f

m

m

f

Lemma 12.14. F' = F m

F.

m

Proof. Consider a set S whh s £ i and tgS. Let c((S, S ) ) denote the capacity of t h e s - r cut ( 5 , 5 ) in N, and c ((S, S)) denote the capacity of the s-t cut (S, S) in N . Consider an edge e = (u, ύ) in TV with « e S and υ G S . The contribution of e to c ((S,lS)) is one_ if /(e) = 0, and zero if /(e) = 1. Thus, the contribution of e to c ((S, S)) can be expressed as (1 - / ( e ) ) . Similarly the contribution from an edge e = (u, v) with uES and υG5 to c ({S, S) can be written as simply /(e). So, we have f

f

f

f

f

^«5,5»=

Σ

(1-/W)+

e£(S.S)

Re)

ee(S.S)

= c({S,S))-f(S,'S) =

Σ

+ f(S,S)

c((S,S))-F.

This implies that a minimum cut in N corresponds to a minimum cut in N. By the maximum flow minimum cut theorem (Theorem 12.3), the capacities of these cuts are, respectively, F' and F . Thus F' = F - F. • f

m

m

m

m

Lemma 12.15. Given a transport network N = (V, E) with zero flow on every edge, let / denote the length of the corresponding layered network. Then (i)

/ < mlF

(ii)

/<(2n/F^ ),

(iii)

/ < ( n -2IF )

m

if Ν is a 0-1 network .

,

if TV is a 0-1 network of type 1 .

/ 2

m

+ 1,

if Ν is a 0-1 network of type 2 .

(Note: F denotes the value of a maximum flow in N.) m

424

FLOWS IN NETWORKS

Proof. Recall that V, denotes the set of vertices in the i'th layer of the layered network, and E, denotes the set of edges in N directed from the vertices in V to those in V . Since/(e) = 0 for every_e e E, it follows that £, = ( 5 , 5 ) , where 5 = V U V, • · · U V,. So c ( ( 5 , 5 ) ) = \E,\. Then, by Corollary 12.1.1, we have f

f

i+l

0

(12.18)

F *\E,\ m

With these preliminaries we can now prove the results of the lemma. (i) Summing up (12.18) for every i = 1 , 2 , . . . , /, we get F · I < \E\ = m and the result follows. (ii) Since in a 0-1 network of type 1, there are no parallel edges, each E, satisfies m

(12.19)

\E,\^\V,\\V \. i+i

From (12.19) we get, for 1 < / < /, *\V,\\V,+x\-

F m

So either |V,| ^ F^ or |V; | > F ^ . Thus at least L(/ + l ) / 2 j of the K/s satisfy | ν ] . | > £ ^ . But 2

1

2

+1

2

i\V,\ = n. So we get

and the result follows. (iii) In case of a 0-1 network of type 2 a maximum flow can be expressed as the sum of flows along F internally disjoint s-t paths. Since s and t are the only two common vertices in these F paths we have F ( A - l ) s « - 2 where λ is the length of a shortest of these paths. So λ < (η - 2)/F + 1. But / =£ A. Thus, the result follows. • m

m

m

m

We are now ready to establish the complexity of Dinic's maximum flow algorithm for the different types of 0-1 networks considered in Lemma 12.15. Theorem 12.13. Given a transport network N, the complexity of Dinic's maximum flow algorithm is as follows.

MAXIMUM FLOW IN 0-1 NETWORKS

(i)

0(m ),

if TV is a 0-1 network .

(ii)

0(n m),

if Ν is a 0-1 network of type 1 .

(iii)

0(n m),

if TV is a 0-1 network of type 2 .

312

2,3

ll2

425

Proof. Consider a transport network TV with flow / of value F. Let N denote the corresponding residual network. Then, the layered network for TV with the flow / is identical to the layered network for N with zero flow everywhere. Thus, while computing (using Lemma 12.15) the bound for the length of the layered network of Ν with flow /, we can use F' = F - F in place of F . This observation plays a key role in the proofs that now follows. We prove the results by obtaining an appropriate bound on the number of phases Dinic's algorithm would require. f

f

m

m

m

(i) If F < m , the number of phases is bounded by m because by Lemma 12.15 the length of any layered network in such a case is at most m . Thus the required complexity result follows. 1/2

1/2

m

1/2

Suppose F >m . Then consider the phase during which the flow reaches F - m . If F is the value of the total flow when the 112

m

112

m

112

layered network for this phase is constructed, then F m ' . As pointed out at the start of our proof of this theorem, the length of this layered network therefore satisfies m

m

m

1 2

Thus the number of phases up to this point in Dinic's algorithm is at most m - l . The number of additional phases to terminate the algorithm is also at most m ' because the value of a maximum flow in subsequent networks is at most m " . Thus the total number of phases is at most 2 m and the result follows. 1 / 2

1 2

2

112

(ii) If F s n ' , the result follows from Lemma 12.15 since the number of phases required is bounded by n . Otherwise, consider the phase during which the flow reaches F - n . Then the value F of the flow when the layered network for this phase is constructed satisfies F < F — n . Also, F' for the corresponding residual network satisfies F' = F - F > n . As observed before, the layered network at this point is identical to the layered network constructed for N with zero flow everywhere. But N may not be a type 1 network. It may have parallel edges: if e, = (u, v) and e = (i>, u) are two edges with / ( e , ) = 0 and f(e ) = 1, then the corresponding residual 2 3

m

213

213

m

213

m

m

213

m

m

f

f

2

2

426

FLOWS IN NETWORKS

edges in N will be in parallel. But between any two vertices in N , there will be at most two edges in parallel. So as in the proof of the result (ii) of Lemma 12.15, we get f

f

^Ζΐν,,,ΙΙν,Ι

for ! = = / < / .

Thus either |V,| > (F'J2) ' or |V,_,| > (F'J2) . So, at least [1+1/ 2J Vj.'s satisfy |V,| 2= (F'J2) . Since Σ', \V,\< η we get 1 2

U2

V2

=0

and so

Since F' ^ n , we get / = 0(n ). Thus, the number of phases up to this point is 0(n ) . Also, the additional phases to completion of Dinic's algorithm is at most n since the value of a maximum flow in subsequent networks is at most n . So, the total number of the phases required is 0 ( n ) and the result follows, (iii) Given a 0-1 network of type 2 with flow / , we first claim that the corresponding residual network is also of type 2. If there is no flow through a vertex υ, then all the edges incident on ν in TV will also be present in N . On the other hand, suppose that the flow through ν is 1, and this flow enters υ via an edge e, and leaves via an edge e , then the corresponding edges in N will have directions opposite to e and e , respectively. Thus the numbers of incoming edges and outgoing edges at ν remain the same in both Ν and N . Since Ν is of type 2, so is N . 2n

2n

m

21

2 / 3

2/3

2 / 3

f

2

f

x

2

f

f

If F ^n , the required result follows from Lemma 12.15 because the number of phases is bounded by n . Otherwise proceeding along the same lines as in the proof of the result (i) of this theorem and using the result (iii) of Lemma 12.15 we can show that the total number of phases required is 0(n ). The result in this case would then follow. • v2

m

112

112

12.9

MAXIMUM MATCHING IN BIPARTITE GRAPHS

In this section we present an application of the maximum flow algorithm in solving a combinatorial problem. We show how this algorithm can be used to obtain a maximum matching in a bipartite graph. Given a bipartite graph G = (V, E) with bipartition (X, Y), let us construct a transport network N = (V*,E*) as follows: TV has vertex set

MENGERS THEOREMS AND CONNECTIVITIES

427

V* = V U {s, r). And TV has a directed edge (s, x) for each χ EX, a directed edge (y, t) for each y E Y, and a directed edge (JC, y) for each undirected edge (JC, y), χ Ε X, y Ε Y of G. Also c(x, y) = l for every edge in TV. And ί and t are, respectively, the source and the sink of N. Clearly TV is a 0-1 network of type 2. Consider now a matching Μ in G. Let us now define a flow / as follows. For each (JC, y) Ε Μ, let / ( J C , y) = l,f(s, x) = l a n d / ( y , t) = 1. Then we can see that v a l ( / ) = |M|. In other words each matching Μ in G defines a flow of value \M\. Let / be a flow in Ν of value F. Then there are exactly F edges of the form (s, JC) for which /(*, JC) = 1. For each such edge (s, x) there is exactly one vertex y Ε Y such that / ( J C , y) = 1. Since c(y, r) = 1 and (y, i) is the only outgoing edge at y, it follows that / ( y , i) = 1 and so there is no JC' Φ χ such that / ( J C ' , y) = 1. In other words the edges (JC, y) with /(JC, y) = 1 define a matching Μ of cardinality F. Summarizing, there is a one-to-one correspondence between the set of matchings in G and the set of flows in TV. Thus a maximum matching in G corresponds to a maximum flow in N. Since Ν is a 0-1 network of type 2, we get the following result from Theorem 12.13. Theorem 12.14. A maximum matching in a bipartite graph G = (V, E) with η vertices and m edges can be constructed in 0{mn ) time. • l

2

The above result was also proved in [12.29] by Hopcroft and Karp. Their algorithm can also be viewed as one of constructing a maximum flow in a 0-1 network of type 2 using Dinic's algorithm. A discussion of the theoretical basis of Hopcroft and Karp's approach may also be found in Swamy and Thulasiraman [12.30]. See also Exercise 12.10. Generalizing Hopcroft and Karp's approach, Even and Kariv [12.31] have developed an 0(n ) algorithm for determining a maximum matching in a general graph. Later Micali and Vazirani [12.32], using the same ideas, obtained a simplified algorithm of complexity 0(mVn). See also Kameda and Munro [12.33]. For a basic approach to the maximum matching problem see Edmonds [12.34]. Gabow [12.35] has given an 0(n ) implementation of Edmonds' algorithm. A discussion of this algorithm may also be found in Swamy and Thulasiraman [12.30]. For algorithms for the maximum weighted matching problem see Edmonds [12.36], Lawler [12.37], Papadimitriou and Steiglitz [12.38], Gabow [12.39], and Galil, Micali, and Gabow [12.40]. See also Tarjan [12.41] and Melhorn [12.42]. 25

3

12.10

M E N G E R S THEOREMS AND CONNECTIVITIES

In this section we prove Menger's theorems for both directed and undirected graphs (Theorems 12.16 through 12.19) using the max-flow min-cut

428

FLOWS IN NETWORKS

theorem. We may recall that we had stated earlier Menger's theorems, without proofs, for undirected graphs (see Theorems 8.9 and 8.12). In the following discussions, for the sake of generality, we shall permit the source to have nonzero in-degree and the sink to have nonzero out-degree. Our proof of Menger's theorems is based on the following result. Theorem 12.15. Let Ν be a transport network with source s and sink t and in which each edge has unit capacity. Then: 1. The value of a maximum flow in Ν is equal to the maximum number r of edge-disjoint directed s-t paths in N. 2. The capacity of a minimum cut in Ν is equal to the minimum number q of edges whose removal destroys all directed s-t paths in N. Proof 1. Let / * be a maximum flow in TV and let N* be the directed graph obtained by removing from TV all its /*-zero edges. Since the capacity of every edge is unity, it is clear that/*(e) = 1 for every edge e in TV*. Thus:

dM -

a. dN.(s) = val(/*) = d-.(t) - d;.(t). b. d„.(v) = d .(v) for all υ Φ s, t. Here άχ.(χ) and d„.(x) denote, respectively, the out-degree and the in-degree in the graph TV* of the vertex x. Therefore (Exercise 5.8) there are val(/*) edge-disjoint directed s-t paths in N* and hence also in N. Thus N

val(/*)
(12.20)

Now let { P P ,..., P ) be a collection of r edge-disjoint directed s-t paths in TV. Define a flow / such that u

2

r

1, f(e) = 0 ,

if e is in some P, ; otherwise.

