Topics on Domination

TOPICS ON DOMINATION ANNALS OF DISCRETE MATHEMATICS General Editor: Peter L. HAMMER Rutgers University, New Brunswic...

Author: S.T. Hedetniemi | R.C. Laskar

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TOPICS ON DOMINATION

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University, New Brunswick, NJ, USA

Advisory Editors: C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, USA V: KLEE, University of Washington, Seattle, WA, USA J.H. VAN LINT; California Institute of Technology, Pasadena, CA, USA G.C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, USA

NORTH-HOLLAND - AMSTERDAM

NEW YORK

OXFORD

TOKYO

48

TOPICS ON DOMINATION

S. T.HEDETNlEMl R. C. LASKAR Department of Computer Science and Mathematical Sciences Clemson University Clemson, SC 29634 USA

1991

NORTH-HOLLAND- AMSTERDAM

NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 2 1 1 1000 AE Amsterdam Netherlands Sole distributors for the USA and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, NY 10010 USA

Library o f Congress Cataloging-in-Publication

Data

/ [ e d i t e d b y ] S . S . Hedetnler?:. R . C . L a s k a r . T o p i c s on domina:ion p. c m . -- ( A n n a l s o f d i s c r e t e m a t h e m a t i c s ; 4 8 ) I n c l u d e s b i b l l o g r a p h l c a l r e f e r e n c e s and index. ISBN 0-444-99006-8 !. G r a p h t h e c r y . I. H e d e t n i e m l . S . T . II. L a s k a r . R. C. III. T i t l e . D o m i n a t i o n . IV. S e r l e s . QAi6G.T65 1991 51:',5--d~C@ 90-26901) CIP

Reprinted from the Journal Discrete Mathematics, Volume 86, Numbers 1-3,1990 Elsevier Science Publishers B.V. ISBN: 0 444 89006 8

0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers 6.K, PO. Box 21 1, 7000 AE Amsterdam, Netherlands. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

Printed in Northern Ireland

Printed on acid-free paper

SPECIAL VOLUME TOPICS ON DOMINATION

Guest Editors: S. T. HEDETNIEMI and R. C. LASKAR

CONTENTS PART II INTRODUCTION

1

S. 7: Hedetniemi and R. C. Laskar Introduction

3

PART II. THEORETICAL

11

E. J. Cockayne Chessboard domination problems

13

C. M. Grinstead, B. Hahne and D. Van Stone On the queen domination problem

21

C. Berge and f! Duchet Recent problems and results about kernels in directed graphs

27

D. I? Summer Critical concepts i n domination

33

J. IF Fink, M. S. Jacobson, L. IF Kinch and J. Roberts The bondage number of a graph

47

M. S. Jacobson and K. Peters Chordal graphs and upper irredundance, upper domination and independence

59

B. Zelinka Regular totally domatically full graphs

71

D. Rall Domatically critical and domatically full graphs

81

B. Bollobas, E. J. Cockayne and C. M. Mynhardt On generalised minimal domination parameters for paths

89

PART Ill. NEW MODELS

99

M. B. Cozzens and L. L. Kelleher Dominating cliques in graphs

Z.Tuza

101

Covering all cliques of graph

117

R. C. Brigham and R. Dutton Factor domination in graphs

127

E. Sarnpathkumar The least point covering and domination numbers of a graph

137

PART IV. ALGORITHMIC

143

D. G. Corned and L. K. Stewart Dominating sets in perfect graphs

145

B. N. Clark, C. J. Colbourn and D. S. Johnson Unit disk graphs

165

C. J. Colbourn and L. K. Stewart Permutation graphs: connected domination and Steiner trees

179

V. Chvataland W. J. Cook The discipline number of a graph

191

J. McHugh and L: Per1 Best location of service centers in a tree-like network under budget constraints

199

M. Skowronska and M. M. Sydo Dominating cycles in Halin graphs

215

D. Kratsch Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs

225

D. L. Grinstead and F! J. Slater On minimum dominating sets with minimum intersection

239

PART V. BIBLIOGRAPHY

255

S. T: Hedetnierniand R. C. Laskar Bibliography on domination in graphs and some basic definitions of domination parameters

257

AUTHOR INDEX TO VOLUME 86

279

Part I. Introduction

This Page Intentionally Left Blank

Discrete Mathematics 86 (1990) 3-9 North-Holland

3

INTRODUCTION S.T. HEDETNIEMI and R.C. LASKAR Department of Computer Science and Mathematical Sciences, Clemson University, Clemson, SC‘29634. USA

Received 2 December 1988

1. Introduction

If one were to browse through the table of contents of several well-known books on general graph theory, for example: “Theorie der Endlichen and Unendlichen Graphen” by D. Konig (Leipzig, 1936 and Chelsea, New York, 1954), “Theory of Graphs and its Applications” by C. Berge (Dunod, Paris, 1958 and Methuen, London, 1962), “Theory of Graphs” by 0. Ore (American Mathematical Society Colloquium Publications 38, Providence, RI, 1962), “Graph Theory” by F. Harary (Addison-Wesley, Reading, MA, 1968), “Introduction to the Theory of Graphs” by M. Behzad and G. Chartrand (Allyn and Bacon, Boston, 1971), “Graphs and Hypergraphs” by C. Berge (North-Holland, Amsterdam, 1975), “Graph Theory with Applications” by J. A. Bondy and U.S.R. Murty (North-Holland, Amsterdam, 1976), “Graph Theory, An Introductory Course” by B. Bollobas (Springer, Berlin, 1976), “Graphs and Digraphs” by M. Behzad, G. Chartrand and L. Lesniak-Foster (Woodsworth and Brooks/Cole, 1979), “Graph Theory” by W.T. Tutte (Encyclopedia of Mathematics and its Applications, Vol. 21, Addison-Wesley, Reading, MA, 1984), one would find a variety of areas of study, on many of which entire books have been written, for example: “Connectivity in Graphs” by W.T. Tutte (Toronto Univ. Press, Toronto, 1967), “Graphical Enumeration” by F. Harary and E.M. Palmer (Academic Press, New York, 1973), “Extremal Graph Theory” by B. Bollobhs (Academic Press, London, 1978), “Topics on Perfect Graphs”, C. Berge and V. Chvatal, eds. (Annals of Discrete 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V. (North-Holland)

1

S . T. Hedetniemi, R. C. Laskar

Mathematics, Vol. 21, North-Holland, Amsterdam, 1983), “Cycles in Graphs”, B.R. Alspach and C.D. Godsil, eds. (Annals of Discrete Mathematics, Vol. 27, North-Holland, Amsterdam, 1985), “Matching Theory” by L. Lovasz and M.D. Plummer (Annals of Discrete Mathematics, Vol. 29, North-Holland, Amsterdam, 1986), “Planar Graphs: Theory and Algorithms”, T. Nishizeki and N. Chiba, eds. (Annals of Discrete Mathematics, Vol. 32, North-Holland, 1988). With the publication of this volume, we can add Domination in Graphs to this list. The most common definition given of a dominating set is that it is a set of vertices D c V in a graph G = (V, E ) having the property that every vertex z1 E V - D is adjacent to at least one vertex in D. The domination number y(G) is the cardinality of a smallest dominating set of G.

2. History

The earliest ideas of dominating sets, it would seem, date back to the origin of the game of chess in India over 400 years ago, in which one studies sets of chess pieces which cover or dominate various opposing pieces or various squares of the chessboard. In more recent times the Eight Queens and Five Queens Problems rekindled interest in dominating concepts, e.g., in the books of Ahrens* in 1901 and Konig* in 1936. Finally with the publications of the books by Berge* in 1958 and Ore* in 1962 the topic of domination was given formal mathematical definition. But by 1972 relatively little work had been done on this topic until Cockayne and Hedetniemi began to study it and ultimately published a 1975 survey of the results that had been obtained by that time. This seems to have brought the subject sufficiently into focus to set research on a much wider scale into motion. In the thirteen years since then over 300 papers have been published on the subject, and in a very real sense domination theory has arrived. Hence this volume. We divide the contributions in this volume into three sections, entitled ‘theoretical’, ‘new models’ and ‘algorithmic’. The nine theoretical papers retain a primary focus on properties of the standard domination number y ( G ) . The four papers which we classify as ‘new models’ are concerned primarily with new variations on the domination theme. The eight algorithmic papers are primarily concerned with finding classes of graphs for which the domination number, and several other domination-related parameters, can be computed in polynomial time. * All bibliographic citations in this introduction can be found in the Bibliography on Domination in Graphs at the end of this volume, by first author and date of publication.

1. Introduction

5

3. Theoretical For a variety of reasons we lead off this volume with the paper “Chessboard domination problems” by Cockayne, because of the historical roots of domination in the game of chess, and because Cockayne has done the most definitive work in this area. The follow up paper “On the queen domination problem” by Grinstead, Hahne and Vanstone contains what we believe to be the best approximation to the old problem of placing a minimum number of queens on an arbitrary n X n chessboard so that all squares are ‘covered’ by at least one queen. We are pleased to be able to present next a reprint of a paper by Berge and Duchet entitled “Recent problems and results about kernels in directed graphs”, which surveys work that has been done for dominating sets in directed graphs, called kernels. Claude Berge has done more than anyone, we think, to establish the mathematical foundations not only of graph theory in general, but in particular of domination theory. To Claude we extend both our appreciation for his willingness to contribute to this volume and our apologies for not adopting his terminology of “Coefficient of external stability” and choosing instead “The domination number”. David Sumner was one of the early researchers in domination theory and was perhaps the first one to consider the question of domination critical graphs. In this paper “Critical concepts in domination” he considers the problem of characterizing graphs for which adding any edge e decreases the domination number, i.e. y(G e ) < y(G). He also considers the problem of characterizing graphs having minimum dominating sets D which are independent, i.e. no two vertices in D are adjacent. A related notion, by Fink, Jacobson, Kinch and Roberts in “The bondage number of a graph”, is that of finding a set of edges F of smallest order (called the bondage number), whose removal increases the domination number, i.e. Y(G - F ) > y ( G ) . It was in a 1978 paper by Cockayne, Hedetniemi and Miller that the following chain of inequalities appeared:

+

where ir and IR are the irredundance and upper irredundance numbers, respectively, y and r are the domination and upper domination numbers and i and P o are the independent domination and vertex independence numbers. This sequence has been the focus of a large number of papers since then. Among these was a paper by BollobAs and Cockayne in which they proved that for bipartite graphs, P o = r = IR. The present paper “Chordal graphs and upper irredundance, upper domination and independence” by Jacobson and Peters expands considerably upon this by presenting several other classes of graphs for which

6

S. T. Hedetniemi, R . C . Laskar

= r = IR. We note that this problem has since been made considerably richer by some, as yet unpublished, new results of Cheston and Fricke. In their original survey paper on domination Codkayne and Hedetniemi introduced the domatic number of a graph, denoted d ( G ) , which equals the maximum order of a partition {V,, V,, . . . , V,} of V ( G )such that every set V, is a dominating set. Today Zelinka has become the world’s foremost authority on the domatic number and a variety of related partition numbers. He has published nearly two dozen papers on this topic. We are pleased to have a contribution from Zelinka entitled “Regular totally domatically full graphs” and another from Rall, entitled “Domatically critical and domatically full graphs”, on the domatic number of a graph. We complete the theoretical section of this volurne with what we consider to be a particularly noteworthy and significant contribution to domination theory by Bollobas, Cockayne and Mynhardt, who challenge our understanding of the fundamental notion of a minimal dominating set, by introducing the new and challenging notion of a k-minimal dominating set in their article “On generalized minimal domination parameters for paths”.

4. New models

The concepts of domination, covering and centrality are so interrelated and so general that it is not at all surprising that so many different types of domination exist; e.g. in a 1985 paper, Hedetniemi, Hedetniemi and Laskar list 30 different types of domination. As of now we know of twice as many types of domination problems which have been studied. Definitions of many of these can be found in the introduction to the Bibliography on Domination at the end of this volume. In this section entitled New Models we present a small sample of some of the newer domination problems currently being studied. The paper “Dominating cliques in graphs” by Cozzens and Kelleher, studies the existence of families of graphs which contain a complete subgraph whose vertices form a dominating set. They present several forbidden subgraph conditions which are sufficient to imply the existence of dominating cliques and they present a polynomial algorithm for finding a dominating clique for a certain class of graphs. The paper “Covering all cliques of a graph” by Tuza considers a different kind of domination, in which one seeks a minimum set of vertices which dominates all cliques (i.e. maximal complete subgraphs) of a graph. The paper by Brigham and Dutton entitled “Factor domination in graphs” considers, among other things, the general problem of finding a minimum set of vertices which is a dominating set of every subgraph in a set of edge-disjoint subgraphs, say G,, G2, . . . , G,, whose union is a given graph G.

1. Introduction

7

The final paper in this section “The least point covering and domination numbers of a graph” by Sampathkumar is one of many papers in which one imposes additional conditions on a dominating set, e.g. the dominating set must induce a connected subgraph (connected domination), a complete subgraph (dominating clique), or a totally-disconnected graph (independent domination). In Sampathkumar’s paper the domination number of the subgraph induced by the dominating set must be minimized.

5. Algorithmic

The algorithmic study of domination has exploded onto the scene even more suddenly than the theoretical study of domination. Nearly 100 papers containing domination algorithms o r complexity results have been published in the last 10 years; we add another eight papers in this section. Perhaps the first domination algorithm was an attempt by Daykin and Ng in 1966 to compute the domination number of an arbitrary tree; we say ‘attempt’ because their algorithm seems to have an error that cannot be easily corrected. Cockayne, Goodman and Hedetniemi apparently constructed the first domination algorithm for trees in 1975 and, at about the same time, David Johnson constructed the first [unpublished] proof that the Domination problem for arbitrary graphs is NP-complete. Since then domination algorithms, of ever increasing sophistication, have been published at a steady rate. We are pleased to present an excellent collection of algorithmic papers on domination in this section. The first paper by Corneil and Stewart entitled “Dominating sets in perfect graphs” presents both a brief survey of algorithmic results on domination and a discussion of the dynamic-programming-style technique that is commonly used in designing domination algorithms, especially as they are applied to the family of perfect graphs. The paper “Unit disk graphs” by Clark, Colbourn and Johnson discusses the algorithmic complexity of such problems as domination, independent domination and connected domination, and several other problems, on the intersection graphs of equal size circles in the plane. We think this paper is especially significant since it contains the result that the Domination problem for grid graphs, a subclass of unit disk graphs, is NP-complete. The family of grid graphs includes arbitrary subgraphs of grids as well. As far as we know, however, the complexity of the Domination problem on arbitrary m x n complete grids is still not known. The paper “Permutation graphs: Connected domination and Steiner trees” by Colbourn and Stewart considers a third class of graphs. To-date a nice variety of NP-complete problems have been shown to have polynomial solutions when

8

S. T. Hedetnierni, R. C. Laskar

restricted to permutation graphs. To this collection of problems we can now add connected domination. Almost as if by coincidence, it seems, we received the paper on “The discipline number of a graph” by Chvhtal and Cook, soon after we had received the paper on “The bondage number of a graph” by Fink, Jacobson, Kinch and Roberts (in the theoretical section of this volume). In their paper, Chvhtal and Cook address the question of the computational complexity of the bondage number and show, among other things that it can be formulated as an integer linear program. This paper also provides an example of the relatively recent study of fractional (i.e. real-valued) parameters of graphs. These are the values obtained by real relaxations of the integer linear programs corresponding to various graphical parameters like domination, matching, covering and independence. Our next paper, “Best location of service centers in a tree-like network under budget constraints” by McHugh and Perl, provides both a nice applications perspective on domination and an illustration of the many papers that have been published on the topic of centrality in graphs. It also provides an example of a pseudo-polynomial domination algorithm and another example of the dynamic programming technique applied to domination problems. The paper “Dominating cycles in Halin graphs” by Skowronska and Syslo, discusses both a fourth class of graphs on which polynomial time domination algorithms can be constructed, and the notion of a dominating cycle, i.e. a cycle C in a graph such that every vertex not in C lies at distance at most one from some vertex in C . The following paper, “Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs” by Kratsch is an algorithmic mate of the paper by Cozzens and Kelleher on dominating cliques (in the new models section of this volume). It discusses two more classes of graphs that permit polynomial domination algorithms, in this case, for finding dominating cliques of minimum size. We conclude the algorithmic section of this volume with a paper “On minimum dominating sets with minimum intersection” by Grinstead and Slater, which is a good representative of the fast developing area of polynomial, and even linear, algorithms on partial k-trees. Grinstead and Slater introduce a difficult, new type of problem, prove that it is in general NP-complete, and give a linear time solution when restricted to trees. This solution also uses a dynamic programmingstyle approach and a methodology created by Wimer in his 1987 Ph.D. Thesis.

Acknowledgements There can be little doubt that this special issue really has many editors; many people have contributed their time and energy to produce this collection of papers and have in the process left their mark on what is contained herein. We

1. Introduction

9

want to thank them all, the contributors, the referees, our assistants, every one (in alphabetical order). Finally, we want to thank Peter Hammer for inviting us to compile such a volume on ‘Topics on Domination’: Barkauskas, A.; Bascci, B.; Berge, C.; Boland, J.; BollobBs, B.; Brigham, R.C.; Chandrasekharan, N.; Cheston, G.; ChvAtal, V.; Clark, B.N.; Cockayne, E.J.; Colbourn, C.J.; Cook, W.J.; Corneil, D.G.; Cozzens, M.B.; Duchet, P.; Dutton, R.; Fink, J.F.; Grinstead, C.M.; Grinstead, D.L.; Hahne, B.; Harary, F.; Hare, E.O.; Hare, W.R.; Jacobson, M.S.; Johnson, D.S.; Kelleher, L.L.; Kinch, L.F.; Kratsch, D.; Livingston, M.; Majurndar, A.; McHugh, J.; Mynhardt, C.M.; Peters, K.; Proskurowski, A.; Rall, D.; Ringeisen, R.D.; Roberts, J.; Sampathkumar, E.; Skowronska, M.; Slater, P.J.; Stewart, L.K.; Suffel, C.L.; Summer, D.P.; Syslo, M.; Tuza, Z.; Vanstone, D.; Wimer, T.; Zelinka, B. This work was partly sponsored by the Office of Naval Research for the University Research Initiative Program, Contract No. N00014-86-K-0693.

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Part 11. Theoretical

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Discrete Mathematics 86 (1990) 13-20 North-Holland

13

CHESSBOARD DOMINATION PROBLEMS E .J . COCKAYNE Uniuersiv of Victoria,Victoria,B.C . , Canada

Received 2 December 1988 A graph may be formed from an n X n chessboard by taking the squares as the vertices and two vertices are adjacent if a chess piece situated on one square covers the other. In this paper we survey some recent results concerning domination parameters for certain graphs constructed in this way.

1. Introduction The classical problems of covering chessboards with the minimum number of chess pieces were important in motivating the revival of the study of dominating sets in graphs, which commenced in the early 1970’s. These problems certainly date back to de Jaenisch [7] and have been mentioned in the literature frequently since that time (see e.g. [2,8, 131). A graph P, may be formed from an n x n chessboard and a chess piece P by taking the n2 squares of the board as vertices and two vertices are adjacent if piece P situated at one of the squares is able to move directly to the other. For example the Queen’s graph Q, has the n2 squares as vertices and squares are adjacent if they are on the same line (row, column or diagonal). In this paper we survey recent results which involve various domination parameters for graphs which are constructed in this way. Outlines of some of the proofs are given, although most appear elsewhere.

2. Domination of the queens’ graph 2.1. An upper bound for the domination number of the queens’ graph The domination number y ( G ) (independent domination number i ( G ) of a graph G = (V, E ) is the smallest cardinality of a subset (independent subset) D of V such that each vertex of V - D is adjacent to at least one vertex of D. Obviously y ( G ) S i ( G ) for any graph G. The determination of y(Q,), which is the minimum number of queens required to cover the entire n x n chessboard, is perhaps the best known chessboard covering problem. The following experimental values of y(Q,) for n 6 17 are due mainly to Kraitchik (see [8]). We have corrected (by computer) values for n = 5, 6, 7. 0012-365X/90/$03.50 0 1990-Elsevier Science Publishers B.V. (North-Holland)

E . J . Cockayne

Fig. 1 . Minimum dominating sets of Q x with 5 queens.

n : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 y ( Q , ) : 1 1 1 2 3 3 4 5 5

5

5

6

7

8

9

9

9

Two optimal queen coverings for the 8 x 8 chessboard are depicted in Fig. 1. Welch [15] has established an upper bound for y(Q,). Theorem 1 (Welch). Let n

= 3q

+ r where 0 G r < 3.

Then y(Qn)

2q

+r

Proof. We first describe the placement of queens which shows that y ( Q n ) s 2n/3, for the case n = 3q. The n x n board is divided into 9 q X q sub-boards (numbered 1 through 9 in Fig. 2). Queens are placed on the main diagonal of sub-board 3 and on the diagonal immediately above the main diagonal of sub-board 7. Finally a queen is placed in the bottom left hand corner of sub-board 7. It is easily seen that these 2q queens cover the entire n x n board. If n = 3q r, where I = 1 or 2, then consider the configuration of Fig. 2 augmented with r extra rows (cols) added on the bottom (right) and place extra queen(s) at position(s) ((3q + i, 3q + i) i = 1, r } . This covering by 2q r queens completes the proof. 0

+

+

I

3

Fig. 2. Example to show y ( Q 3 , )

5

29.

15

Chessboard domination problems

2.2. An upper bound for the independent domination number of the queens’ graph In this section we consider i(Q,), the minimum number of queens which cover the entire n X n board, with the additional requirement that the queens do not cover each other. Spencer and Cockayne [6] have established the following upper bound for i(Q,).

Theorem 2 (Spencer and Cockayne). For any n, i(Q,) < 0.70%

+ 0.895.

Idea of Proof. Consider an infinite square chessboard. A queen placed on any square x covers the infinite set of squares which are collinear with x and completely covers the 3 x 3 board Bl which surrounds square x (see Fig. 3). The placement procedure is then continued iteratively. For each n > 1, a completely covered chessboard B, is found as follows. Let A l = { x } . Four queens are symmetrically placed on the set X , of squares. These four squares are not covered by the queens of B,-, and lie immediately outside the board B,-,. The new board B, is the largest square chessboard symmetrically containing B n p 1 ,which is completely covered by queens placed on the set of squares A , = A , - , U X,. The construction implies that A,, is an independent vertex subset of B,. The 9 x 9 board B2 and the sets X z and X 3 are depicted in Fig. 3. In the diagram small dots denote squares covered by A2 = { x } U X1. The size of the board B, depends on the following pair of recursively defined integer functions: f ( 1 ) = g(1) = 1, f (n 1) = the least integer greater than f (n) which does not equal f ( k ) 2 g ( k ) for any k S n,

+

+

............. 1

. . .

1

.

1

I

I

I

.....

1

.

1

1

I

..

Fig. 3. Illustration of the construction used in proof of Theorem 2.

E.J . Cockayne

16

and

g ( n + 1) = the least integer greater than g ( n ) which does not equal f(k) + g ( k ) for any k G n.

In fact the size of B, is 2(f(n) + g ( n ) ) - 1 and we have 4n - 3. (1) The remainder of the proof is a lengthy estimation of f ( n ) , g ( n ) and the details may be found in [6]. i(Qqf(n)tg(n))-~)

2.3. A lower bound for y(Q,)

Theorem 3 (Spencer [14]). For any n, y(Q,)

3 (n -

1)/2.

Proof. Consider a covering of the n x n board using y = y(Q,) queens. Suppose that the rows and columns are sequentially labelled 1, . . . ,n from top to bottom and feft to right respectively. A row or column is said to be occupied if it contains a queen. Let column a, ( b ) be the left most (right most) unoccupied column and let row c ( d ) be the unoccupied row closest to the top (bottom). Further we set 6, = b - a and 6, = d - c and assume without lost of generality that 6 , a a2. Consider the sets S,and S, of squares in columns a and b respectively, which lie between rows c and c + 6, - 1 inclusive and let S = S, U S,. Since 6 , s d,, no diagonal intersects both S1 and S,. Hence every queen diagonally dominates at most two squares of S (i.e. at most one per diagonal). Further queens situated above row c or below row c 6, - 1 d o not dominate squares of S by row or column. By definition of c , there are at least c - 1 queens above row c. Each row below row d is occupied and d = c + 8, d c + 6,. Therefore all the n - c - 6, rows below row c 6, are occupied. Hence there are at least n - c - 6, queens below row c + 6, - 1. It follows that at least ( c - 1) ( n - c - 6,) = n - 6, - 1 queens dominate at most 2 squares of S. The remaining queens of which there are at most y - ( n - 6, - l ) , may cover at most 4 squares of S. Since all the 26, squares of S must be dominated we have

+

+

+

2(n - 6, - 1) + 4(y - ( n - 6, - 1)) 3 2d1, which gives y 3 ( n - 1)/2 as required.