Clearly, val(/) = r . Since / * is a maximum flow, we have val(/*)>r . Combining (12.20) and (12.21), we get val(/*) = r .

(12.21)

MENGERS THEOREMS AND CONNECTIVITIES

429

2. Let K* = ( 5 , 5 ) be a minimum s-t cut in N. Suppose we remove from Ν the set of edges (S, S). Then in the resulting directed graph there will be no directed s-t paths. So cap(A:*) = | ( S , S ) | > c 7 .

(12.22)

Now let Ζ denote a set of q edges whose removal destroys all directed s-t paths in N, and let S denote the set of all vertices reachable from s by a directed path containing_no edge from Z. Clearly, K= (S,S) is an s-t cut in N. Further, (S, S) C Z. So, cap(A:*)
= q.

(12.23)

Combining (12.22) and (12.23), we get cap(K*) = q . U Theorem 12.16. Let s and t be two vertices in a directed graph G. Then the maximum number of edge-disjoint directed s-t paths in G is equal to the minimum number of edges whose removal destroys all directed s-t paths in G. Proof. From G construct a transport network Ν with s as the source and t as the sink by assigning unit capacity to each edge of G. The theorem then follows from Theorem 12.15 and the max-flow-min-cut theorem. • For an undirected graph G, let D(G) denote the directed graph obtained by replacing each edge e of G by a pair of oppositely oriented edges having the same end vertices as e. It can be easily shown that 1. There exists a one-to-one correspondence between the paths in G and the directed paths in D(G). 2. For any two vertices s and t, the minimum number of edges whose removal destroys all s-t paths in G is equal to the minimum number of edges whose removal destroys all directed s-t paths in D(G). The undirected version of Theorem 12.16 now follows from the above observations. Theorem 12.17. Let s and t be two vertices of an undirected graph G. The maximum number of edge-disjoint s-t paths in G is equal to the minimum number of edges whose removal destroys all s-t paths in G. • Vertex analogs of Theorems 12.16 and 12.17 are proved next. Let s and t be any two nonadjacent vertices in a directed graph G = (V, E). From G construct a directed graph G' as follows:

430

FLOWS IN NETWORKS

1. Split vertex vEV{s, t} into two new vertices v' and v" and connect them by a directed edge (υ', υ"). 2. Replace each edge of G having υ £ V - {s, t) as terminal vertex by a new edge having υ' as terminal vertex. 3. Replace each edge of G having υ £ V- {s, t) as initial vertex by a new edge having v" as initial vertex. A graph G and the corresponding graph G' are shown in Fig. 12.7. It is not difficult to prove the following: 1. Each directed s-t path in G' corresponds to a directed s-t path in G that is obtained by contracting all edges of the type (υ', ν"); conversely, each directed s-t path in G corresponds to a directed s-t path in G' obtained by splitting all the vertices other than s and t of the path. 2. Two directed s-t paths in G' are edge-disjoint if and only if the corresponding paths in G are vertex-disjoint. 3. The maximum number of edge-disjoint directed s-t paths in G' is equal to the maximum number of vertex-disjoint directed s-t paths in G.

w'

to"

z

ζ"

Figure 12.7. (a) Graph G. (b) Graph G'.

MENGERS THEOREMS AND CONNECTIVITIES

431

4. The minimum number of edges whose removal destroys all directed s-t paths in G' is equal to the minimum number of vertices whose removal destroys all directed s-t paths in G. From these observations we get the following vertex analog of Theorem 12.16. Theorem 12.18. Let s and t be two nonadjacent vertices in a directed graph G. Then the maximum number of vertex-disjoint directed s-t paths in G is equal to the minimum number of vertices whose removal destroys all directed s-t paths in G. • The vertex analog of Theorem 12.17 now follows. Theorem 12.19. Let s and / be two nonadjacent vertices in an undirected graph G. Then the maximum number of vertex-disjoint s-t paths in G is equal to the minimum number of vertices whose removal destroys all s-t paths in G. Proof. Apply Theorem 12.18 to the graph D(G).

•

Given an undirected graph G and two nonadjacent vertices s and t, let k(S, r) denote the minimum number of vertices whose removal disconnects s and t. The observations leading to Theorem 12.19 suggest the following algorithm for computing k(S, t). First construct the directed graph D(G) as described before. Then construct the graph D'(G) (see Fig. 12.7). Assign unit capacity to all the edges in D'(G), and in the resulting transport network Ν compute the maximum flow from s to t. k(S, t) is then equal to the value of this maximum flow. Since the network Ν is a 0-1 network of type 2 it follows from Theorem 12.13 that k(S, t) can be computed in 0(mn ). Note that the network has at most 2/i vertices and has at most (2m + n) edges, where m and η are, respectively, the numbers of vertices and edges of G. Recall that the connectivity of an η-vertex complete graph is equal to n — l. We now state and prove the correctness of a procedure to compute κ, the connectivity of a simple η-vertex undirected graph G = (V, E), which is not completely connected. We follow Even [12.43]. The procedure is as follows. Order the vertices as v , v ,..., v such that (u,, υ , ) ^ £ for some i Φ 1. Compute κ(υ , ι> ) for every vertex v, such that (v , 0^0E. Let γ be the minimum of these values. Repeat this computation with v , u , . . . , (each time updating the value of y) and terminate with v , once k exceeds the current value of γ. We now prove that this procedure terminates with y = κ. v

x

χ

;

2

n

x

2

k

3

432

FLOWS IN NETWORKS

Clearly, after the first computation of κ ( υ υ,.) for some (υ,, satisfies π

κ<γ < «- 2.

v , ) 0 E ,

γ

(12.24)

From this point on, γ can only decrease. By definition κ is the minimum number of vertices whose removal disconnects two vertices. Let R denote such a set of vertices. Since \R\ = κ, at least one of the vertices u,, v ,...,v is not in R. Let u, be one such vertex. Then removing from G the vertices in R will separate the remaining vertices into at least two sets, such that each path from a vertex of one set to a vertex of another passes through at least one vertex of R. Thus there exists a vertex υ such that K ( U , V) =£\R\ = κ. Therefore γ ^ κ. Thus γ = κ. Since each step of this procedure requires at most η maximum flow computations, each of complexity 0(n m), and at most κ + 1 steps will be performed, it follows that the complexity of this procedure is 0(κη τή). But by Theorem 8.2, κ ^ 2mln. Thus we have the following. 2

k

(

1/2

3η

Theorem 12.20. The complexity of computing the vertex connectivity of an undirected graph G is 0(n m ), where m and η are, respectively, the numbers of edges and vertices of G. • ll2

2

A discussion of implementations that improve on this complexity result may be found in Galil [12.44]. For early algorithms to test whether the connectivity of an undirected graph is at least k, see Kleitman [12.45], Even [12.46], and Galil [12.44]. Recently Cheriyan and Thurimella [12.47] and [12.48] and Nagamochi and Ibaraki [12.49] and [12.50] have improved upon the complexity of the Λ-vertex connectivity problem. The complexity of their algorithms is 0{k n ) for k< Vn and 0(k n ) , otherwise. Cheriyan and Thurimella also present distributed as well as parallel algorithms. Recall that in Section 11.8 we discussed an algorithm (based on DFS) to test whether a given undirected graph is 2-connected. For an O(n) algorithm to test the 3-connectivity of an η-vertex graph, see Hopcroft and Tarjan [12.51]. As can be seen from the result of Theorem 12.19 edge connectivities of an undirected graph G can also be computed using the network flow techniques. The corresponding directed graph D{G) will be of type 1. In fact, it can be easily shown that edge connectivity computation would require at most η - 1 network flow computations. Thus, from Theorem 12.13, we can show that the complexity of computing the edge connectivity of an undirected graph is 0 ( m i n { m n , n m}). Matula [12.52] has given an 0{mn) edge connectivity algorithm. He has also developed (private communication) an 0(n) algorithm that determines an edge-cut of cardinality at most 2 + ε times the cardinality of a minimum edge-cut. However, these algorithms are not based on network flow techniques. 2

2

3

3/2

5

l

5/3

NP-COMPLETENESS

433

For algorithms for computing connectivities of directed graphs see Even [12.43] and Galil [12.44]. Recently Gabow [12.53] has given an algorithm of complexity 0(km \og(n /m)) to test fc-edge connectivity of directed graphs. This algorithm obviously extends to undirected graphs too. 2