0

2.4. The diagonal queens’ domination problem Inspection of Fig. 1 shows that one can cover the 8 X 8 board with a minimum number of queens by restricting the placement of queens to the main diagonal, hence the following definition: diag(n) = minimum number of queens which may be placed on the main diagonal of an n X n chessboard and which dominate the entire board.

Chessboard domination problem

17

Cockayne and Hedetniemi [ 5 ] have related diag(n) to the following difficult and well-studied number-theoretic function. Let r3(n) be the largest cardinality of a subset of N = (1, . . . ,n} which contains no 3-term arithmetic progression.

Theorem 4 (Cockayne and Hedetniemi). For any n,

(13.

diag(n) = n - r3

Indication of Proof. This theorem is proved by way of the following lemma. Define K G N to be diagonal dominating if queens placed in the positions {(k, k): k E K} on the main diagonal cover the entire board. A subset of N is called midpoint-free if it contains no 3-term arithmetic progression. Finally a subset of N is called even-summed if all its elements have the same parity. Lemma 1. K N is diagonal dominating if and only if N - K is midpoint-free and even-summed. Theorem 3 is easily deduced from this lemma. Several estimates for r3(n)have appeared in the literature [l,9-12] and Roth [ l l ] has proved (r3(n)/n)= 0. The latter result implies the following corollary.

Corollary 1. (diag(n)/n)

= 1.

Using Theorem 1 we deduce the following:

Corollary 2. For n sufficiently large, y(Q,) < diag(n). 2.5. Domination of Q, b y queens in a single column Denote by col(n), the minimum number of queens on any single column which are required to dominate the entire n x n chessboard. (It is easy to see that a column nearest the centre is as good as any other.) Cockayne, Gamble and Shepherd [3] have related col(n) to the same function r3(n)mentioned in Section 2.4. Let

and

E.J . Cockayne

18

Theorem 5 (Cockayne, Gamble and Shepherd). col(n) = min[A(n), B ( n ) ] ( n 2 2). The proof of this theorem is highly technical and we give no details here. However, one may deduce from the proof:

Corollary 3. For any n, col(n) 2 diag(n). 2.6. Unsolved problems concerning queens

Problem 1. Is y(Q,,)

y(Q,+J for all n?

We now refer to equations (2) and (3) of Section 2.5. The computer has determined that A(n) 3 B ( n ) for n < 150 and we therefore ask:

Problem 2. Is A(n) 3 B ( n ) for all n? Finally, could (3) be simplified by evaluation of the maximum?

Problem 3. Find

3. Domination parameters for the bishops’ graph The bishops’ graph D,, has the n2 squares for vertices and two squares are adjacent if they lie on the same diagonal. In (41, Cockayne, Gamble and Shepherd have calculated three parameters for D,. 3.1. Domination and independent domination numbers

Theorem 6 (Cockayne, Gamble and Shepherd). For any n, y(D,) = i(D,,) = n. Indication of Proof. The set of squares of a nearest column to the centre is an independent dominating set of D, hence y(D,) c i(L>,) n and it remains to show y(D,) > n. The North-West to South-East running diagonals are labelled sequentially 1, . . . , 2n - 1 in the North-East direction, and w (and 6 ) are the labels of the white (black) diagonal closest to the main diagonal which has no bishop. Without

Chessboard domination problems

19

losing generality, one may assume {w, b } G (1, . . . , n}. Diagonal w has w squares and these must be dominated. Further by definition of w , there are bishops on each diagonal strictly between w and 2n - w. Hence n, the number of white bishops in any dominating set satisfies

n, Similarly,

2 max(w,

n - w - 1).

nb 2 max(b, n

- b - 1).

The result is simply deduced from (4) and (5). 0 3.2. The total domination number

The total domination number t ( G ) of a graph G = (V, E) is the minimum cardinality of a subset T of vertices, such that each vertex of V is adjacent to at least one vertex of T.

Theorem 7 (Cockayne, Gamble and Shepherd). For any n 2 3, t(D,,) = 2 [$(n1>1. Outline of Proof. D,,is the disjoint union of the white bishops graph W,, and the black bishops graph B,,. We summarize only the proof that t(B,,)= [$(n- 1)1 for n even. Notice that a total bishop dominating set of B, is precisely a total rook dominating set of the diamond shaped chessboard S, which has n rows and n - 1 columns. We exhibit S, in Fig. 4. For ease of presentation, we use rooks, rows and columns, rather than bishops and diagonals. 0 Lemma 1. For any n, S,, has a minimum total rook dominating set with the rooks on consecutive rows and columns. Proof. See [4].

0

It follows from Lemma 1 that some minimum total rook dominating set of S,, may be used to construct a total rook dominating set of an m x p rectangular board with property REL, i.e. a rook on every line (row or column). It is shown

Fig. 4. The diamond-shaped chessboard S,

E.J. Cockayne

20

that such a board satisfies m + p an - 1 and hence, if s(m, p ) =minimum number of rooks in an REL total dominating set of an m x p board, we have t(B,,)=

min

s(m,p).

m + p a n -1

Lemma 2.

Proof. By establishing and solving a recurrence for s ( m , p ) . For details see [41. One may deduce from (6) and (7) that t(B,,)3 [ $ ( n - 1)1 and the final part of the proof exhibits a total rook dominating set of S,, with [{(n - 1)1 rooks. This completes the outline of the proof of Theorem 6.

Acknowledgement The author gratefully acknowledges the research support of the Natural Sciences and Engineering Research Council of Canada grant A7544.

References F.A. Behrend, On sets of integers which contain no 3 terms in arithmetic progression. Proc. Nat. Acad. Sci. U.S.A. 32 (1946) 331-332. C. Berge, Theory of Graphs and its Applications (Methuen, London, 1962) 40-51. E.J. Cockayne, B. Gamble and B. Shepherd, Domination of chessboards by queens on a column, Ars. Combin., to appear. E.J. Cockayne, B. Gamble and B. Shepherd, Domination parameters for the bishops graph. Discrete Math., to appear. E.J. Cockayne and S.T. Hedetniemi, A note on the diagonal queens domination problem, J . Combin. Theory Ser. A 41 (1986). E.J. Cockayne and P.H. Spencer, An upper bound for the independent domination number of the queens graph, submitted. C.F. De Jaenisch, Applications de I’Analyse Mathematique an Jeu des Echecs (Petrograd, 1862). R.K. Guy, Unsolved Problems in Number Theory, Vol. 1 (Springer, Berlin, 1981). L. Moser, On non-averaging sets of integers, Canad. J . Math. 5 (1953). J. Riddell, On sets of numbers containing no 1 terms in arithmetic progression, Nieuw Arch. Wisk. (3) XVII,(1969) 204-209. K.F. Roth, Sur quelques ensembles d’entiers, C.R. Acad. Sci. Paris 234 (1952) 388-390. K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953) 104-109. W.W. Rouse Ball, Mathematical Recreations and Problems of Past and Present Times (Macmillan. London. 1892). [ 141 P.H. Spencer, private communication 1151 L. Welch, private communication.

21

Discrete Mathematics 86 (1991) 21-26 North-Holland

ON THE QUEEN DOMINATION PROBLEM* Charles M. GRINSTEAD, Bruce HAHNE and David VAN STONE Deparfment of Mathematics, Swarthmore College, Swarthmore, PA 19081, USA Received 2 December 1988

A configuration of queens on an rn X m chessboard is said to dominate the board if every square either contains a queen or is attacked by a queen. The configuration is said to be non-attucking if no queen attacks another queen. Let f ( m ) and g ( m ) equal the minimum number of queens and the minimum number of non-attacking queens, respectively, needed to dominate an m x m chessboard. We prove that: (1) f(m)

< grn + 0(1), and

(2) g ( m ) s $n

+ O(1).

These are the best upper bounds known at the present time for these functions.

Introduction In this paper, we study the problems of finding, on an m x m chessboard, the smallest configurations of queens and non-attacking queens, respectively, with the property that every square of the board either contains a queen or is sttacked by a queen. We define f ( m ) and g ( m ) to be the minimum sizes of these two types of configurations. Clearly, since any one queen can attack at most 4m - 3 squares, both f(m)and g(m) are bounded below by am. It is also obvious that each function is bounded above by m. Until quite recently, no non-trivial lower or upper bounds for these two functions were known, in spite of the fact that the problems of determining f ( m ) and g ( m ) are classical recreational problems dating as far back as the 1850's. In [l], Spencer shows that f(m)2 $(m- 1). Of course, the same lower bound holds for g(rn), since f(m)s g(rn). Also Welch [I] has provided a construction which shows that f ( m )s $rn + O(1). In addition, Cockayne and Spencer have shown that g ( m ) s 0.705m 0(1)(see [l]). In this paper we give explicit configurations which show that if rn = 46n 26, then f ( m )< 28n 16 = Zrn - 8, and if m = 9n - 1, then g ( m ) $m - 4. We note that these configurations require restrictions on m, but the set of allowable values for m in each case is an arithmetic progression. For all other values of m, we can create a dominating configuration as follows: We start with a configuration of the largest allowable board size less than m. We add queens, one at a time, always placing them on currently undominated squares, until no

+

+

+

*All three authors were partially supported by NSF Grant DMS-8406451, a grant from the Swarthmore College Research Support Fund, and a Greenwall Grant. 0012-365X/90/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland)

22

C.M . Grinsteud

er ul

undominated squares remain. We thus obtain a dominating configuration for the m x m board. This new configuration preserves the non-attacking property required by the second problem. It is easy to check that this procedure adds at most one queen for each new row and each new column. Therefore, the number of added queens is never more than a constant. This will show that f ( m ) s Em + 0 ( 1 ) and g ( m )C 3m + O(1).

Descriptions of the configurations The standard integer lattice in R 2 defines an infinite tiling of R 2 with unit squares. An m x m chessboard is simply an m x m subset of squares from this tiling. We note that the centers of the squares, which are useful in identifying squares, have coordinates both of which are odd multiples of It is also helpful to define rows and columns to be sets of squares whose centers lie on horizontal and vertical lines, respectively. Frequently, a row will be described by its row number, which is the y-coordinate shared by all of the centers of the squares in a row. The column number of a column is defined in a similar manner. Positive and negative diagonals are sets of squares whose centers lie on lines of slope 1 and -1, respectively. The diagonal number of a diagonal is the y-intercept of the line containing the centers of the squares on the diagonal. (We note that for positive diagonals, this is the difference between the y - and x-coordinates of the center of any square on the diagonal, and for negative diagonals, this is the sum of these coordinates.) Fig. 1 shows an example (when n = 2) of a general dominating configuration of queens on an m x m chessboard, where m = 46n + 26. We will describe the positions of the queens in the upper half of Fig. 1, and stipulate that a rotation by n radians should take the configuration to itself. The four lines of queens labelled A l , B , , C1, and D, have the common property that their centers lie on lines of slope 4. The number of queens in each of the lines C , and D,is denoted by the parameter n. This parameter is allowed to take on any positive integer as a value, and each positive integer gives a configuration. We will have completely described this configuration if we give the positions of the first and the last queens in each of these four lines. These positions are given in the following table:

4.

Name of line A1 B1 CI DI

Coordinates of last queen

Coordinates of first queen

(-Ion - 4 , -2n

- 2)

(-6n - ;,6n + $) (6n + 9, 10n y ) (8n + 9, 22n + $)

+

Number of queens

(6n+$, 6 n + z ) (2n 4, 10n y ) (8n $, l l n 9) (10n $, 23n + 9 )

+ + +

+ +

8n+5 4n + 3

n n

The number of queens in this configuration is 28n + 16. In the next section, we show that this configuration dominates a board with 46n 26 squares on a side.

+

23

On the queen domination problem

Fig. 1. A configuration of queens which shows that f ( m ) G grn + O(1) Note that the grid shown has basic squares of size 4 X 4, except near the origin, where the grid size is 1 x 1 .

Fig. 2 shows an example (when n = 4) of a general configuration of queens on an m x m chessboard, where m = 9n - 1. The origin is at the lower left-hand corner of the figure. In this configuration, there are two lines of queens. These lines have the common property that their centers lie on lines of slope 4. There are 3n queens on the lower line and (3n - 1) queens on the upper line. The L4

L3

.L 1

Fig. 2. A non-attacking configuration which shows that g ( m ) s

+

24

C .M . Grinstead et al.

z,

endpoints of the lower line are the squares (4, f)and (6n - 3n - i), and the enpoints of the upper line are the squares (3, 3n + f)and (6n - 3, 6n - 3). The number of queens in this configuration is (6n - 1). In ihe next section, we show that this configuration is a non-attacking, dominating configuration.

Proofs

Theorem 1. We have f(m)C grn + O(1). Proof. To prove this theorem, we will show that a configuration of the type shown in Fig. 1, with parameter n, dominates a board with center at the origin and with (46n 26) squares on a side. Since this configuration is symmetric under a rotation by n radians, we need only show that the upper half of this board is dominated. To aid in the proof, we will define five regions of the upper half-plane. These regions, together with their bounding lines, are shown, for the case n = 2, in Fig. 3. The equations of the bounding lines are given the following table:

+

Line

Equation

L,

y=o y=lln+6 y - x = - 12n - 7 y-x=12n+7 x = 10n + 5 x = - 10n - 5 y + x = 12n + 7 y + X = -12n - 7

L2 L3 L4 L5 L6

L7 LR

Region I consists of all squares whose centers lie between lines L, and Lz. Region I1 consists of all squares whose centers lie on or between lines L3 and L4. Region I11 consists of all squares whose centers lie between lines L , and L,. Region IV consists of all squares whose centers lie on or between lines L7 and L g . Finally, Region V consists of all squares whose centers lie above the L4 and to the right of line L5, together with all squares whose centers lie above line L7 and to the left of line L6. All squares in Region I lie on rows containing queens in lines A l , B 1 and C,. This can easily be checked by noting that the y-coordinates of these queens run from (-2n - +) to ( I l n y). The squares in Region I1 all lie on positive diagonals containing queens. This can be checked by noting that the difference of the coordinates of the queens in lines BZ,A*, A, and B, include all integers between (-12n - 7) and (12n + 7) inclusive.

+

25

Fig. 3.

The squares in Region I11 all lie on columns containing queens. The queens in lines A l , C , , and D1dominate all columns with numbers which are congruent to $ (mod 2) and which are between (-lOn - and (lOn inclusive. The queens in lines A*, Cz, and D2dominate all columns with numbers which are congruent to 4 (mod 2) and which are between (-10n - $) and (10n + 4) inclusive. The squares in Region IV all lie on negative diagonals containing queens. The queens in line A , dominate all negative diagonals with numbers which are congruent to 1 (mod 3) and which are between (-12n - 5) and (12n + 7) inclusive. The queens in line A2 dominate all negative diagonals with numbers which are congruent to 2 (mod 3) and which are between (-12n - 7) and (12n 5) inclusive. Finally, the queens in lines B , and B2 dominate all negative diagonals with numbers which are congruent to 0 (mod 3) and which are between (-12n - 6) and (12n + 6) inclusive. The squares in Region V are not all dominated by queens. However, the queens in line D, dominate all squares in Region V with centers whose y-coordinates are less than (23n 13). The point in Region V with coordinates (10n + 9, 23n + 9 ) is not attacked by any queen. This point is the point in the upper half-plane with the smallest y-coordinate which is not attacked. The point with the smallest x-coordinate which is not attacked is the point (23n + 4, l l n + q). (This point is above line L2 and to the right of line L3.)This means that in the upper half-plane, all squares with centers (x, y ) , where 1x1 < 23n 14 and Iyl 23n + 13, are attacked by queens. Thus this configuration domainates a rectangular board with dimensions

s)

+

+

+

+ z)

C . M . Grinstead et al.

26

+

+

46n 26 and 46n 28, so it dominates a square board with 46n + 26 squares on a side. Thus, if m = 46n 26, then an m x m board can be dominated with 28n + 16 = g m - & queens. This statment, combined with the discussion in the introduction, shows that f ( m )< grn O(1). 0

+

+

Theorem 2. We have g(m)S f m + O(1). Proof. To prove this theorem, we will show that a configuration of the type shown in Fig. 2, with parameter n, dominates a board with its lower left-hand corner at the origin and with 9n - 1 squares on a side, and that this configuration is non-attacking. The queens in this configuration dominate all squares whose centers lie between the x-axis and the line L1 (y = 6n - $), and all squares whose centers lie in columns between the y-axis and the line L4 (x = 6n - 1). Furthermore, they dominate all positive diagonals between and including lines L2 and LJ, which have diagonal numbers (-3n + 1) and (3n - 1) respectively. Thus the center of the square in the first quadrant which is unattacked and which has the smallest x-coordinate is (6n - 4, 9n The center of the square in the first quadrant which is unattacked and which has the smallest y-coordinate is the point (9n - 4, 6n - 4). Thus, all squares in the first quadrant whose centers have both corrdinates less than (9n - 4) are attacked. Therefore, this configuration dominates a board with (9n - 1) squares on a side. It is clear that no two queens in the same line attack each other. In order for two queens in different lines to attack each other, they must do so on a negative diagonal. This will happen if and only if there are two queens whose coordinates sum to the same number. The queens in the lower line have coordinate sum congruent to 1 (mod3) and the queens in the upper line have coordinate sum congruent to 2 (mod 3 ) . Therefore this is a non-attacking configuration. Thus, if m = ( 9 n - l), an m X M board can be dominated with 6n - 1 = ( $ ) m- 5 queens. This statement, combined with the discussion in the introduction, shows that g(m)S #m O(1). 0

4).

+

References [I] E. Cockayne, Chessboard domination problems, Discrete Math. 86 (this Vol.) (1990) 13-20.

Discrete Mathematics 86 (1990) 27-31 North-Holland

27

RECENT PROBLEMS AND RESULTS ABOUT KERNELS IN DIRECTED GRAPHS* C. BERGE** and P. DUCHET Universite‘ de Paris, Centre National de Recherche Scientifique, Paris, France Received 2 December 1988

In Section 1, we survey the existence theorems for a kernel; in Section 2, we discuss a new conjecture which could constitute a bridge between the kernel problems and the perfect graph conjecture. In fact, we believe that a graph is ‘quasi-perfect’ if and only if it is perfect.

1. Kernel-perfect graphs Let G be a directed graph. Its vertex-set will be denoted by X,and its arcs (or ‘directed edges’) are a subset of the Cartesian product X x X . A kernel of G is a subset S of X which is ‘stable’ (independent, i.e.: a vertex in S has no successor in S) and ‘absorbant’ (dominating, i.e. a vertex not in S has a successor in S). This concept has found many applications, for instance in cooperative n-person games, in Nim-type games (cf. [l]), in logic (cf. [2]), etc. So, the main question is: Which structural properties of a graph imply the existence of a kernel? By ‘subgraph’, we shall always mean ‘induced subgraph’. A graph G whose all subgraphs have kernels is called kernel-perfect. Otherwise, G is kernel-imperfect. The classical results (see [l]) are: (1) A symmetric graph is kernel-perfect (trivial); (2) A transitive graph is kernel-perfect, and all kernels have the same cardinality (Konig); ( 3 ) A graph without circuits is kernel-perfect, and its kernel is unique (von Neumann) ; (4) A graph without odd circuits is kernel-perfect (Richardson). Many extensions of Richardson’s Theorem have been found in the last ten years. An easy one is:

Proposition 1.1. Let G be a graph such that every odd circuit has all its arcs belonging to pairs of parallel arcs (‘double-edges’). Then G is kernel-perfect. *Reprinted with permission from Ringeisen, Richard D., and Roberts, Fred S. (1988), Applications of Discrete Mathematics, Proceedings of the Third Conference on Discrete Mathematics, Clemson University, Clemson, South Carolina, May 14-16, 1986, SIAM Publications, Philadelphia, PA. Copyright 1988 Society for Industrial and Applied Mathematics. * * The first author gratefully acknowledges the partial support of the Air Force Office of Scientific Research under grant number AFOSR-0271 to Rutgers University. 0012-365X/90/$03.50

01990-

Elsevier Science Publishers B.V. (North-Holland)

28

C . Berge, P. Duchet

Proof. It suffices to show that G possesses a kernel. Let x , , x 2 , . . . ,x, be the vertices of G. Let G ' be the graph obtained from G by removing the arc (xi, xi) if i >j and ( x i , xi) belongs to a pair of parallel arcs. Clearly, G' has a kernel S'. In G , the set S' is both stable and absorbant. Hence S' is a kernel of G. Other theorems have been found recently, in particular the following: (1) If every odd circuit [ x , , x 2 , . . . , x Z k + , ,x , ] has two chordr of the type ( x i , xi+,), ( x i + , , xiCJ) then the graph is kernel-perfect (Duchet and Meyniel [ll]). ( 2 ) If every odd circuit has at least two arcs belonging to pairs of parallel arcs, then the graph is kernel-perfect (Duchet [8]). ( 3 ) If every odd circuit has two chordr whose heads are consecutive vertices of the circuit, then the graph is kernel-perfect (Neuman-Lara and Galeana-Sanchez 1131). However, it is false that a graph G such that all odd circuits have two chords is kernel-perfect (Neuman-Lara and Galeana-Sanchez [ 121). Other related results are due to Meyniel (unpublished), or to Neuman-Lara and Galeana-Sanchez (unpublished). A critical kernel-imperfect graph is a graph G without kernel such that every strict subgraph is kernel-perfect. We have:

Proposition 1.2. A critical kernel-imperfect graph is strongly connected. Proof. Otherwise, let G be a critical kernel-imperfect graph which is not strongly connected. There exists a strong component C , of G such that no arcs go from C , to X - C , ('terminal component'). Let S, be a kernel of Gc,. Consider the subgraph of G induced by C2= X - S , - {x I x E X , x has a successor in S,}. Clearly, C2 is a strict subset of X ; consequently, G,, has a kernel S,. The set S, U S, is stable, because no arc goes from S, to S2 (because S2 does not meet the terminal component C , ) , and no arc goes from S, to S, (by the definition of C , ) . Therefore, the set S, US,, which is also absorbant for G, is a kernel of G: a contradiction. 0 Remark. This proposition yields a very simple proof for the theorem of Richardson. Let G be a graph with no odd circuits which would not be kernel-perfect. Let G' be a critical kernel imperfect subgraph of G. Since G' has no odd circuits, its vertices can be colored with two colors by the following procedure: color with blue a given vector xo. Color with red every successor of blue vertex. Color with blue every successor of a red vertex. Clearly, no vertex can be colored with both colors (otherwise there would be an odd circuit). By Proposition 1.2, G' is strongly connected, and therefore all its vertices will be colored when the procedure terminates. Then, the set consisting of all blue vertices is both stable and absorbant for G': a contradiction.

Recent problem and resulis about kernels in directed graphs

29

Various examples of critical kernel-perfect graphs exist in the literature, but no structural characterization has been found so far. However, we must keep in mind the following remark:

Proposition 1.3. Let G be a kernel-perfect graph. Then every complete subgraph (‘clique’) has a vertex which is successor of all its other vertices.

2. Quasi-perfect graphs Let us recall that a simple graph G is perfect if every induced subgraph GA satisfies a(GA)= O(GA), where a ( G ) denotes the stability number of G (maximum number of independent vertices), and 0 denotes the minimum number of cliques needed to cover the vertex-set of G. The perfect graph conjecture is: G is perfect if and only if there is no induced odd cycle C2k+l (with k 2 2), and no induced Czk+](complement of a CZk+]rk 2 2 ) . Recall that an ‘orientation’ of an edge consists in replacing this edge either by an arc or by two parallel arcs in opposite directions; in a directed graph, the orientation is normal if every clique contains a vertex which is successor of all its other vertices. A simple graph G is quasi-perfect (or ‘solvable’) if every normal orientation of its edges results in a kernel-perfect directed graph. Thus, a clique K,, is a quasi-perfect graph; furthermore, we have:

Proposition 2.1 A complete directed graph has a normal orientation if and only if every circuit has at least one arc belonging to a pair of parallel arcs.