12.11

NP-COMPLETENESS

Following Edmonds [12.34], we consider an algorithm for a problem efficient if there exists a polynomial p(k) such that an instance of the problem whose input length (length of the data describing the instance) is k takes at most p(k) elementary computational steps to solve. That is, we accept an algorithm as efficient only if it is of polynomial-time complexity. Thus all the graph algorithms discussed in this book are efficient because they have complexities polynomial in m and n, namely, the parameters that reflect the length of the data representing the input graph. However, there exists a large family of problems that defy polynomial-time algorithms in spite of massive efforts to solve them efficiently. This family includes several important problems such as the travelling salesman problem, the graph coloring problem, the problem of simplifying Boolean functions, scheduling problems, and certain covering problems. Interestingly, these problems, called the NP-complete problems, are tied by a strong structural relationship. / / one can find a polynomial-time algorithm for an NP-complete problem, then polynomial-time algorithms can be found for all the NPcomplete problems. Though the mathematical machinery required to develop the theory of NP-completeness is beyond the scope of this book, we attempt in this section a brief introduction to this theory. Our presentation follows the elegant and easy-to-read and yet rigorous discussions given by Even [12.43] and Wilf [12.54]. A decision problem is one that asks only for a yes or no answer. For example, the question "Can this graph be 5-colored?" is a decision problem. Similarly, the question "Is there a tour of this graph of length less than 20 miles?" is also a decision problem. Many of the important optimization problems can be phrased as decision problems. Usually, if we find a fast algorithm for a decision problem, then we will be able to solve the corresponding original problem also efficiently. For instance, if we have a fast algorithm to solve the decision problem for graph coloring, then by repeated applications (in fact, η log η applications) of this algorithm, we can find the chromatic number of an η-vertex graph. A decision problem belongs to the class Ρ if there is a polynomial-time algorithm to solve the problem. A decision problem belongs to the class Ν Ρ (nondeterministic polynomial) if there exists a polynomial-time algorithm to check the correctness of a claimed answer. The reason for calling such a "computation" nondeterministic is that we do not claim to have a (deterministic) method to guess the correct answer. We merely say that one exists. For example, for the problem "Is it true that TV is not prime?" the

434

F L O W S IN N E T W O R K S

guess is simply a specification of a factor F. We can easily divide TV by F and check that the remainder is zero. Thus this problem is in NP. Similarly, the decision problems "Can this graph be -colored?" and "Is there a tour of this graph of length less than 20 miles?" are in NP. Summarizing, in Ρ are problems for which it is easy to find a solution, and in NP are problems for which it is easy to check a solution that may have been very tedious to find. We should emphasize that there are problems that are not even known to belong to NP, let alone to P. Given two decision problems Q and Q', Q' is quickly reducible to Q if whenever we are given an instance / ' of the problem Q' we can convert it, with only a polynomial amount of work, into an instance / of Q, in such a way that / ' and I both have the same answer (Yes or No). A decision problem is NP-hard if every problem in NP is quickly reducible to it. An NP-hard problem Q is NP-complete if Q is in NP. Thus a decision problem is NP-complete if it belongs to NP and every problem in NP is quickly reducible to it. Thus a fast algorithm for some NP-complete problem implies a fast algorithm for every problem in NP. The question now is: Is there any NP-complete problem at all? In other words, is there a single computational problem with the property that every one of the diverse problems in NP is quickly reducible to it? In 1971 Stephen Cook [12.55] proved that there is an NP-complete problem. The problem is called the satisfiability problem. Cook's work is based on the theory of Turing machines. A Turing machine is an extremely simple finite-state computer, and when it performs a computation, a unit of computational work can be very clearly and unambiguously described. What is important is the thesis (called Church's thesis) that a problem can be solved by an algorithm if and only if it can be solved by some Turing machine that halts for every input. See standard tests such as Minsky [12.56] and Hopcroft and Ullman [12.57] for a discussion of Turing machines and Church's thesis. The satisfiability problem is defined next. Given a finite set X= ( J C , , J C , . . . , JC„} °f Boolean variables. A literal is either a variable JC, or its complement x,. Thus the set of literals is L = { J C , , J C , . . . , J C „ , 3c,, xl,..., x„). A clause C is a subset of L . The satisfiability problem (SAT) is: Given a set of clauses, does there exist a set of truth values (Tor F), one for each variable, such that every clause contains at least one literal whose value is T. Cook's proof that SAT is NP-complete opened the way to demonstrate the NP-completeness of a vast number of problems. To prove the NPcompleteness of some problem X that is in NP, it is sufficient to show that SAT reduces to X. If that is true, then every problem in NP can be reduced to the problem X by first reducing it to an instance of SAT and then to an instance of X. The second problem to be proved NP-complete is 3-SAT the 3-satisfiability problem, which is the special case of SAT in that only three literals are permitted in each clause. 2

2

NP-COMPLETENESS

435

The list of NP-complete problems has grown very rapidly since Cook's work. Karp [12.58] and [12.59] demonstrated the NP-completeness of a number of combinatorial problems. Garey and Johnson [12.60] is the most complete reference on NP-completeness and is highly recommended. Other good textbooks recommended for this study include Horowitz and Sahni [12.61], Melhorn [12.42], Aho, Hopcroft, and Ullman [12.62], and Reingold, Nievergelt, and Deo [12.63]. For updates on NP-completeness see the articles titled "NP-Completeness: An Ongoing Guide" in the Journal of Algorithms. Since the concept of reducibility plays a dominant role in establishing the NP-completeness of a problem, we shall illustrate this with an example. Consider the graph G = (V, E) in Fig 12.8 and the decision problem "Can the vertices of G be 3-colored?" We now reduce this problem to an instance of SAT. We define 12 Boolean variables x (i = 1, 2, 3, 4; / = 1, 2,3) where the variable x , corresponds to the assertion that "vertex i has been assigned color / . " The clauses are defined as follows: t ;

t

C(i) = {x ,x, ,x } lA

t2

lsi<4;

,

lt3

Γ(ί) = {*,,,, * , , } ,

ls/<4;

tf(0 = {*,.i, * . , 3 > \

ls/<4;

V(i) = {x,, ,x,, },

1 ^ 4 ;

2

2

3

D(e, j) = {x , x J, u y

u

for every e = (u, υ) e Ε, 1 =s / < 3 .

Whereas C(i) asserts that each vertex i has been assigned at least one color, the clauses T(i), U(i), and V(i) together assert that no vertex has been assigned more than one color. The clauses D(e, ;')'s guarantee that the coloring is proper (adjacent vertices have been assigned distinct colors). Thus, the graph of Fig. 12.8 is 3-colorable if and only if there exists an assignment of truth values Τ and F to the 12 Boolean variables JC, , . . . , J C 3 such that the each of the clauses contains at least one literal whose value is T. 2

4

Figure 12.8

FLOWS IN NETWORKS

436

Now, we are ready to conclude with the following quote from Lawler [12.37, page 13] attributed to Vilenkin [12.64, page 11]: A mathematician asked a physicist: "Suppose you were given an empty teakettle and an unlit gas plate. How would you bring water to boil?" "I'd fill the teakettle with water, light the gas, and set the teakettle on the plate." "Right," said the mathematician, "and now please solve another problem. Suppose the gas were lit and the teakettle were full. How would you bring the water to a boil?" "That's even simpler. I'd set the teakettle on the plate." "Wrong," exclaimed the mathematician. "The thing to do is to put out the flame and empty the teakettle. This would reduce our problem to the previous problem!" That is why, whenever one reduces a new problem to problems already solved, one says in jest that one is applying the "teakettle principle."

12.12

FURTHER READING

Standard references on network optimization include Ford and Fulkerson [12.3], Lawler [12.37], Papadimitriou and Steiglitz [12.38], Chvatal [12.65], and Rockefeller [12.66], For a guided tour of combinatorial optimization (through the travelling salesman problem) including approximation algorithms and the theory of NP-completeness, see Lawler, Lenstra, Rinnooy Kan, and Shmoys [12.67]. For recent work on minimum-cost circulation algorithms, see Goldberg [12.19] and Goldberg, Plotkin, and Tardos [12.68]. Goldberg, Tardos, and Tarjan [12.69] and Ahuja, Magnanti, and Orlin [12.70] are excellent surveys of algorithmic developments on the network flow problem. They also contain exhaustive lists of references related to this problem. The recent review by Goldberg and Gusfield [12.71] of a Russian book, Flow Algorithms by Adel'son-Vel'ski, Dinic, and Karzanov deserves special mention. This book published in 1975 has not yet been translated into English. Quoting from the review, "this book describes many major results obtained in the Soviet Union (and originally published in papers by 1976) that were independently discovered later (and in some cases much later) in the West." Of special interest to us in the context of discussions in this chapter is the fact that Podderyugin published in 1973 an 0(mn) edge connectivity algorithm. However, Matula's work [12.52] has a stronger result. Also, Karzanov, independently of Hopcroft and Karp [12.29], published in 1971 the 0(mn ) bipartite matching algorithm. Network flow techniques have been extensively used in formulating and solving several important problems. A recent application to systems level diagnosis may be found in Sullivan [12.72], xn

EXERCISES

437

In the past decade considerable progress has been made in parallel and distributed computing. Akl [12.73], Quinn [12.74], and Lakshmivarahan and Dhall [12.75] are very good references on parallel computing principles. They also treat some graph problems. Bertsekas and Tsitsiklis [12.76] give a detailed coverage of distributed algorithms for several network optimization problems. Random graphs play a dominant role in average-case and probabilistic analysis of graph algorithms. Two excellent sources of information on random graphs are Palmer [12.77] and Bollobas [12.78]. See also Bollobas [12.79].

12.13

EXERCISES

12.1

Show that if there exists no directed s-t path in a transport network N, then the value of a maximum flow and the capacity of a minimum cut are both zero.

12.2

If ( 5 , 5 ) and (Τ, T) are minimum cuts in a transport network TV, show that ( S U T, SL> T) and ( 5 Π Τ, 5 Π Τ) are also minimum cuts in N.

12.3

Show that in any transport network with integer capacities, there is a maximum flow / such that f(e) is an integer for every edge e in N.

12.4

Consider a transport network Ν in which each vertex ν Φ s, t is associated with a nonnegative integer m(v). Show how a maximum flow in which the flow into each vertex υ Φ s, t is not greater than m{v) can be found by applying the maximum flow algorithm to a modified network. (See the construction used in establishing Theorem 12.18.)

12.5

Consider a transport network TV in which a lower bound is also specified for flow in each edge. (a) Find a necessary and sufficient condition for the existence of a flow in N. (b) Modify the maximum flow algorithm to find a maximum flow in N.

12.6

Prove that in a transport network with a lower bound for flow in each edge, there exists a flow if and only if every edge e is in a directed circuit or is in a directed path from s to t or is in a directed path from t to s.

12.7

Describe a method for locating in a transport network Ν an edge e that has the property that increasing its capacity increases the maximum flow in N.

438

FLOWS IN NETWORKS

12.8

Let G = (V, E) be an acyclic digraph. We wish to find a minimum number of directed vertex-disjoint paths that cover all the vertices; that is, every vertex is on exactly one path. The paths may start anywhere and end anywhere, and their lengths are not restricted in any way. A path may be of zero length; that is, consist of one vertex. (a) Describe an algorithm for achieving this goal and make it as efficient as possible. (b) Is the condition that G is acyclic essential for the validity of your algorithm? Explain. (c) Give the best upper bound you can on the time complexity of your algorithm (Note: This exercise is from Even [12.43].)

12.9

This problem is similar to 12.8 except that the paths are not required to be vertex (or edge) disjoint. (a) Describe an algorithm for finding a minimum number of covering paths. (b) Is the condition that G is acyclic essential for the validity of your algorithm? Explain. (c) Give the best upper bound you can on the time complexity of your algorithm. (Hint: 0(\V\-\E\) is achievable.) (d) Two vertices u and ν are called concurrent if no directed path exists from Μ to υ or from υ to u. A set of concurrent vertices is such that every two in the set are concurrent. Prove that the minimum number of paths that cover the vertices of G is equal to the maximum number of concurrent vertices. (Note: This exercise is from Even [12.43].)

12.10

Prove the following results which lead to the complexity 0(n ) of Hopcroft and Karp's [12.29] maximum matching algorithm for a bipartite graph. 1. Let Μ and TV be two matchings in a graph G. If \M\ = s and \N\ = r with r>s, then ΜΦΝ contains at least r-s vertexdisjoint augmenting paths relative to M. 2. Let Μ be a matching. Let \M\ = r and suppose that the cardinality of a maximum matching is s. Then there exists an augmenting path relative to Μ of length at most 25

3. Let Μ be a matching, Ρ a shortest augmenting path relative to M, and P' an augmenting path relative to Μ Φ P. Then | P ' | 2 : |P| + | P n P ' | .

REFERENCES

439

4. Suppose that we compute, starting with a matching M = 0 , a sequence of matchings A/,, M ,..., Λ ί , , . . . , where M, = Μ, Θ Ρ, and Ρ, is a shortest augmenting path relative to M,. Then 0

+ 1

2

|Ρ,·Ι^ΙΛ

+ 1

Ι·

5. For all i and / such that |P,| = | P j , P, andP, are vertex disjoint. 6 . Let s be the cardinality of a maximum matching. The number of disjoint integers in the sequence | P | , | P j , . . . , | P , | , . . . is less than or equal to 2 [ V J J + 2 . (See Swamy and Thulasiraman 0

[12.30])

12.14

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10

REFERENCES

L. R. Ford and D. R. Fulkerson, "Maximal Flow through a Network," Canad. J. Math., Vol. 8, 399-404 (1956). P. Elias, A. Feinstein, and C. E. Shannon, "A Note on the Maximum Flow through a Network," IRE Trans. Information Theory, Vol. IT-2, 117-119 (1956). L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., 1962. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973. J. Edmonds and R. M. Karp, "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems," J. ACM, Vol. 19, 248-264 (1972). N. Zadeh, "Theoretical Efficiency of the Edmonds-Karp Algorithm for Computing Maximal Flows," J. ACM, Vol. 19, 184-192 (1972). E. A. Dinic, "Algorithm for the Solution of a Problem of Maximum Flow in a Network with Power Estimation," Soviet Math. Dokl., Vol. 11, 1277-1280 (1970). V. M. Malhotra, M. Pramodh-Kumar, and S. N. Maheswari, "An 0(V ) Algorithm for Maximum Flows in Networks," Information Processing Lett., Vol. 7, 277-278 (1978). Α. V. Karzanov, "Determining the Maximal Flow in a Network by the Method of Preflows," Soviet Math. Dokl., Vol. 15, 434-437 (1974). R. V. Cherkasky, "Algorithm of Construction of Maximal Flow in Networks with Complexity of 0(V VE) Operations," Mathematical Methods of Solution of Economical Problems, Vol. 7, 112-125 (1977) (in Russian). Z. Galil, "An 0(V E ' ) Algorithm for the Maximal How Problem," Acta Informatica, Vol. 14, 221-242 (1980). Z. Galil and A. Naamad, "An 0(EV ' \og V) Algorithm for the Maximal How Problem," J. Comput. System Sci., Vol. 21, 203-217 (1980). Y. Shiloach, An Ofnllog !) Maximum-Flow Algorithm, Technical Report STAN-CS-78-802, Computer Science Department, Stanford University, Stanford, Calif., 1978. D. D. Sleator, An Ofnm logn) Algorithm for Maximum Network Flow, 3

2

12.11 12.12 12.13 12.14

5,3

2 3

5 3

2

2

440

12.15 12.16 12.17 12.18 12.19

12.20 12.21 12.22 12.23

12.24 12.25 12.26 12.27

12.28 12.29 12.30 12.31

FLOWS IN NETWORKS

Technical Report STAN-CS-80-831, Computer Science Department, Stanford University, Stanford, Calif., 1980. D. D. Sleator and R. E. Tarjan, "A Data Structure for Dynamic Trees," J. Comput. System Sci., Vol. 26, 362-391 (1983). Y. Shiloach and U. Vishkin, "An 0(n log n) Parallel Max-How Algorithm," Journal of Algorithms, Vol. 3, 128-146 (1982). Η. N. Gabow, "Scaling Algorithms for Network Problems," J. of Comput. and Sys. Sci., Vol. 31, 148-168 (1985). R. E. Tarjan, "A Simple Version of Karzanov's Blocking Flow Algorithm," Operations Research Letters, Vol. 2, 265-268 (1984). Α. V. Goldberg, "Efficient Graph Algorithms for Sequential and Parallel Complexity," Ph.D. Thesis, Laboratory for Computer Science, M.I.T., Cambridge, Mass., 1987. Also available as Technical Report MIT/LCS/TR374. Α. V. Goldberg and R. E. Tarjan, "A New Approach to the Maximum Flow Problem," Λ ACM, Vol. 35, 921-940 (1988). J. Cheriyan and S. N. Maheshwari, "Analysis of Preflow Push Algorithms for Maximum Network Flows," SIAMJ. Computing, Vol. 18,1057-1086 (1989). R. K. Ahuja, J. B. Orlin, and R. E. Tarjan, "Improved Time Bounds for the Maximum Flow Problem," SIAM. J. Computing, Vol. 18, 939-954 (1989). R. K. Ahuja and J. B. Orlin, "A Fast and Simple Algorithm for the Maximum Flow Problem," Working Paper, Sloan School of Management, M.I.T., Cambridge, Mass. Available as Sloan W.P. No. 1905-87 (revised), Oct. 1988 (to appear in Operations Research). L. Tuncel, "On the Complexity of Preflow Push Algorithms for Maximum Flow Problems," Technical Report No. 901, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, April 1990. J. Cheriyan and T. Hagerup, "A Randomized Maximum Flow Algorithm," Proc. 30th Annual IEEE Symp. on Foundations of Computer Science, 1989, pp. 118-123. N. Alon, "Generating Pseudo-Random Permutations and Maximum Flow Algorithms," Information Processing Letters, 1990. J. Cheriyan, T. Hagerup, and K. Melhorn, "Can a Maximum Flow Be Computed in 0(mn) time?," Proc. 17th International Colloquium on Automata, Languages and Programming, Warwick University, England, July 1990. (See Springer-Verlag Lecture Notes in Computer Science, No. 443, 235-248.) S. Even and R. E. Tarjan, "Network Flow and Testing Graph Connectivity," SIAM. J. Computing, Vol. 4, 507-518 (1975). J. E. Hopcroft and R. M. Karp, "An n Algorithm for Maximum Matching in Bipartite Graphs," SIAM J. Computing, Vol. 2, 225-231 (1973). Μ. N. S. Swamy and K. Thulasiraman, Graphs, Networks and Algorithms, Wiley-Interscience, New York, 1981. S. Even and O. Kariv, "An 0(n" ) Algorithm for Maximum Matching in General Graphs," Proc. 16th Annual IEEE Symp. on Foundations ofComp. Science, 1975, pp. 100-112. 2

sn

2

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12.32 S. Micali and V. V. Vazirani, "An 0(yf\V\-\E\) Algorithm for Finding Maximum Matching in General Graphs," Proc. 16th Annual IEEE Symp. on Foundations of Comp. Science, 1980, pp. 17-27. 12.33 T. Kameda and I. Munro, "A 0(\V\ · \E\) Algorithm for Maximum Matching of Graphs," Computing, Vol. 12, 91-98 (1974). 12.34 J. Edmonds, "Paths, Trees and Flowers," Canad. J. Math., Vol. 17, 449-467 (1965). 12.35 Η. N. Gabow, "An Efficient Implementation of Edmonds' Algorithm for Maximum Matching on Graphs," /. ACM, Vol. 23, 221-234 (1976). 12.36 J. Edmonds, "Maximum Matching and a Polyhedron with 0, 1 Vertices, " / . Res. Nat. Bur. Std., Vol. 69B, 125-130 (1965). 12.37 E. L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976. 12.38 C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982. 12.39 Η. N. Gabow, "Implementation of Algorithms for Maximum Matching on Nonbipartite Graphs," Ph.D. Thesis, Dept. of Electrical Engineering, Stanford University, Stanford, Calif., 1973. 12.40 Z. Galil, S. Micali, and H. Gabow, "Maximal Weighted Matching on General Graphs," Proc. 23rd Annual IEEE Symp. on Foundations of Computer Science, 1982, pp. 255-261. 12.41 R. E. Tarjan, Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, 1983. 12.42 K. Melhorn, Graph Algorithms and NP-Completeness, Springer-Verlag, New York, 1984. 12.43 S. Even, Graph Algorithms, Computer Science Press, Potomac, Md., 1979. 12.44 Z. Galil, "Finding the Vertex Connectivity of Graphs," SIAM J. Computing, Vol. 9, 197-199 (1980). 12.45 D. J. Kleitman, "Methods for Investigating Connectivity of Large Graphs," IEEE Trans. Circuit theory, CT-16, 232-233 (1969). 12.46 S. Even, "Algorithm for Determining Whether the Connectivity of a Graph Is at Least k," SIAM J. Computing, Vol. 4, 393-396 (1977). 12.47 J. Cheriyan and R. Thurimella, "Algorithms for Parallel Λ-Vertex Connectivity and Sparse Certificates," Proc. 24th ACM Symp. on Theory of Computing (1991). 12.48 J. Cheriyan and R. Thurimella, "On Determining Vertex Connectivity," Technical Report UMIACS-TR-90-79 CS-TR-2485, University of Maryland, College Park, Md, June 1990. 12.49 H. Nagamochi and T. Ibaraki, "Linear Time Algorithms for Finding a Sparse Ac-Connected Spanning Subgraph of a Ac-Connected Graph," to appear. 12.50 H. Nagamochi and T. Ibaraki, "Computing Edge-Connectivity in Multiple and Capacitated Graphs," Proc. SIGAL Conference, Lecture Notes in Computer Science, Springer-Verlag, Berlin, August 1990. 12.51. J. E. Hopcroft and R. E. Tarjan, "Dividing a Graph into Triconnected Components," SIAM J. Computing, Vol. 2, 135-158 (1973).