Proof. Let G be a complete directed graph with a normal orientation; then a circuit p has a vertex x which is successor of all its other vertices; therefore the arc of the circuit which is incident from x belongs to a pair of parallel arcs. Conversely, assume that G = (X, U ) is a complete (directed) graph whose circuits satisfy the condition of Proposition 2.1. We shall assume that its orientation is not normal, to obtain a contradiction. Let C be a clique of G having no kernel, and let xle C. Since {xl} is not a kernel of C, there exists a vertex x1 E C with (x2, x l ) @ U , and ( x , , x 2 ) E U. Also, there exists a vertex x j E C with ( x 3 , x 2 ) @U and ( x z , x3)e U , etc.. . . ; so we define a sequence xl, x 2 , x 3 , . . . ,xi,. . . of distinct vertices with (xi, xi+l)E U , and ( x i + l ,xi)@ U . Since the graph is finite, the sequence ( x l , x 2 , . . . ,x,) terminates with x,, and for some p < q , we have (x,, x p ) € U , (x,,, x,) @ U. Then the sequence (xp, . . . ,x,, x,,) constitutes a circuit with no arc belonging to a pair of parallel arcs. The contradiction follows. 0 Proposition 2.2. The graph C2k+l,with k

2 2, is

not quasi-perfect.

30

C. Berge, P. Duchet

Proof. The orientation c 2 k + l of as a directed circuit is a normal orientation. Since t 2 k + l has no kernel, the graph c Z k + 1 is not quasi-perfect. Proposition 2.3. The graph

C2k+l,

with k

3 2, is

not quasi-perfect.

Proof. Let [xl,x 2 , , . . , x Z k + l = x , ] be the cycle C 2 k + I . We can provide C 2 k + l with the following normal orientation: join xi and xi with two parallels arcs if J # i - 2, i - 1, i, i + 1, i + 2; join xi and x j with only one arc ( x i , xi) if j = i + 2. Clearly, this is a normal orientation of c z k + , . Furthermore, the set { x i } is not a kernel, because (xi+2,x i ) $ U ; neither is the set { x i , x i t l } because (xi+2,x i ) C$ U. Hence, the directed graph has no kernel; so the graph c , k + 1 is not quasiperfect. 0 Proposition 2.4. Every induced subgraph of a quasi-perfect graph G is quasiperfect. Proof. Let GA be the subgraph of G induced by A c X ;let GA be a normal orientation of GA. We have to show that has a kernel. Let 6 be a normal orientation of G obtained by directing every edge [x, y] as follows: If x, y E A , direct [x, y] as in GA. If x $ A , y E A , direct [x, y] from x to y. If x C$ A , y 4 A , direct [x, y ] in both directions. Clearly, this is a normal orientation. Since G is quasi-perfect, has a kernel S . Clearly, the set S nA is a kernel of GA;this achieves the proof.

eA

It follows from the Propositions 2.2, 2.3, 2.4 that a quasi-perfect graph has no induced C 2 k + l and no induced C 2 k + l and we do not know any other minimal prohibited configuration. This justifies the name of ‘quasi-perfect graph’. In the last two years, several people tried to prove the quasi-perfectness for the main classes of perfect graphs. To summarize these results, consider the following list of properties: (1) G is chordal (triangulated): every cycle has a chord; (2) G is weakly chordal (weakly triangulated) (Hayward): no induced C , , k 3 5 and no induced C k , k s 5 ; (3) G is i-triangulated (Gallai): every odd cycle has two non-crossing chords; (4) G is a parity graph (Olaru-Sachs): every odd cycle has two crossing chords; ( 5 ) G is a Meyniel graph (Meyniel): every odd cycle has two chords; (6) G is quasi-perfect; (7) G has no induced k # 2, and no induced C 2 k + , , k # 2; (8) G is perfect. It is well known (see [l]) that (1)+(2)+(8), or (1)+(3)+(5)+(8), or (4) 3 ( 5 ) 3 (8). Maffray [15] has proved that (1)+(6); it follows from Jacob [14] and Maffray [15] that (3) j (6). We do not know if (5) (6), or if (2) j (6).

+

Recent problems and results about kernels in directed graphs

31

Is it true that (8) .$ (6)? (Conjecture of Berge-Duchet [3]). Is it true that (7) j (8)? (Perfect Graph Conjecture). Is it true that (6) j (8)? (Weak form of the Perfect Graph Conjecture). Is it true that the odd circuits are the only connected kernless graphs such that the removal of any arc results in a graph with a kernel? (Conjecture of Duchet, [91) References [l] C. Berge, Graphs, North-Holland Mathematical Library, Vol. 6 (North-Holland, Amsterdam, 1985) Chapter 14. [2] C. Berge, Nouvelles extensions du noyau d‘un graphe et ses applications en theorie des jeux, Publ. Economttriques 6 (1977). [3] C. Berge and P. Duchet, ProblBme, SCminaire MSH, 1983. [4] C. Berge and A. Ramachandra Rao, A combinatorial problem in logic, Discrete Math. 17 (1977) 23-26. [S] C. Berge and P. Duchet, Perfect graphs and kernels, Bull. Inst. Math. Acad. Sinica 16 (1988) 263-274. [6] M. Blidia, Contribution ? I’btude i des noyaux dans les graphes, Thesis, Paris, 1984. [7] M. Blidia, Every parity digraph has a kernel, Combinatorica 6 (1986) 23-27. [8] M. Blidia, P. Duchet and F. Maffray, On kernel of perfect graphs, Rutcor Res. Report 4-88, 1988. (91 P. Duchet, Graphes noyaux parfaits, Ann. Discrete Math. 9 (1980) 93-101. [lo] P. Duchet and H. Meyniel, A note on kernel-critical graphs, Discrete Math. 22 (1981) 103-105. [ l l ] P. Duchet and H. Meyniel, Une gCnCralisation du thCortme de Richardson sur I’existence de noyaux dans le graphes orientts, Discrete Math. 43 (1983) 21-27. [12] H. Galeana-Sanchez and V. Neuman-Lara, A counterexample to a conjecture of Meyniel on kernel-perfect graphs, Discrete Math. 41 (1982) 105-107. [131 H. Gdleana-Sanchez and V. Neuman-Lara, On kernels and semikernels of diagraphs, Discrete Math. 48 (1984) 67-76. [14] H. Jacob, Etude thtorique du noyau d’un graphe, Thesis, Paris 6, 1979. [15] F. Maffray, Sur I’existence des noyaw dans les graphes parfaits, Thesis, Paris 6, 1984. [16] F. Maffray, On kernels in i-triangulated graphs, Discrete Math. 61 (1986) 247-251.

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Discrete Mathematics 86 (1990) 33-46 North-Holland

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CRITICAL CONCEPTS IN DOMINATION David P. SUMNER Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA Received 2 December 1988

Introduction Graphs which are minimal or critical with respect to a given property frequently play an important role in the investigation of that property. Not only are such graphs of considerable interest in their own right, but also a knowledge of their structure often aids in the development of the general theory. In particular, when investigating any finite structure, a great number of results are proven by induction. Consequently it is desirable to learn as much as possible about those graphs that are critical with respect to a given property so as to aid and abet such investigations. In this paper we will survey two such concepts. The first, domination critical graphs, deals with those graphs that are critical in the sense that their domination number drops when any missing edge is added. The other, domination perfect graphs, is analogous to the idea of perfect graphs in the chromatic sense, and deals with those graphs that have all their induced subgraphs satisfying y ( G ) = i ( G ) where i ( G ) is the independent domination number of G. All our graphs will be finite, undirected and without loops or multiple edges. We will write x I y to indicate that x is adjacent to y and x Y y when x is not adjacent to y. We will denote the neighborhood of a vertex v by N ( v ) . The minimum degree and maximum degree of G will be indicated by 6 ( G ) and A(G) respectively. We will use n to represent the number of vertices in a graph and q to denote the number of edges.

Domination critical graphs A graph is said to be dominution critical if for every edge e 4 E ( G ) , y(G e) < y ( G ) . If G is a domination critical graph with y ( G ) = k we will say G is k-domination critical or, in the context of this paper, just k-critical. The 1-critical graphs are (vacuously) K , for n 3 1. It is also a simple matter to characterize the 2-critical graphs as was shown in Sumner-Blitch [121.

+

0012-365X/90/$03.50 0 1990 -Elsevier Science Publishers B.V. (North-Holland)

34

D . P . Sumner

Theorem 1. A graph G is a 2-domination critical iff G = star K , , n , n 2 1. Here G denotes the complement of G .

u Hiwhere each Hiis a

A similar concept was studied by Bauer, Harary, Niemenin and Suffel[3]. They defined a graph to be critical if the deletion of any edge increased the domination number; i.e. y ( G ) < y ( G - e) for every e E E ( G ) . These graphs turn out to be precisely the complements of the 2-domination-critical graphs defined here. Hence Theorem 1 above also provides a characterization of these graphs. Walikar and Acharya [1] also characterized this class of graphs. For k > 2, the structure of the k-critical graphs is more complex. We will concentrate here primarily on the concept of 3-critical graphs. Many of the difficulties in understanding critical graphs are already present at this level. The disconnected 3-critical graphs are easily characterized and so we will generally assume that our 3-critical graphs are connected. Theorem 2. G is a disconnected 3-critical graph iff G = A U B where either A is trivial and B is any 2-critical graph or A is complete and B is a complete graph minus a 1-factor.

The characterization in Theorem 2 involves the concept of a complete graph with a 1-factor removed. This type of graph occurs frequently in the study of 3-critical graphs. This phenomenon may be better understood when it is pointed out that these graphs are precisely those with the property that y ( G ) = 2, and y ( G - v ) = 1 for every vertex v E V ( G ) (i.e. the 2-vertex-critical graphs). It is possible to generate by computer a large number of 3-critical graphs at random. We have done this and many conjectures appearing in this paper are supported by heuristic evidence complied from these examples. Since the procedure used is not truly random in the sense of producing each 3-critical graph on n vertices with equal likelihood, we will explain how the examples are generated. Algorithm to generate a connected random 3-critical graph. For a given number of vertices n : Generate a random permutation n of 1 , 2 , 3 , . . . , n(n - 1)/2. Associate with each integer i = 1 , 2, 3 , . . . , n(n - 1)/2 an edge e ( i ) . Pick a tree T on n vertices at ‘random’ subject only to y ( T ) > 2. For each value i = 1 to n ( n - 1)/2, add the edge n ( e ( i ) ) to T if the domination number does not drop to 2 as a consequence. At the completion of this loop, the resulting graph must be 3-critical.

While not every connected 3-critical graph on n vertices is equally likely to be produced by this procedure, every connected 3-critical graph can result, and after a long run, it is reasonable to expect a good collection of examples for study. This seems to hold up in practice. Of course essentially the same procedure can also be used to generate random examples of k-critical graphs for k > 3.

Critical concepts in domination

35

Fig. 1 shows a selection of examples of 3-critical graphs of small order (at most 9 vertices).

The fundamentals. If x , y are non-adjacent vertices of the 3-critical graph G, then adding the edge xy to G must result in a graph having domination number 2. Consequently, there must be a vertex a in G such that either { x , a } dominates G - y or else { y , a } dominates G - x . If { x , a } dominates G - y , we write [x, a ] M Y . This relation imposes a natural ordering on the complement G of G. We simply orient the edge xy of G from x to y in case [ x , a ] ~ y Note . that it is possible to have both the arcs ( x , y ) and ( y , x ) for a given edge in G.We refer to this orientation of G as the domination ordering on G. The following lemmas, established in Sumner-Blitch [12] have proven to be of considerable utility in dealing with 3-critical graphs.

Lemma 1. If G is a 3-critical graphs and S is an independent set of r 2 4 vertices in G, then the elements of S can be ordered us a,, a2, a3, . . . ,a, in such a way that there exists a path x , , x 2 , x 3 , . . . , xrW1with each xi4 S, and such that f o r each i = 1, 2, . . . , r - 1, [ai,x i ] -ai+l. The proof of this lemma appears in [12] and is a direct consequence of considering the domination ordering on G restricted to S, and noting that since this ordering on the edges in contains a tournament, there is a spanning directed path for

s.

s

Lemma 2. If v is a cutpoint of the 3-critical graph G , then v is adjacent to an endpoint of G. Lemma 3. If G is a 3-critical graph, then no two endpoints of G have a common neighbor. Lemma 4. If S is an independent set of r vertices in the connected, 3-critical graph G, then S contains a vertex v with degree 6(v)2 r - 2.

Degree sequences It is reasonable to expect that a 3-critical graph on n vertices, where n is large, cannot have many vertices of small degree. In fact letting d , denote the number of vertices in G having degree at most k, we can show that when n is sufficiently

36

D . P. Sumner

Fig. 1.

Fig. 1. (contd.)

Critic01 concepts in domination

37

larger than k, dk must be bounded by a simple linear function of k. In Sumner-Blitch [12], the following bound was determined for dk.

Theorem 3. If G is 3-critical on n >> k vertices then dk 6 k + 1. We first believed that this bound was best possible. In fact it is easy to see that d, can be 2 for arbitrarily large graphs, and the example in Fig. 2 shows that for every odd k 2 3, dk can be k + 1 for arbitrarily large graphs. The most general graphs of this type are formed by taking a 1-factor F out of a complete graph Kur of even order and adjoining a set S of three independent vertices adjacent to all of the vertices in KZk- F. Finally, add a complete graph K , with each vertex of S adjacent to at least one vertex of K, and so that for each a, b E S, N ( a ) n N ( b ) n K , = O . The graphs in Fig. 2 are special in that two of the vertices in S are required to be adjacent to exactly one vertex of K,. This is to force those two vertices to have degree 2k 1. However, the result of Theorem 3 is not best for k = 2, as the next theorem shows.

+

Theorem 4. d2 < 2 for n >> 2.

Proof. Suppose that G is a 3-critical graph with S = (a, b, c} the set of vertices having degree at most 2. Let M = {v: is adjacent to some element of S } , and let W = V ( G )- M U S . We first note that no element of S can be adjacent to the other two. For suppose that b Ia and b Ic , then since G is connected, there exists x E M such that x Ia. But now x Y c as otherwise x would have to be a cutpoint of G that is not adjacent to an endpoint. Thus there would have to exist

(1 0 ........0 I 0 .........0 0 .........0

1 nd

K - F (F a 1-factor)

1

Zk

t A 3 - c r 1 t 1 c a I graph w i t h c o n l e c t u r e d m l n i m u m n u m b e r 01 edqes

The v e r t i c e s I n A and t h e t o p t w o

i n S a l l have degree 2 k + 1

Fig. 2.

=

n-2k-3

38

D . P . Sumner

or (ii) [ y , c ] HX. In either case, a simple consideration of the possibilities-realizing that each of the vertices a and b have to be dominated-shows that y must be a. So either [ x , a ] c which is impossible since then we would have { x , b } dominating G, or [a, c ] HX which is impossible since {a, c } can dominate at most one vertex other than those discussed so far. Hence we will suppose that no vertex in S is adjacent to the other two. An extension of this argument shows that in fact we may assume that S is independent. Now for any x E W , x is not adjacent to a, and so there exists a y E M such that either [ x , y ] * a or [a, y ] HX. In the latter case it must be that y is not adjacent to x but is adjacent to all of W - { x } . Hence there can be at most one such x for each element of M. But since M can have only at most six elements and n >> 2, we can assume that there exists x E W and y E M with [ x , y ] * a . Similarly we can assume that there is some r E W and s E M such that [r, s ] H b, and a u E W and v E M such that [ v ,u ] H C . But then we have N ( a ) = { s , v } , N ( b ) = { v , y } , and N ( c ) = { y , s } . In particular s, v , and y are all distinct. But now since a ,l!b, we may assume that there exists a w in G such that [a, w ] H 6 . But a consideration of the possibilities shows that w must be s, and so [a, s ] ++ b. But then it follows that { s , v } dominates G which is impossible. 0

y such that either (i) [ x , y ] H C

So the question remains as to whether dk S k + 1 for n 3 k is best possible for even values of k 3 4. If not, what is the best bound on dk for even values of k ? Fig. 3 shows the degree sequences which are known to be possible for 3-critical graphs of order at most 9. It would be desirable to completely characterize k-critical degree sequences. A reasonable conjecture relating to degree sequences of 3-critical graphs is:

Conjecture. If d , s d2 s . . < d, is the degree sequence of a 3-critical graph G, then for each i = 0, 2, . . . , [ n / 2 ] ,di+l + d,-; 3 n - 3. +

Vizing [15] gave an upper bound on the number of edges in a graph on n vertices and having domination number k. In Blitch [ 4 ] his result was improved for connected 3-critical graphs by the next theorem.

Theorem 5. If G is connected and 3-critical on n vertices then q s (" ;'). This result is best possible since the graphs of the form in Fig. 4 all have (" 2') edges. A much harder problem is to determine the minimum number of edges in a connected domination critical graph. For 3-critical graphs we can conjecture that the graphs in Fig. 2 have as few edges as possible.

Critical concepts in domination n=6

n19 114666666 123666666 124456666 133456666 134455666 144445666 222G66666 223355666 223455566 224444455 224445555 224555566 224556666 233333555 233444446 233444455 233444466 233445555 233455666 233556666 234444456 234444566 234445556 234455566 244444444 244444455 244444666 244455556 333344444 333444445 333444456 333444555 333445556 333455555 334444455 334445555 344444445 444444444

111333

n =? 1123344 2223333 n=6 11355555 12255555 12334555 13334455 22334444 22334455 22344555 23333345 33333333 33334444

Fig. 3.

R

b

A U B U C is Complete

Fig. 4.

39

D . P . Sumner

40

Conjecture. If G is a connected, 3-critical graph on n vertices, then the number of edges in G is at least n-k

min{

(

) + (i)+ (k

-

4)/2)

where the minimum is taken over all even k , 2 is achieved at essentially k = n / 2 .

k

=S n.

For n 3 10, this minimum

Table 1 shows the conjectured minimum number of edges for 3-criticai graphs on at most 14 vertices. This table has been verified by computer search for n s 9.

1-Factors. A 1-factor or a perfect matching of a graph is a partition of its vertices into adjacent pairs. Clearly such a graph must have even order. For 3-critical graphs this obvious condition was shown to be also sufficient in Sumner-Blitch [121. The next theorem is the foundation for this result. Theorem 6. If G is a connected, 3-critical graph, and S is a separating set of vertices for G , then G - S has at most IS1 + 1 components. The following version of Tutte’s characterization of graphs with 1-factors appears in Sumner [13].

Theorem I (Tutte). A connected graph of even order has a 1-factor ifs it does not contain a set S such that G - S has at least (SJ 2 components of odd order.

+

The next theorem is an immediate consequence of this version of Tutte’s theorem and the previous theorem. Table 1. The minimum number of edges in a connected 3-critical graph n

Minimum edges

6 7 8 9 10 11 12 13 14

6 10 12 16 21 26 31 37 44

Critical concepts in domination

41

Theorem 8. If G is a connected 3-critical graph of even order, then G has a l-factor. Question. Heuristic evidence from computer-generated examples and degree considerations lead us to speculate that perhaps more is true. And in fact we ask if it could be that every connected, 3-critical graph on n 2 7 vertices contains a Hamiltonian path. Another piece of heuristic evidence for this is that, as a consequence of Theorem 6, every 3-critical graph is at least ;-tough (see ChvAtal

[61)* Diameters. For general graphs if y(G) = k, then the diameter of G can be at most 3k - 1. The situation is even more restrictive for k-critical graphs as is shown by the next theorem from Sumner-Blitch [12]. Theorem 9. The diameter of a k-critical graph is at most 3k - 4. This result is certainly not best possible.

Problem. What is the maximum diameter of a k-critical graph? For small values of k we can be more precise. The next two results are simple to establish.

Theorem 10. The diameter of a 3-critical graph is at most 3. Theorem 11. The diameter of a 4-critical graph is at most 7 . Independent sets and cliques. We will denote the maximum size of an independent set of vertices in a graph G by /3(G), and we will denote the size of a largest clique by o ( G ) . Theorem 12. Zf G is 3-critical, then P(G) s A(G). It might seem that there should be a limit on the size of independent sets in critical graphs. However this is not the case. A construction to demonstrate this was first provided by Trotter [14]. The following improvements on his construction are due to Blitch [4].

Theorem W. For every n 2 3, there exists a 3-critical graph G with 3n vertices and P(G) = n.

This theorem provides half of the proof of the next one.

D.P. Sumner

42

Theorem 14. Zf G is a 3-critical graph with P(G) = t and as few vertices as possible, then 2t s IG(s 3t. Another simple class of 3-critical graphs having arbitrarily large independent sets, and different from those described above, is the family G,, defined by V ( G , ) = {a, b } U S U T, where T = {xi,j:i, j = 1,2, . . . , n, i # j } is complete, and S = {1,2,3, . . . , n} is independent. Let N(a) = N ( b ) = S, and for each i, j = 1, 2, . . . , n, i # j , xi,jis adjacent to all of S - {i, j } . It is simple to check that G,, is 3-critical. See Fig. 5 for illustration. In [2], Allan and Laskar studied graphs for which the domination number and independent domination number are the same. The independent domination number for G -denoted by i(G) -is the size of a smallest independent dominating set for G. Since every maximal independent set in G is also a dominating set for G , i(G) exists for every graph. Clearly y(G) s i(G). There is a great deal of heuristic evidence for the next conjecture.

Conjecture. If G is a connected, 3-critical graph then y(G) = i(G). This result is known to be true if G has diameter 3 or 6 ( G )< 3. For awhile we hoped that an even stronger result might be true; namely that every vertex in a 3-critical graph was contained in an independent dominating set of order 3. This however is not true as we have determined by computer search. The smallest known counter-examples have 10 vertices. The extreme values for w ( G ) are easy. On the one hand, the next theorem from Sumner-Blitch [121 shows that every 3-critical graph contains a triangle.

Theorem 15. If G is connected and 3-critical, then w(C)

3.

On the other hand, the largest possible size for a clique in a 3-critical graph is clearly n - 3. The next result from Blitch [4] shows that this size is readily achievable.

a

b

complete

Independent

N(x ) fl S = S - ( I , ] )

'1

A 3-Crihcal Graph W i t h A

Fig. 5 .

Large Independent Set

Critical concepts in domination

43

Theorem 16. If G is connected and 3-critical with w ( G ) = n -3, then G = {a, b, c } UA U B U C, where A U B U C is complete and N ( a ) = A , N ( b ) = B , and N ( c ) = C. (See Fig. 4). Vertex deletions. In general removing a vertex from a graph can increase the domination number of the graph dramatically. For instance removing the central vertex of Kl.n raises the domination number from 1 to n. The k-critical graphs are much better behaved than graphs in general.

Theorem 17. Zf G is a k-critical graph k 3 1, then for every vertex v in G , y(G - V ) S k. The 3-critical graph in Fig. 6 shows that it may not always be possible to find a vertex v such that y ( G - v ) = k. For this example y(G - v ) = 2 for every v. Of course every k-critical graph contains a vertex v with y(G - v ) = k - 1. We call a graph k-vertex-domination critical if y(G) = k and y ( G - v ) = k - 1 for every v E V ( G ) . As we pointed out earlier, the 2-vertex-domination critical graphs are obtained by removing a 1-factor from a complete graph of even order. Note then that every 2-vertex-critical graph is also 2-critical. Thus the example in Fig. 6 shows that it is possible for a graph to be both 3-critical and 3-vertex critical. However most 3-critical graphs are not vertexcritical. Also, it is not necessary that a 3-vertex-critical graph be 3-critical; The cycle C, is 3-vertex-critical, but not 3-critical.

Other connections with the independence number There are many relationships between the independence number, p(G), and y(G). This should not be too surprising since every maximal independent set in G

is also a dominating set for G. In [2], Allan and Laskar showed that every

Fig. 6 .

D.P. Sumner

44

claw-free graph G (i.e. G has no induced subgraph isomorphic to K1,J satisfies y ( G ) = i ( G ) . We give a somewhat different proof of their result. The technique of our proof can be used to obtain slightly more general results.

Theorem 18. Zf G has no induced K 1 , 3 ,then y ( G ) = i ( G ) .

Proof. Suppose that C is a claw-free graph and let S be a minimum dominating set for G such that q ( S ) , the number of edges in S, is as small as possible. If S is independent, i.e. q ( S ) = 0, then we are finished. So suppose that q ( S ) > 0. Let a, b be adjacent vertices of S. Then since S is minimum, S - { a } is not a dominating set for G. Thus there must exist a vertex x in G - S such that x is adjacent to a, but x is not adjacent to any other vertex of S. But now, let T = S - { a } U { x } . Then q( T) < q(S), and hence T can not be a dominating set for G. Hence there must exist a vertex y which is adjacent to a, but not to any vertex in S U { x } . Thus, the set { a , b , x , y } induces a Kl,3in G with a as the center. This completes the proof. 0 The Allan-Laskar theorem was generalized by BollabAs and Cockayne [5].