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12.52 D. Matula, "Edge Connectivity in 0(mn) Time," Proc. 28th IEEE Annual Symp. on Foundations of Computer Science, Los Angeles, 1987, pp. 249-251. 12.53 Η. N. Gabow, "A Matroid Approach to Finding Edge Connectivity and Packing Arobrescences," Preprint, University of Colorado, Boulder, Co, Oct. 1990, Proc. 24th ACM Symp. on Theory of Computing, 1991. 12.54 H. S. Wilf, Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, N.J., 1986. 12.55 S. A. Cook, "The Complexity of Theorem Proving Procedures," Proc. 3rd ACM Symp. on Theory of Computing, 1971, pp. 151-158. 12.56 M. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, N.J., 1967. 12.57 J. E. Hopcroft and J. D. Ullman, Formal Languages and Their Relation to Automata, Addison-Wesley, Reading, Mas., 1969. 12.58 R. M. Karp, "Reducibility among Combinatorial Problems," Complexity of Computer Communications, R. E. Miller and J. W. Thatcher, Eds., Plenum Press, New York, 1972, pp. 85-104. 12.59 R. M. Karp, "On the Computational Complexity of Combinatorial Problems," Networks, Vol. 5, 45-68 (1975). 12.60 M. R. Garey and D. S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness, Freeman, San Francisco, Calif., 1979. 12.61 E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, Computer Science Press, Potomac, Md., 1978. 12.62 Α. V. Aho, J. E. Hopcroft, and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974. 12.63 Ε. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Prentice-Hall, Englewood Cliffs, N.J. 1977. 12.64 N. Ya. Vilenkin, Combinatorics (Trans, from Russian by A. Schenitzer and S. Schenitzer), Academic Press, New York, 1971. 12.65 V. Chvatal, Linear Programming, Freeman Press, San Francisco, 1983. 12.66 R. T. Rockafellar, Network Flows and Monotropic Optimization, WileyInterscience, New York, 1984. 12.67 E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, Eds., The Travelling Salesman Problem, Wiley-Interscience, New York, 1985. 12.68 Α. V. Goldberg, S. A. Plotkin, and E. Tardos, Combinatorial Algorithms for the Generalized Circulation Problem, Research Report, Lab. for Computer Science, M.I.T., Cambridge, Mass., 1988. 12.69 Α. V. Goldberg, E. Tardos, and R. E. Tarjan, "Network Flow Algorithms," Technical Report #860, School of Operations Research and Industrial Engineering, Cornell University, Itaca, N.Y., Sept. 1989. 12.70 R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, "Network Flows," Working Paper, Sloan School of Management, M.I.T., Cambridge, Mass. Available as Sloan W.P. No. 2059-88, 1989. 12.71 Α. V. Goldberg and D. Gusfield, "Book Review: Flow Algorithms by G. M. Aderson-VePski, E. A. Dinic and Α. V. Karzanov," Technical Report No.

REFERENCES

12.72 12.73 12.74 12.75 12.76 12.77 12.78 12.79

443

STAN-CS-90-1313, Dept. of Computer Science, Stanford University, Stanford, Calif., 1990. G. F. Sullivan, "A Polynomial-Time Algorithm for Fault Diagnosability," Proc. 16th Annual IEEE Symp. on Foundations of Comp. Science, 1984, pp. 148-150. S. G. Akl, The Design and Analysis of Parallel Algorithms, Prentice-Hall, Englewood Cliffs, N.J., 1989. M. J. Quinn, Designing Efficient Algorithms for Parallel Computers, McGraw-Hill, New York, 1987. S. Lakshmivarahan and S. K. Dhall, Analysis and Design of Parallel Algorithms, McGraw-Hill, New York, 1990. D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1989. E. Palmer, Graphical Evolution: An Introduction to Theory of Random Graphs, Wiley-Interscience, New York, 1985. B. Bollobas, Random Graphs, Academy Press, London, 1985. B. Bollobas, Graph Theory: An Introductory Course, Springer-Verlag, New York, 1979.

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

AUTHOR INDEX

Aardenne-Ehrenfest, T. van, 144 Abe, S., 379 Adams, J. M., 387 Aho, Α. V., 121, 346, 368, 380, 386-387, 435 Ahuja, R. K., 422, 436 Akl, S., 437 Alagar, V. S., 230 Allen, F. E., 362, 387 Alon, N., 422 Amin, A. T., 230 Anderson, I., 228 Appel, Κ. I., 258 Arlazorov, V. L., 382 Avis, D., 341 Balabanian, N., 173 Bedrosian, S. D., 172 Behzad, M., 67 Bellman, R. E., 320 Belmore, M., 67 Berge, C, 25-26, 67, 121, 172, 225, 230, 239, 258-259, 298, 396 Bermond, J. C, 230 Bertsekas, D. P., 437 Bickart, R. Α., 173 Biggs, N., 172 Birkhoff, G., 72 Birkhoff, G. D., 258 Bland, R. G., 298 Bock, F. C, 332 Boesch, F. T., 173, 230, 258

Boisvert, M., 173 Bollobas, B., 25-26, 68, 230, 258, 437 Bondy, J. Α., 25-26, 65, 67-68, 121, 196, 205, 248, 258, 336 Booth, K. S., 374 Bose, Ν. K., 172, 196 Brooks, R. L., 252 Brualdi, R. Α., 230 Bruijn, N. G. de, 114 Bruno, J., 94, 298 Camerini, P. M, 332 Carre, B., 324 Cartwright, D., 121 Cayley, Α., 151 Cederbaum, I., 173, 371, 374, 379 Chartrand, G., 25, 67-68, 121, 196, 208, 230, 258 Chen, W. K., 51, 91, 93, 150, 172-173, 370 Cheriton, D., 327 Cheriyan, J., 420-422, 432 Cherkasky, R. V., 410 Chiba, N., 379 Chien, R. T., 51, 173 Christofides, N., 258, 322 Chu, Y., 332 Chua, L. O., 298 Chung, F. R. K., 230 Chvatal, V., 63, 68-69, 124, 436 Coates, C. L., 163 Cockayne, E. J., 259

445

446

AUTHOR INDEX

Cocke, J., 362, 386-387 Colbourn, C. J., 230 Cole, R., 342 Comeau, Μ Α., 322 Cook, S. Α., 434, 484 Cummins, R. L., 53, 70 Dahbura, A. T., 346 Dantzig, G. B., 324, 380 Demoucron, G., 374 Deo, N., 26, 121, 172, 324, 374, 435 Derigs, V., 341 Dhall, S. K., 437 Dijkstra, E. W., 315 Dinic, Ε. Α., 382, 404 Dirac, G. Α., 65, 68 Dreyfus, S. E., 322 Dror, M., 346 Duffin, R. J., 298 Dulmage, A. L., 227 Dzikiewicz, J., 314

Gallai, T., 70, 211, 232, 243 Garey, M. R., 230, 380, 435 Ghouila-Houri, Α., 117 Goldberg, Α. V., 411-412, 422, 436 Golomb, S. W., 124 Golumbic, M. G, 259 Gordon, M., 322, 383 Gould, R., 93 Graham, R. L., 327 Graver, J. E., 298 Greene, C, 299 Greene, D. M, 298 Gries, D., 307 Grigoriadis, M. D„ 341 Gusfield, D., 436

Hagerup, T., 422 Haken, W., 258 Hakimi, S. L., 209, 230, 232 Hall, P., 218, 327 Halmos, P. R., 72, 218 Harary, F., 25, 121, 160, 172, 186-187, 196, 202, 211, 230, 258, 298 Eades, P., 379 Edmonds, J., 211, 297-298, 322, 327, 344, 380, Havel, V., 209 Heawood, P. J., 258 401, 427, 433 Hecht, M. S., 362, 382, 387 Elias, P., 215, 396 Hedetniemi, S. T., 259 Erbert, J., 373 Held, M, 67 Erdos, P., 70, 211, 232, 258 Hohn, F. E., 72, 148 Eve, J., 383 Homobono, N., 230 Even, S., 342, 371, 374, 379, 388, 422, 427, 431-433, 438 Hopcroft, J. E., 121, 342, 354, 362, 368, 374, 386, 427, 432, 434-436, 438 Horowitz, E., 368, 435 Faradzev, I. Α., 382 Hu, T. C, 322 Fary, I., 181 Feick, R., 172 Ibaraki, T., 432 Feinstein, Α., 215, 396 Fischer, M. J., 368, 382 Iri, M, 298 Fleury, 344 Ishizaki, Y., 94 Floyd, R. W., 322 Itai, Α., 336, 342 Folkman, J., 298 Jayakumar, R., 173, 379 Ford, L. R., 215, 320, 383, 396, 401, 436 Johnson, D. J3., 322 Fournier, J. C, 248 Johnson, D. S., 435 Frank, H., 230, 258, 322 Johnson, E. L., 344 Fratta, L., 332 Fraysseix, H. de, 379 Kainen, P. C, 258 Frederickson, G. N., 324, 346 Kajitani, Y., 94, 173, 370 Fredman, M. L., 322 Kalantari, B., 341 Frisch, I. T., 230, 258, 322 Fulkerson, D. R., 215, 298, 383, 396, 401, 436 Kameda, T., 427 Kariv, O., 342, 427 Furman, Μ. E., 382 Karivaratharajan, P., 258 Gabow, Η. N., 173, 342, 410-411, 427, 433 Karp, R. N., 67, 322, 327, 341, 380, 401, 427, Gale, D., 297 435-436, 438 Karzanov, Α. V., 410, 436 Galil, Z., 410-411, 427, 432-433

AUTHOR INDEX

Kendall, M. G., 121 Kennedy, K., 386 Kernighan, B. W., 67 Kerschenbaum, Α., 327 Kim, W. H., 51, 173 Kirchhoff, G., 150 Kishi, G., 94 Kleitman, D. J., 209, 211, 213, 232, 432 Knuth, D. E., 121, 298 Konig, D., 222, 239 Koren, I., 374 Krishnamoorthy, V., 230 Kronrod, Μ. Α., 382 Kruskal, J. B., 297, 324 Kuhn, H. W., 336 Kuratowski, C, 186 Kurki-Suonio, R„ 383 Kwan, M.K., 343 Lai, C. W., 341 Lakshmivarahan, S., 437 Las Vergnas, M., 298 Lawler, E. L., 67, 298, 322, 370, 427, 436 Lawrence, J., 298 Lee, D., 346 Lehman, Α., 298 Lempel, Α., 371, 374, 379 Lengauer, T., 322, 379 Lenstra, J. K., 67, 346, 436 Lesniak, L., 25, 68, 121, 196, 230, 258 Lewis, P. M, 67 Liao, Y., 322 Lin, P. M., 94 Lin, S., 67 Lipton, R., 389 Liu, C. L., 26, 66, 258, 261 Liu, T„ 332 Lovasz, L., 25-26, 124, 230, 252, 258-259 Lueker, G. S., 374 MacLane, S., 72, 188 Maffioli, F., 332 Magnanti, T. L., 436 Maheswari, S. N., 408, 420-422 Malgrange, Y., 374 Malhotra, V. M., 408 Marek-Sadowska, M., 379 Mason, J. H., 301 Mason, S. J., 163 Matula, D., 432, 436 Maxwell, L. M., 91, 93 Mayeda, W., 51, 153, 172, 370 Megiddo, N., 341 Melhorn, K., 314, 324, 374, 422, 427, 435

447

Melnikov, L. S., 252 Mendelsohn, N. S., 227 Menger, K., 213 Meyer, A. R., 382 Meyniel, M., 123 Micali, S., 427 Miller, R. E., 387 Minieka, E., 322 Minoux, M., 322, 383 Minsky, M., 434 Minty, G. J., 280, 282, 292-293, 299, 303 Mirsky, L., 230, 298 Moffat, Α., 324 Moon, J. W., 115, 121, 151 Moore, E. F., 320 Morley, T. D., 298 Munkres, J., 336 Munro, I., 383, 427 Murti, V. G. K., 153 Murty, U. S. R., 25-26, 68, 121, 196, 248, 258, 336 Myers, B. R., 172-173 Myers, E. W., 173 Naamad, Α., 410-411 Nagamochi, H., 432 Narayanan, H., 298 Nash-Williams, C. St. J. Α., 53, 68, 121 Nemhauser, G. L., 67 Nievergelt, J., 121, 374, 435 Nishioka, 1., 379 Nishizeki, T., 379 Norman, R. Z., 121 Ohtsuki, T., 94 O'Neil, E. J., 383 O'Neil, P. E., 383 Opatrny, J., 230 Ore, O., 65, 69-70, 196, 222, 258 Orlin, J. B., 422, 436 Orloff, C. S., 346 Otten, R. H. J. M., 379 Overbeck-Larisch, M., 123 Ozawa, T., 379 Palmer, Ε. M., 172, 437 Pan, Α., 322 Pang, C. Y., 324 Papadimitriou, C. H., 298, 327, 346, 427, 436 Parsons, T. D., 196 Parthasarathy, K. R., 259 Perfect, H., 230 Pertuiset, R., 374 Petersen, B., 298

448

AUTHOR INDEX

Peyrat, C, 230 Phelan, J. M., 387 Pierce, A. R., 324 Plotkin, S. Α., 436 Polimeni, A. D., 68 Posa, L., 65 Powell, Μ. B., 260 Prabhu, Κ. Α., 196 Pramodh Kumar, M., 408 Prim, R. C, 302, 325-326 Priifer, H., 151 Pullman, N. J., 121 Purdom, P., 383 Quinn, M., 437 Rado, R., 223, 297-298 Randow, Rabe von, 298 Rao, K. S., 153 Rao, V. V. B., 153, 173 Ravindra, G., 259 Read, R. C, 258 Recski, Α., 298 Reed, Μ. B., 43-44, 51, 172, 196 Reingold, Ε. M., 121, 374, 435 Rinnooy-Kan, A. H. G., 67, 346, 436 Robert, F. S., 173 Robert, J., 121 Robichaud, L. P. Α., 173 Rockefeller, R. T., 436 Rodeh, M., 336 Rose, D. J., 38 Rosenkrantz, D. J., 67 Rosenstiehl, P., 379 Rubin, F., 374 Saaty, T. L., 258 Sahni, S., 368, 435 Schaefer, M, 386 Schnorr, C. P., 383 Seshu, S., 43, 51, 172, 196 Shamir, Α., 342 Shank, H., 70 Shannon, C. E., 215, 396 Shiloach, Y., 410 Shinoda, S., 370 Shirakawa, 1., 379 Shmoys, D. B., 67, 436 Sleator, D. D., 410-411, 440 Spira, P. M, 322 Srinivasan, N., 230 Stark, R. H., 387 Stearns, R. E., 67 Steiglitz, K., 298, 427, 436

Stein, S. K, 181 Stern, H., 346 Stewart, M. J., 68 Strassen, V., 382 Sullivan, G. F., 436 Sun, F. K., 172 Swamy, Μ. N. S., 26, 51, 96, 109, 172-173, 230, 379, 427, 439 Syslo, Μ. M., 314 Szekeres, G., 261 Tabourier, Y., 324 Takahashi, H., 379 Takoka, T., 324 Tamassia, R., 379 Tamir, Α., 341 Tanimoto, S. L., 336 Tardos, E., 436 Tarjan, R. E., 324, 327, 332, 354, 362, 368, 370-371, 374, 379, 386, 389, 410-412, 422, 427, 432, 436 Thomas, R. E., 230 Thulasiraman, K„ 26, 51, 96, 109, 172-173, 230, 258, 322, 379, 427, 439 Thurimella, R., 432 Toida, S., 68 Tomizawa, N., 298 Trudeau, P., 346 Tsitsiklis, J. N., 437 Tucker, Α., 259 Tuncel, L., 422 Turan, P., 258, 259 Tutte, W. T„ 53, 155, 186-187, 197, 214, 228, 258, 285, 298-299, 389 Ueno, S., 173 Ullman, J. D., 121,362,368, 380, 382,386-387, 389, 434-435 Uyar, M. U., 346 Valdes, J., 370 Vandewalle, J., 298 van Slyke, R., 327 Vaughan, Η. E., 218 Vazirani, V. V., 427 Vilenkin, Ν. Y., 436 Vishkin, U., 410 Vizing, V. G., 246, 248, 252, 260 Wagner, K., 181, 187 Wagner, R. Α., 322 Wang, D. L., 209, 211, 213, 232 Warren, H. S., 307, 312, 314 Warshall, S., 307

AUTHOR INDEX

Watanabe, Η., 94 Watkins, Μ., 298 Weinberg, L., 94, 298 Welsh, D. J. Α., 260, 287, 297-298, 300 White, G. P., 324 Whitney, H., 21, 25, 53, 182, 188, 192, 196, 214, 258, 265, 276, 298, 300 Wijk, J. C. van, 379 Wilf, H. S„ 261, 433 Williams, Τ. Α., 324 Williams, T. W., 91, 93 Wilson, R. J., 26, 233, 298, 302, 344 Wimer, S., 374

Wolaver, D. H., 298 Wong, C. K., 322 Wood, D. C, 258 Woodall, D. R., 123 Yannakakis, M., 327 Yao, A. C, 327 Yasuda, T., 370 Yen, J. Y., 324 Youla, D. G, 173 Zadeh, N., 404

449

Graphs: Theory and Algorithms by K. Thulasiraman and M. N. S. Swamy Copyright © 1992 John Wüey & Sons, Inc.

SUBJECT INDEX

Abelian group, 73 Ackermann's function, 368 Acyclic graph, 31 directed, 100 Adjacency list, 347 Adjacency matrix: of directed graph, 159 of weighted directed graph, 163 Adjacent edges, 2, 98 Adjacent vertices, 2, 98 Algorithm(s): biconnectivity, 356 breadth-first search, 381 Chinese postman tour, 344 connectivity, 427-433 depth-first search of directed graph, 351 depth-first search of undirected graph, 348 Dinic's maximum flow, 404-408 Dinic-MPM maximum flow algorithm, 404-410 Edmonds and Karp's modification of labeling algorithm, 400-404 Euler trail, 345 FIFO maximum flow, 419 generic maximumflow(Goldberg and Tarjan), 413 graphs with prescribed degrees, 209-213 greedy, 294 greedy coloring, 260 highest label preflow push, 420-421 layered network, 406

LIFO preflow push, 420 maximal excess preflow push, 421 maximal flow in a layered network, 409-410 maximum bipartite matching, 426-427 maximumflowin transport network (labeling algorithm), 397 maximum flow in 0-1 networks, 422-426 minimum weight spanning tree, 325-326 optimal assignment, 339 optimum branching, 332 path finding, 372 perfect matching, 335 planarity testing, 378 program graph reducibility, 366 shortest paths between all pairs of vertices, 323 shortest paths from a specified vertex to all vertices, 318-320 i/-numbering, 372 strong connectivity, 360 timetable scheduling, 341-342 transitive closure, 310-312 All-vertex incidence matrix: of directed graph, 126 of undirected graph, 126 Alternating chain (with respect to a matching), 224 Alternating sequence (relative to an independent set), 239 maximal, 239 Λί-AIternating tree, 334

451

452

SUBJECT INDEX

Ancestor (of a vertex), 349 proper, 349 Arborescence, 106 Arc coloring lemma, 122, 292 Associated directed graph, 231 Associative property, 72 Augmentation theorem, 269 Augmenting path (relative to a matching), 226 /-Augmenting path, 394 Back edge, 347, 351 Backward edge, 394 Backward labeling, 397 Base axioms for matroid, 273 Base in matroid, 266 Base orderable matroid, 301 Basis of vector space, 76 Basis vectors, 76 Bellman-Ford-Moore algorithm, 320 Berge's (alternating chain) theorem, 225 Biconnected component, 354 Biconnectivity algorithm, 356 Binary: matroid, 287 relation, 104 Binary tree, 108 Binet-Cauchy theorem, 148 Bipartite graph, 16 Bipartite matroid, 301 Bipartition, 16, 239 Block of graph, 19 Bond matroid, 267 Bottleneck, 401 Branch, 31 Branching, 327 algorithm, 332 optimum, 327 Breadth-first search algorithm, 381 Bridge, 28 Brooks' theorem, 252 Bush form, 376 Capacity: of cut, 392 of edge, 391 Capacity constraint, 391 Cayley's theorem, 150 Center, 53 Cheriyan and Maheswari's maximum flow algorithm, 420-421 long trajectory push in, 421 phase in, 420 short trajectory push in, 421