Theorem 19. If G does not contain K,,,+l ( n > 1) as an induced subgraph, then i ( G )s y(G)(n- 1) - (n - 2). As a consequence of this theorem we can bound the domination number in terms of the independence number. First we observe (see Sumner-Moore [ll]) that the cardinality of any maximal independent set must be at least an appropriate fraction of that of a maximum independent set.

Theorem 20. Zf G does not contain K1,,+l as an induced subgraph, then every maximal independent set S in G satisfies IS1 3 B ( G ) / n .

As a consequence of this we get a bound on the size of y ( G ) .

Theorem 21. I f G does not contain Kl,,+l as an induced subgraph, then y(G)z

P(G) + n(n - 2 )

n(n -1)

Nebesky [9] defined the concept of a partial square of a graph. Two extreme cases of partial squares of the graph G are G2 and the line graph L(G). It is possible to bound y(G) between the independence numbers of these two types of partial squares.

Theorem 22. p(G2)< y ( G ) s P ( L ( G ) ) .

Critical concepts in domination

45

The lower bound was established in Meir-Moon [8] and the upper bound in Hedetniemi [7].It is an interesting problem to determine whether for any graph G, there exists a partial square H of G such that y(G) = /3(H). There are other ramifications of the Allan-Laskar theorem. The following result of Sumner (unpublished) is easy to prove using the proof technique of Theorem 18.

Theorem 23. If G is a graph such that at least one of G and G2 does not contain an induced K 1 , 3 ,then there is a minimum dominating set S f o r G2 such that S is independent in G. Question. A natural question that arises at this point is: When does G" ( n z= 3 ) contain a minimum dominating set which is independent in G? Domination perfect graphs. Motivated by the Allan-Laskar theorem and the concept of a perfect graph, Sumner-Moore [ll]defined a graph to be domination perfect if y ( H ) = i ( H ) for every induced subgraph H of G. The proofs of the results mentioned in this section will appear in [111. As a consequence of the Allen-Laskar theorem, we have

Theorem 24 (Allan-Laskar). Every claw-free graph is domination perfect. It turns out that it is not necessary to check every induced subgraph of a graph in order to determine if it is domination perfect.

Theorem 25. A graph G is domination perfect if y ( H ) = i ( H ) f o r every induced subgraph H of G with y ( H ) = 2. There are many classes of domination perfect graphs.

Theorem 26. If G is chordal, then G is domination perfect iff G does not contain an induced subgraph isomorphic to the graph in Fig. 7. Let Y = {H: IH( s 8, y ( H ) = 2, i ( H ) > 2).

Theorem 27. If G does not contain any member of Y as an induced subgraph and also does not contain an induced copy of either of the graphs in Fig. 8, then G is domination perfect. It is possible to completely characterize the planar domination perfect graphs by a forbidden subgraph condition.

D. P. Sumner

46

Fig. 7 Fig. 8

Theorem 28. A planar graph is domination perfect iff it does not contain any graph from 9 as an induced subgraph. This theorem is a direct consequence of Theorem 26 and the observation that the graphs in Fig. 8 are both nonplanar. Unfortunately it is impossible to provide a finite forbidden characterization of the entire class of domination perfect graphs.

References [l ] B.D. Acharya and H.B. Walikar, Domination critical graphs, preprint. [2] R.B. Allan and R. Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978) 73-76. [3] D. Bauer, F. Harary, J. Nieminen and C. Suffel, Domination alteration sets in graphs, preprint. [4] P. Blitch, Domination in graphs, Dissertation Univ. of S. C., 1983. [5] B. Bolobas and E.J . Cockayne, Graph theoretic parameters concerning domination, independence, and irredundace, J. Graph Theory 3 (1979) 271-278. [6] V. Chvital, Tough graphs and Hamiltonian circuits, Discrete Math. 2 (1973) 215-228. [7]E.J. Cockayne, Domination in undirected graphs, in: Y. Alavi and D.R. Lich, eds., Theory of Graphs in American Bicentennial Year (Springer, Berlin, 1978). [8] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree. Pacific J. Math. 61 (1975) 225-233. [9] L. Nebesky, On the existence of I-factors in partial squares of graphs, Czechoslovak Math. J. 29 (1979) 349-352. (lo] J. Nieminen, Two Bounds for the Domination Number of a Graph, J . Inst. Maths. Applics 14 (1974) 183-187. [I l l D.P. Sumner and J.I. Moore, Domination perfect graphs, preprint. [I21 D.P. Sumner and P. Blitch, Domination critical graphs. J . Combin. Theory Ser. B 34 (1983) 65-76. [13] D.P. Sumner, 1-factors and antifactor sets. J. London Math. Soc. 13 (2) (1976) 351-359. [14] W.T. Trotter, personal communication. [15] V.G. Vizing, A Bound on the External Stability Number of a Graph, Dokl. Akad. Nauk. 164 (1%5) 729-731.

Discrete Mathematics 86 (1990) 47-57 North-Holland

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THE BONDAGE NUMBER OF A GRAPH John Frederick FINK, Michael S. JACOBSON, Lael F. KINCH and John ROBERTS Department of Mathematics, University of Louisville, Louisville, K Y 40292, USA Received 2 December 1988 A set D of vertices in a graph G is a dominating set if each vertex of G that is not in D is adjacent to at least one vertex of D. The minimum cardinality among all dominating sets in G is called the domination number of G and denoted a(G). We define the bondage number b ( G ) of a graph G to be the cardinality of a smallest set E of edges for which o(G - E) > a ( G ) . Sharp bounds are obtained for b ( G ) , and the exact values are determined for several classes of graphs.

Introduction A set D of vertices in a graph G is a dominating set if each vertex of G that is not in D is adjacent to at least one vertex of D. A dominating set of minimum cardinality in G is called a minimum dominating set, and its cardinality is termed the domination number of G and denoted a(G). Except as indicated otherwise, all terminology and notation follows [2]. Among the various applications of the theory of domination that have been considered (see e.g. [3]), the one that is perhaps most often discussed concerns a communications network. This network consists of existing communication links between a fixed set of sites. The problem at hand is to select a smallest set of sites at which to place transmitters so that every site in the network that does not have a transmitter is joined by a direct communication link to one that does have a transmitter. This problem reduces to finding a minimum dominating set in the graph, corresponding to this network, that has a vertex corresponding to each site, and an edge between two vertices if and only if the corresponding sites have a direct communications link joining them. We now carry the foregoing example further and examine a question concerning the vulnerability of the communications network under link failure. In particular, suppose that someone (a saboteur) does not know which sites in the network act as transmitters, but does know that the set of such sites corresponds to a minimum dominating set in the related graph. What is the fewest number of communication links that he must sever so that at least one additional transmitter would be required in order that communication with all sites be possible? With this in mind, we introduce the bondage number of a graph. 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V. (North-Holland)

J . F. Fink et al.

48

The bondage number b ( G ) of a nonempty graph G is the minimum cardinality among all sets of edges E for which a(G - E ) > o ( G ) . Thus, the bondage number of G is the smallest number of edges whose removal will render every minimum dominating set in G a “nondominating” set in the resultant spanning subgraph. Since the domination number of every spanning subgraph of a nonempty graph G is at least as great as a ( G ) , the bondage number of a nonempty graph is well defined. In what follows, we investigate the value of the bondage number in progressively more general settings.

Some exact values We begin our investigation of the bondage number by computing its value for several well known classes of graphs. In several instances we shall have cause to use the ceiling function of a number x ; this is denoted [XI and takes the value of the least integer greater than or equal to x. We begin with a rather straightforward evaluation of the bondage number of the complete graph of order n.

Proposition 1. The bondage number of the complete graph K , ( n

2 ) is

b(K,) = [ n / 2 ] .

Proof. If H is a (spanning) subgraph of K , that is obtained by removing fewer than [n/21 edges (possibly none) from K,,, then H contains a vertex of degree n - 1, whence a ( H ) = 1. Thus, b(K,,) 2 [n/21. If n is even, the removal of n / 2 independent edges from K,, reduces the degree of each vertex to n - 2 and therefore yields a graph H with domination number a ( H ) = 2. If n is odd, the removal of ( n - 1)/2 independent edges from K, leaves a graph having exactly one vertex of degree n - 1; by removing one edge incident with this vertex, we obtain a graph H with a ( H ) = 2. In both cases ( n even, n odd) the graph H resulted from the removal of [n/21 edges from K,. Thus, b(K,) = [ n / 2 ] . 0 We next determine the bondage numbers of the n-cycle C, and the order n path P,. We shall make use of the following lemma whose proof (which we omit) is straightforward.

Lemma 1. The domination numbers of the n-cycle and the path of order n are respectively a(C,) = [ n / 3 ] for n 3 3, and a(P,) = [ n / 3 ] f o r n 3 1.

The bondage number of a graph

49

Theorem 1. The bondage number of the n-cycle is 3 i f n = 1 (mod 3), 2 otherwise.

Proof. Since a(C,,) = a(Pn)for n 2 3, we see that b(C,,)2 2. If n = 1 (mod 3), the removal of two edges from C,, leaves a graph H consisting of two paths P and Q. If P has order n1 and Q has order n2, then either n , = n2 = 2 (mod 3), or, without loss of generality, n , = 0 (mod 3) and n2= 1 (mod 3). In the former case,

a ( H ) = a ( P ) + a(Q)= [n1/31 + [n2/31 = ( n l + 1)/3 + (n2 + 1)/3 = (n + 2)/3 = [n/31 = a(C,,). In the latter case,

a ( H ) = n1/3 + (n2+ 2)/3 = ( n + 2)/3 = [n/3] = a(C,,). In either case, when n = 1 (mod 3) we have a(C,,) 2 3. To obtain the upper bounds that, by trichotomy, will yield the desired equalities of our theorem’s statement, we consider two cases. Case 1. Suppose that n = 0 , 2 (mod 3). The graph H obtained by removing two adjacent edges from C,, consists of an isolated vertex and a path of order n - 1. Thus,

a ( H ) = 1 + a(P,,-,) = 1 + [(n - 1)/3] = 1 + [n/3] = 1 + a(C,,), whence b(C,,)S 2 in this case. Combining this with the upper bound obtained earlier, we have b(C,,) = 2 if n = 0, 2 (mod 3). Case 2. Suppose now that n = 1(mod 3). The graph H resulting from the deletion of three consecutive edges of C,, consists of two isolated vertices and a path of order n - 2. Thus,

+ [ ( n - 2)/3] = 2 + (n - 1)/3 = 2 + ( I d 3 1 - 1) = 1 + ~(c,,),

a ( H )= 2

so that b(C,,)S 3. With the earlier inequality we conclude that b(C,,)= 3 when n~l(mod3).0 As an immediate corollary to Theorem 1 we have the following. Corollary 1.1. The bondage number of the path of order n ( 2 2 ) is given by 2 i f n = 1 (mod 3), b(Pn) = 1 otherwise.

1.F. Fink et at.

50

The next theorem establishes the bondage number of the complete t-partite graphs K ( n , , n 2 , . . . , n J .