Chinese postman tour: algorithm, 344 problem, 343 rural, 346 Chord, 31 jfc-Chromatic graph, 251 Chromatic index, 245 Chromatic number, 251 Chromatic polynomial, 254 Circuit(s): axioms for matroid, 275 directed, 100 in directed graph, 100 directed Hamilton, 115 edge, 8 even,8 fundamental, 41 fundamental set of, 41 in graph, 8 Hamilton, 62 length of, 8 matroid, 266 in matroid, 266 odd, 8 subspace, 82 vector, 134 weight of, 67 Circuit matrix, 134 of directed graph, 134 fundamental, 135 of matroid, 289 of undirected graph, 134 Class NP, 433 Class P, 433 Clique, 262 Closed set: in matroid, 299 relative to binary relation, 72 (n + l)-CIosure, 70 Closure of graph, 69 Coates' flow graph, 164 Coates' gain formula, 167 Coates' graph, 164 Coates' method, 163 Cobase in matroid, 277 Cocircuit, 277 fundamental, 278 matrix, 286 Code-optimization problem, 362 Co-forest, 40 Cographic matroid, 268 Collapsing a vertex, 362 Coloop in matroid, 277

SUBJECT INDEX

Column-oriented, 310 Commutative property, 73 Complement: of simple graph, 6 of subgraph, 7 Complete bipartite graph, 16 Complete directed graph, 115 Complete graph, 16 Complete matching (in bipartite graph), 216 Complete λ-partite graph, 16 Completely scanned, 347 Component(s): biconnected, 354 of directed graph, 100 even, 228 of graph, 9 odd, 228 Condensed graph, 101 Condensed image, 101 Connected directed graph, 100 Connected graph, 9 Λ-Connected graph, 201 Connected matroid, 300 Connected vertices, 9 Connectivity, 200 algorithms, 427-433 edge, 207 vertex, 200 Conservation condition, 391 Contractible, 15 Contraction: of edge, 15 of graph, 282 of matroid, 284 Coordinates of vector, 78 Corank function of matroid, 277 Cospanning tree, 31 Cospanning k-tree, 38 Critical edge, 327 Critical subgraph, 327 Cross edge, 349, 351 Crossing number, 196 Cubic graph, 27 Cut, 44 capacity of, 392 directed, 122 matrix, 130 minimum, 394 vector, 130 Cut matrix, 130 of directed graph, 130 of undirected graph, 130 Cutset(s), 42

453

fundamental, 46 fundamental set of, 46 matroid, 267 subspace, 85 Cut-vertex, 19 splitting, 23 Dantzig's algorithm, 380 Dark edge, 224 de Bruijn sequence, 111 Decision problem, 433 Class NP, 433 Class P, 433 Deficiency, 217 Degree: of region, 184 of vertex, 2, 98 Degree matrix, 150 Degree sequence, 63 Dependent set in matroid, 266 Depth-first number, 347 Depth-first search algorithms, 348, 351 for directed graph, 351 for undirected graph, 348 Descendant (of vertex), 349 proper, 349 DFS tree, 347 Diameter, 8 Dijkstra's algorithm, 318 Dimension of vector space, 76 Dinic's maximum flow algorithm, 404-408 phase in, 407 Dinic-MPM maximumflowalgorithm, 404-410 Directed circuit, 100 Directed cut, 122 Directed Eulerian graph, 110 Directed Euler trail, 110 open, 110 Directed graph, 1, 97 Directed Hamilton circuit, 115 Directed Hamilton path, 115 Directed Hamiltonian graph, 115 Directed path, 100 Directed spanning tree, 108 Directed trail, 100 closed, 100 open, 100 Directed tree, 106 Directed walk, 99 closed, 99 open, 99 Direct sum of subspaces, 76 Discharge operation, 418

454

SUBJECT INDEX

Disconnected matroid, 300 Disconnecting set, 201 Distance: between spanning trees, 93 between vertices, 8 Dominating set, 258 Dot product of vectors, 79 Doubly stochastic matrix, 232 Dual matroid(s), 277 Dual of graph, 188 Edge(s), 1, 97 adjacent, 2, 98 back, 347, 351 backward, 394 circuit, 8 critical, 327 cross, 349, 351 current, 416 current residual, 416 eligible, 328 forward, 350, 394 independent, 216 noncircuit, 8 parallel, 1, 98 pendant, 3 residual, 411 tree, 347, 351 virtual, 375 fc-Edge chromatic graph, 245 Edge chromatic number, 245 Λ-Edge colorable graph, 245 yt-Edge coloring, 245 optimum, 247 proper, 245 λ-Edge connected: directed graph, 231 graph, 207 Edge connectivity, 207 Edge cover, 243 minimum, 243 Edge covering number, 243 Edge-disjoint graphs, 23 Edge-induced subgraph, 5 Edge-removal, 13 Edmonds and Johnson's Chinese postman tour algorithm, 344 Edmonds and Karp's modification of labeling algorithm, 400-404 Edmonds and Karp's theorem on labeling algorithm, 403 Edmonds' optimum branching algorithm, 332 Edmonds' theorem on A-edge connectivity, 212 Eligible edge, 328

Embeddable, 179 Empty graph, 2 End vertex: of directed walk, 99 of edge, 1, 98 of walk, 7 Equality subgraph, 338 Equi-cofactor matrix, 150 Equivalence: classes, 104 relation, 104 Euler trail, 57 directed, 110 open, 57 open directed, 110 Euler trail algorithm, 345 Eulerian graph, 57 directed, 110 randomly, 61, 62 Eulerian matroid, 301 Euler's formula, 184 Even and Tarjan's analysis of maximum flow algorithms, 422-426 Even and Tarjan's s/-numbering algorithm, 372 Examine (an edge in a DFS), 346 Extremal graph theory, 258 A:-Factor, 233 Ar-Factorable graph, 233 1-Factorial connection in directed graph, 164 1-Factor in directed graph, 160 Father (of vertex), 346, 349 Feasible vertex labeling, 338 Feedback loop, 171 Feedback path, 171 Field, 73 Five-color theorem, 257 Fleury's algorithm, 345 Flow, 391 blocking, 407 maximal, 407 maximum, 392 maximum algorithms, 390-426 pre-, 411 pseudo-, 411 Floyd's algorithm, 323 Ford-Fulkerson's maximumflowalgorithm, 397 Forest, 40 Forward edge: relative to DFS, 350 relative to directed path, 394 Forward labeling, 397 Forward path, 171

SUBJECT INDEX

Four-color conjecture, 258 Four-color problem, 257 Four-color theorem, 258 Fundamental circuit, 41 matrix, 135 matrix of matroid, 288 in matroid, 271 Fundamental cocircuit, 278 matrix, 288 Fundamental cutset, 46 matrix, 133 Fundamental set of circuits, 41 Fundamental set of cutsets, 46 Gale optimal, 295 Galois field, 74 Gammoid, 301 Girth, 27 Goldberg and Tarjan's maximum flow algorithm, 411-420 Graph(s), 1 acyclic, 31 associated directed, 231 bi-connected, 22 bipartite, 16 A-chromatic, 251 Coates, 164 complete, 16 complete bipartite, 16 complete directed, 115 complete fc-partite, 16 connected, 9 2-connected, 22 A;-connected, 201 connected directed, 100 cubic, 27 directed, 1, 97 directed acyclic, 100 directed Eulerian, 110 directed Hamilton connected, 123 directed Hamiltonian, 115 λ-edge chromatic, 245 k-edge colorable, 245 A-edge-connected, 207 Λ-edge-connected directed, 231 edge-disjoint, 23 empty, 2 Eulerian, 57 ^-factorable, 233 Hamilton connected, 70 Hamiltonian, 62 homeomorphic, 186 isomorphic, 23 1-isomorphic, 23

2-isomorphic, 23 Kuratowski's, 182 Mason, 169 maximal planar, 197 minimally connected, 52 minimally connected directed, 102 nonoriented, 1 nonseparable, 19 null, 2 one-terminal-pair, 198 oriented, 1 parallel combination of, 98 A-partite, 16 perfect, 259 Peterson, 187 planar, 179 planar one-terminal-pair, 198 program, 361 quasi-strongly connected, 103 randomly Eulerian, 61, 62 randomly Hamiltonian, 67 reducible program, 362 reflexive directed, 105 regular, 16 r-regular, 16 self-complementary, 27 self-dual, 198 separable, 19 series combination of, 198 series-parallel, 198 simple, 1 i/-numbering of a, 370 strongly connected, 100 symmetric directed, 105 transitive directed, 105 trivial, 19 underlying undirected, 99 undirected, 1 A-vertex colorable, 251 vertex-disjoint, 23 weighted, 175 Graphic matroid, 268 Graphic sequence, 63 Graphoid, 280 Greedy algorithm, 294 Greedy coloring algorithm, 260 Group, 73 Hall's theorems, 218-219 Hamilton circuit, 62 directed, 115 Hamilton connected: directed graph, 123 graph, 70

455

456

SUBJECT INDEX

Hamiltonian graph, 62 directed, 115 randomly, 67 Hamilton path, 62 directed, 115 HIGHPTdO, 365 Homeomorphic graphs, 186 Hopcraft and Tarjan's algorithms, 354, 388 Hungarian tree, 334 Hyperplane, 300 Identifying vertices, 15 Identity of group, 73 Improvement on edge coloring, 248 Incidence matrix, 128 Incidence set, 94 Incidence vector, 127 Incident into, 98 Incident on, 2, 98 Incident out of, 98 In-degree, 98 matrix, 155 Independence axioms for matroid, 266, 272 Independence number, 236 Independent edges, 216 Independent set(s): of graph, 124, 236 of matroid, 266 maximum, 236 Induced subgraph, 5, 99 In-going directed tree, 175 root of, 175 Initial vertex: of directed walk, 99 of walk, 7 Inner product, 79 Internal vertex, 7, 99 Intersection of graphs, 11 Interval analysis, 362 Inverse of element in group, 73 Isolated vertex, 3, 98 Isomorphic: graphs, 23 matroids, 268 vector spaces, 78 1-Isomorphic graphs, 23 2-Isomorphic graphs, 23 Konig-Egervary theorem, 223 Konigsberg bridge problem, 55 Konig's theorem, 222 Kruskal's algorithm, 325 Kuhn-Munkres algorithm, 339

Kuratowski's graphs, 182 Kuratowski's theorem, 186 Label of vertex, 338 Labeled trees, 150 Labeling algorithm, 39 Layered network, 406 algorithm, 406 blocking flow in a, 407 layers of, 406 length of, 406 maximal flow in a, 407 maximal flow in a, algorithm, 409-410 Leaf, 106 Left (of vertex), 349 Lempel, Even and Cederbaum's planarity testing algorithm, 378 Length: of circuit, 8 of edge, 314 of path, 8 Lexicographically greater, 295 Lexicographically maximum, 295 Light edge, 224 Linear combination, 76 Linearly independent vectors, 76 Link, 31 Loop in matroid, 266 LOWXf), 355 LOWLINK(y), 358 Major, 147 Major determinant, 147 Majorise, 63 Malhotra, Pramodh Kumar and Mahaswari's algorithm, 409-410 Marriage problem, 215 Mason graph, 169 Mason's gain formula, 171 Mason's method, 168 Mason's signal flow graph, 169 Matched into, 216 Matching, 216 bipartite, algorithm, 426-427 complete, 216 matroid, 267 maximum, 216 number, 216 optimal, 336 perfect, 228 (0,1) Matrix, 223 Matroid(s), 266 base axioms for, 273 base in, 266

SUBJECT INDEX

base orderable, 301 binary, 287 bipartite, 301 bond, 267 circuit, 266 circuit axioms for, 275 circuit in, 266 circuit matrix of, 289 closed set in, 299 closure in a, 299 cobase in, 277 cocircuit in, 277 cocircuit matrix of, 289 cographic, 268 coloop in, 277 connected, 300 contraction of, 284 corank function of, 277 cutset, 267 dependent set in, 266 disconnected, 300 dual of, 277 elements of, 266 Eulerian, 301 Fano, 301 fundamental circuit in, 271 fundamental circuit matrix of, 286, 288 fundamental cocircuit in, 278 fundamental cocircuit matrix of, 286, 288 graphic, 268 hyperplane in, 300 independence axioms for, 266, 272 independent sets of, 266 isomorphic, 268 loop in, 266 matching, 267 minor of, 285 nonseparable, 300 orientable, 292 rank axioms for, 274 rank function of, 266 rank in, 266 rank of, 266 regular, 298 representable, 286 restriction of, 284 separable, 300 standard representation of, 286 transversal, 300 uniform, 299 union of, 301 Max-flow min-cut theorem, 396 Maximal planar graph, 197 Maximal subgraph, 5

Maximal subset, 5 Maximally distant spanning trees, 93 Mendelsohn and Dulmage's theorem, 227 Menger's theorems, 213-215 proof of, 427-431 Mesh, 183 Minimal subgraph, 5 Minimal subset, 5 Minimally connected: directed graph, 102 graph, 52 Minimum weight spanning tree algorithms, 325, 326 Minor of matroid, 285 Minty's painting theorem, 280 Network: layered, 406 residual, 405 0-1, 422 0-1 Network, 422 of type 1, 422 of type 2, 422 Network optimization, 436 Noncircuit edge, 8 Nonoriented graph, 1 Nonseparable graph, 19 Nonseparable matroid, 300 NP, 433 NP-complete, 434 NP-hard, 434 Null graph, 2 Nullity: of directed graph, 102 of graph, 41 Null space, 286 One-terminal-pair graph, 198 planar, 198 Operations on graphs, 11-15 Optimal assignment algorithm, 339 Optimal assignment problem, 336 Optimum i-edge coloring, 247 Order of graph, 1 Orientable matroid, 292 Orientation of edge, 98 Oriented graph, 1 Orthogonal complements, 80 Orthogonality: of circuit and cutset subspaces, 90 relation, 137 Orthogonal subspaces, 80 Orthogonal vectors, 79

457

458

SUBJECT INDEX

Out-degree, 98 matrix, 175 Painting: of orientable matroid, 292 of set, 280 Painting theorem, 280 Parallel edges, 1, 98 Λ-Partite graph, 16 complete, 16 Partition, 9 principal, 94 Partitioned, 9 Path: augmenting, 226 /-augmenting, 394 directed, 100 in directed graph, 100 directed Hamilton, 115 even, 8 in graph, 8 Hamilton, 62 internally disjoint, 21 length of, 8 odd, 8 Pendant: edge, 3 vertex, 3 Perfect graph conjecture, 259 strong, 259 Perfect matching, 228 algorithm, 335 Permutation, 160 even, 160 matrix, 232 odd, 160 Petersen graph, 187 Planar embedding, 179 Planar graph, 179 region of, 184 Planar one-terminal-pair graph, 198 Planar separator theorem, 382 Planarity testing algorithm, 378 /-Positive edge, 393 Postman tour, 343 Prefix code, 109 Prim's algorithm, 326 Principal partition, 94 Principal subgraphs, 94 Program graph, 361 Program graph reducibility algorithm, 366 Proper subgraph, 4 Pseudo edges, 343 Pseudo vertex, 330

Push, 410 nonsaturating, 410, 412 nonzeroing, 420 saturating, 410, 412 Push (v,w) operation, 412 Push/relabel operation, 412 Quasi-strongly connected graph, 103 Quickly reducible, 434 Rank axioms for matroid, 274 Rank function of matroid, 266 Rank in matroid, 266 Rank of directed graph, 102 Rank of graph, 41 Rank of matroid, 266 Ranking, 121 Reachability matrix, 307 Reducible program graph, 362 Reducibility of program graph, 362 algorithm, 366 Reduction order, 369 Reference vertex, 128 Reflexive directed graph, 105 Reflexive relation, 104 Region of planar graph, 182 Regular graph, 16 r-Regular graph, 16 Relabel operation, 412 Related vertices, 349 Representable matroid, 286 standard, 286 Residual network, 405 Residual sequence, 209 Restriction: of graph, 282 of matroid, 284 Right (of vertex), 349 Ring sum of graphs, 13 Root: of component (in DFS), 348, 351 of DFS, 346 in directed graph, 105 of in-going directed tree, 175 Row-oriented, 311 Satisfiability problem, 434 3-Satisfiability problem, 434 /-Saturated edge, 393 Saturated vertex (in matching), 216 Scalar, 76 Score, 120 Score sequence, 120 Seg, 44

SUBJECT INDEX

Self-complementary graph, 27 Self-dual graph, 198 Self-loop, 1, 98 Separable graph, 19 Separable matroid, 300 Series edges, 186 Series insertion, 186 Series merger, 186 Series-parallel graph(s), 198 parallel combination of, 198 series combination of, 198 Short-circuiting vertices, 15 Shortest path, 314 algorithms, 318, 320, 323, 380 Simple graph, 1 Sink, 391 SNUMBER(t)), 369 Son (of vertex), 349 Source, 391 Spanning subgraph, 4 Spanning tree(s), 31 directed, 108 distance between, 93 maximally distant, 93 maximum weight, 294 minimum weight, 324 minimum weight, algorithm, 325-326 Spanning fc-tree, 38 Splitting cut vertex, 23 Splitting vertex, 430 Stability number, 236 Stable set, 236 Start vertex (in program graph), 361 Stereographic projection, 181 i/-graph, 375 ir numbering, 370 algorithm, 372 rf-number of a vertex, 370 Strong connectivity algorithm, 360 Strongly connected component, 100 Strongly connected graph, 100 Strongly connected vertices, 100 Subbush, 378 SubgrapMs), 4, 99 complement of, 7 critical, 327 edge-induced, 5 induced, 5, 99 maximal, 5 minimal, 5 principal, 94 proper, 4 spanning, 4 vertex-induced, 5

weight of a, 324 /c-Subgraphs, 94 Subspace, 76 circuit, 82 cutset, 85 Subtree, 31 Susceptibility, 230 Symmetric directed graph, 105 Symmetric relation, 104 System of distinct representatives, 223 Tarjan's algorithms, 348, 351, 356, 360, 366 Terminal vertex: of directed walk, 99 of edge, 98 of walk, 7 Thickness, 196 Timetable scheduling algorithm, 341-342 Topological degrees of freedom, 94 Topological sorting, 118 Tournament, 119 Trail, 8, 100 closed, 8 closed directed, 100 directed, 100 directed Euler, 110 Euler, 57 open, 8 open directed, 100 open directed Euler, 110 Trajectory, 421 long, 421 short, 421 Transitive closure: algorithms, 310, 312 of binary relation, 307 of graph, 307 Transitive directed graph, 105 Transitive reduction, 380 Transitive relation, 104 Transport network, 391 Transversal, 223 partial, 300 Transversal matroid, 300 Travelling salesman problem, 67 Tree, 31 binary, 108 cospanning, 38 DFS, 346 directed, 106 in directed graph, 106 directed spanning, 108 in graph, 31 Hungarian, 334

459

460

SUBJECT INDEX

Tree (Continued) spanning, 31 A-Tree, 38 cospanning, 38 spanning, 38 Tree edge, 346, 350 Tree graph, 53 Triangle, 27 Trivial graph, 19 Turan's theorem, 259 Turning around, 24 Tutte's perfect matching theorem, 228 Underlying undirected graph, 99 Undirected graph, 1 Uniform matroid, 299 Unimodular matrix, 145 Union: of graphs, 11 of matroids, 301 Unrelated vertices, 349 /-Unsaturated edge, 393 /-Unsaturated path, 394 Valency, 3 Valid labeling, 412 Vector, 76 η-Vector, 75 Vector space, 74 Vertex, 1, 97 active, 412 adjacent, 2, 98 ancestor of, 349 concurrent, 438 cut-, 19 descendant of, 349 end, 1, 7, 98-99 excess of a, 411 father of, 346, 349 initial, 98-99

in potential of a, 408 internal, 7, 99 isolated, 3, 98 out potential of a, 408 pendant, 3 potential of a, 409 reference, 128, 409 son of, 349 i/-number of a, 370 terminal, 7, 98-99 virtual, 375 ^-Vertex colorable graph, 251 Λ-Vertex coloring, 251 proper, 251 Vertex connectivity, 200 Vertex cover, 237 minimum, 237 Vertex covering number, 237 Vertex-disjoint graphs, 23 Vertex-induced subgraph, 5 Vertex removal, 13 Visit (vertex in DFS), 346 Vizing's theorem, 248 Vulnerability, 230 Walk, 7, 100 closed, 7 closed directed, 99 directed, 99 open, 7 open directed, 99 Warren's algorithm, 312 Washall's algorithm, 310 Weighted graph, 175 Weighted path length, 109 Weight product of subgraph, 163 Wheel, 214 Whitney's theorems, 21, 213 /-Zero edge, 393

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