Theorem 2. If G = K ( n , , n 2 , . . . , nt), where n , 4 n , s . . . S n,, then 2t - 1

i f n , = 1 and 3 2, for some m, 1c m < t, i f n l = I t 2 = . - .= n, = 2, otherwise.

i=l

Proof. The proof of the statement “ b ( G )= [m/21 if n , = 1 and n,,, >2” is similar to the proof of Proposition 1, and is omitted here. Suppose then that n , = n2 = . . . = n, = 2, and note that a(G) = 2. We show first that b ( G ) 3 2t - 1. Assume to the contrary that there is a set E of edges in G such that IE( = 2t - 2 and a ( G - E ) > a ( G ) . Observe that G - E has no isolated vertex, for any subgraph of G that has an isolated vertex and 2t - 2 fewer edges than G is isomorphic to K , U K(1,2,2, . . . , 2) and has domination number 2. Also, if G - E has a vertex of degree 2t - 2, then a(G - E) = 2. Thus, the degree of each vertex in G - E is between 1 and 2t - 3 inclusive. In fact, since IEl = 2t - 2, there is a vertex x1 with deg,-,xl = 2t - 3. Let x 2 be the other vertex of G that belongs to the same partite set as x l , and let y , be the one vertex distinct from x 2 that is not adjacent to xl. Since the 2-element partite set { x l , x 2 } can not be a dominating set of G - E, it must be that edge y , x , ~E. Let y2 be the other member of the partite set in G that contains y , . Then, since o(G - E) > 2 (=a(G)), and since x1 is adjacent in G - E to all vertices but x 2 and y l , each vertex different from x , , x,, y,, y2 must be nonadjacent with at least one of x 2 and y 1 in G - E. Since there are 2t - 4 such 2 = 2 t -2, or all, edges of E. As vertices, we have now accounted for (2t - 4 ) none of these edges was incident with y,, we see that y , has degree 2t - 2 in G - E, a contradiction to our earlier analysis. Thus, b ( G )2 2t - 1. Now, if we let {xl, x 2 } be any partite set of G and let H be the graph obtained by removing from G the 2t - 2 edges incident with x , and one edge incident with x 2 , then o ( H ) = 3. Hence b ( G )= 2t - 1 when n , = n2 = . . . = n, = 2. Suppose now that n , 3 2 and n, 3 3. Note that a(G) = 2 and let s = CII: n,. Assume that there is a set E of edges in G such that IEI < s and a(G - E) > a(G). Then each vertex of G is incident with at least one member of E. For if there is a vertex v such that deg,-, v = deg, v and if V is the partite set containing v, each of the more than s vertices not in V must be nonadjacent in G - E with at least one member of V (otherwise u and one vertex not in V would constitute a 2-element dominating set in G - E). But this then implies that I E l 2 s in contradiction to our assumption. Thus each vertex is incident with at least one edge in E. Note also that since IEl< s, there must be a vertex x1 incident with exactly one edge, say e, of E. Let y be the other vertex of G that is incident with

+

The bondage number of a graph

51

e, and let Y denote the partite set of G that contains y. Furthermore, let . . . ,x, be the other vertices of the partite set X that contains xl. Since x1 is adjacent in G - E to every vertex diffierent from y,, x2, x 3 , . . . , x,, and since a(G - E) > 2, each vertex not in X U Y must be nonadjacent with at least one of y,, x2, x 3 , . . . ,x, in G - E. Since each vertex of Y is also nonadjacent with some vertex in G - E, we conclude that IEl3 IV(G)\(XU Y)l JYIa s ; this is a contradiction. Thus, b ( G ) S s . Since the graph H obtained by removing the s edges incident with a vertex in a partite set of cardinality n, has a ( H ) = 3 , we conclude that b ( G ) =s. 0 x2, x 3 ,

+

The bondage number of trees In the preceding section of this paper we obtained exact values of the bondage number for some graphs whose structure was completely described. In this section we look at the bondage number of a more general class of graphs, namely trees. The principal result of this section is the following.

Theorem 3. If T is a nontrivial tree, then b ( T ) S 2. Proof. The statement is obviously true for trees of order 2 or 3, so we shall suppose that T has at least 4 vertices. Suppose first that T has a vertex u that is adjacent to two end-vertices v and w (and possibly more). If D is a dominating set for T that does not contain u, then D contains both u and w ; but then, ( D \ { v , w } ) U { u } is also a dominating set for T. Thus every minimum dominating set for T contains the vertex u and therefore does not contain v. Since every dominating set of T - uv contains v and is also a dominating set for T, it follows that a(T - u v ) > a ( T ) . Hence b ( T )= 1 in this case. Suppose now that each vertex of T is adjacent with at most one end-vertex. Then T has a vertex u of degree 2 that is adjacent with exactly one end-vertex v . Let w be the other vertex adjacent to u, and let D be a minimum dominating set for T - uv - uw. Then both u and u are in D and D \ { v } is a dominating set for T. Hence a( T) < a( T - uv - u w ) and b( T) S 2. 0 The proof of Theorem 3 verifies the following.

Corollary 3.1. If any vertex of a tree T is adjacent with two or more end-vertices, then b ( T ) = 1. As evidenced by Corollary 1.1, the trees of Corollary 3.1 are not the only trees with bondage number equal to 1. Also, the subdivision graphs of the stars K(1, n) have bondage number equal to 1 and have no vertex that is adjacent with more

52

J.F. Fink et al.

than one end-vertex. The question now arises: “Which trees have bondage number equal to 1, and which have bondage number equal to 2?” This question is unresolved and appears to be difficult. As the following theorem shows, a simple ‘forbidden subgraph’ statement will not answer the question.

Theorem 4. If F is a forest, then F is an induced subgraph of a tree S with b ( S ) = 1 and a tree T with b( T ) = 2.

Proof. Let u be the central vertex of a path of order 3. From each component of F, select one vertex and introduce an edge from that vertex to u. The resulting tree S contains F as an induced subgraph and has a vertex, namely u, adjacent with two end-vertices. By Corollary 3.1, b(S) = 1. We now prove the existence of the tree T with b( T) = 2 that contains F as an induced subgraph. We proceed by induction on the order p of F. The claim is easily verified for p = 2. Assume that the claim is true for every let T be a path forest of order p, and let F be a forest or order p + 1. If F = whose order is congruent to 1 modulo 3 and is at least as large as 2p + 1. Then T contains an independent set of p + 1 vertices (this gives the induced F) and, by Corollary 1.1, has b( T) = 2. Suppose now that F is nonempty. Let u be an end-vertex of F, and let u be the vertex adjacent to u. By the inductive hypothesis, the order p forest F’ = F - u is an induced subgraph of a tree T’ with b ( T ’ ) = 2. Let H be the union of two paths of order 4, and label its vertices as in Fig. l(a). Let T be the tree obtained by taking the union of H and T‘ and adding the vertex u together with edges uu, uy,, and uy, (see Fig. l(b)). Clearly F is an induced subgraph of T. Also, from each pair of vertices

Fig. 1.

The bondage number of a graph

53

{w,,xl}, { y l , z , } , {w2, x 2 } , { y 2 ,z2}, exactly one must be in every dominating set for T, and none of these dominates a vertex of T'; thus, a ( T ) 3 a ( T ' ) 4. If D' is a minimum dominating set for T', then D = D ' U { x 1 , y l , x 2 , y 2 }is a dominating set for T of order a ( T ' ) + 4; thus a ( T ) = a ( T ' ) 4 and D is a minimum dominating set for T. From this line of reasoning we also see that, since b ( T ' ) = 2, if e is an edge of T', then a ( T - e) = a ( T ) . Furthermore, if e belongs to the subgraph J = ( { u , v, wl, x , , y , , z,, w2,x 2 , y2, z 2 } ) (see Fig. l(c)), then, since b(J)= 2, we have a(T - e) = a ( T ) . Thus b ( T ) = 2. I7

+

+

It is clear that the bondage number of a forest is either 1 or 2. We can decide in O ( n 2 ) time by methodically removing each pair of edges, whether the bondage number is 1 or 2. It would be interesting to determine if there is a linear time algorithm to find b ( F ) for a given forest F.

General bounds In this section we shall establish bounds on the bondage number of a graph that are independent of the graph's structure. Our first result relates the bondage number to the order of the graph.

Theorem 5. If G is a connected graph of order p

3 2,

then b ( G )s p - 1.

Proof. Let u and v be adjacent vertices with deg u =sdeg v. If b ( G )S deg u, then b (G) s p - 1, so we suppose that b (G) > deg u. Let Eu denote the set of edges incident with u. Then a(G - E,) = a(G) and a(G - u ) = a(G) - 1. Also, if D denotes the union of all minimum dominating sets for G - u , then u is adjacent in G to no vertex of D . Hence, JE,(s p - 1 - (Dl and v t$ D . Now, if F, denotes the set of edges from v to a vertex in D , then since u t$ D we must have a(G - u - F , ) > a(G - u ) , or equivalently, a(G - u - F , ) > a(G) - 1.

Thus a((; - (E, U 4))> a(G) and we see that

b ( G )S IEU u F,I = lEUl+ lFvl S ( p - 1 - IDl)+ ID1 = p - 1. This completes the proof.

c7

While the bound b ( G )< p - 1 is not particularly good for many classes of graphs (e.g. trees and most cycles), it is an attainable bound. For example, if

J . F. Fink et al.

54

G = K ( 2 , 2 , . . . , 2), then, by Theorem 2, b ( G ) = p - 1. The next theorem provides an upper bound on b ( G ) that is better in many instances than the bound of Theorem 5; in particular, it is better when the graph G has adjacent vertices of relatively low degree.

Theorem 6. If G is a nonempty graph, then b (G) < min { deg u + deg v

- 1: u

and v are adjacent vertices}.

Proof. Let A denote the right hand side of the inequality above, and let u and v be adjacent vertices of G such that deg u deg v - 1 = A. Assume that b ( G ) > A. If E denotes the set of edges that are incident with at least one of u and v, then IEl = A and therefore a(G - E) = a(G). Thus, since u and v are isolated vertices in G - E, we see that a(G - u - v ) = a ( G )- 2. But then, if D is a minimum dominating set for G - u - v , the set D U { u } is a dominating set for G and has cardinality a(G)- 1-a contradiction. Therefore, b ( G ) < A. 0

+

Both Theorem 3 and Theorem 6 are also proved by Bauer, Harary Nieminen and Suffel in [l]. As a corollary to Theorem 6 we have the following easily computed bound.

Corollary 6.1. If A(G) and 6(G) denote respectively the maximum and minimum degree among all vertices of nonempty connected graph G, then

b ( G )S A(G)

+ 6(G) - 1.

Proof. Let u be a vertex of minimum degree 6(C) in G, and let v be any vertex adjacent to u. Then, by Theorem 6,

b ( G ) s deg u S

+ deg v - 1= 6(G) + deg v - 1

6(G) + A(G) - 1.

0

We remark that the bounds stated in Theorem 6 and Corollary 6.1 are sharp. As indicated by Theorem 1, one class of graphs in which the bondage number achieves these bounds is the class of cycles whose orders are congruent to 1 modulo 3. Another bound on the bondage number that involves the maximum degree among the vertices of the graph is given by the following theorem. This bound also indicates a relationship between the bondage number and the domination number.

Theorem 7. If G is a nonempty graph with domination number a ( G )3 2, then b ( G ) S (a(G) - l)A(G)

+ 1.

The bondage number of a graph

55

Proof. We proceed by induction on the domination number a(G). Let G be a nonempty graph with a(G) = 2, and assume that b ( G )3 A(G) + 2, Then, if u is a vertex of maximum degree in G , we have a(G - u ) = o ( C )- 1= 1 and b(C - u ) 2 2. Since o(G) = 2 and a(G - u ) = 1, there is a vertex v that is adjacent with every vertex of G but u ; thus deg, v = A(G) also, and u is adjacent with every vertex of G except v. Since b(G - u) 2 2, the removal from G - u of any one edge incident with u again leaves a graph with domination number 1. Thus there is a vertex w # v that is adjacent with every vertex of G - u. But, since is the only vertex of G that is not adjacent with u, vertex w must be adjacent in G with u. This however implies that a(G) = 1, a contradiction. Thus, b(G)s A(G) + 1 if a(G) = 2. Now, let k ( 2 2 ) be any integer for which the following statement is true: if H is a nonempty graph with a ( H ) = k, then b ( H ) S (k - 1) . A ( H ) + 1. Let G be a nonempty graph with a(G) = k + 1, and assume that b ( G )> k * A(G) + 1. Then, if u is any vertex of G , we have a(G - u ) = a(G) - 1= k, since deg u < b(G). But then, b ( G )s b(G - u ) + deg u, and by the inductive hypothesis we have

b ( G ) s [ ( k - 1).A(G - u ) + 11 + d e g u S ( k - 1). A(G)+ 1 + A(G), or

b ( G )s k * A(G) + 1,

a contradiction to our assumption that b ( G )> k - A(G) + 1. Thus, b ( G )s k A(G) + 1, and, by the principle of mathematical induction, the proof is complete.

0

Again, by considering the complete t-partite graph G = K ( 2 , 2, . . . , 2), we see that Theorem 7 provides a sharp bound on b(G). This same graph can be used to show that the bound given in our next theorem is sharp.

Theorem 8. If G is a connected graph of order p 3 2, then b ( G )“ p - a(G) + 1. Proof. If a(G) 6 2. the desired inequality follows from Theorem 5. Thus, we suppose that a ( G ) 2 3 and assume, contrary to our claim, that b ( G ) a p - a(G) 2. Fix a vertex x in G, let N ( x ) be the set of all vertices adjacent with x, and let Ex denote the set of edges incident with x . Since V ( G ) \N( x ) is a dominating set in G, degx s p - a(G) < b(G),

+

and thus a(G-x)= a(G)- 1. Furthermore, if D denotes the union of all minimum dominating sets for G - x, we see that N ( x ) n D = 0; thus IExI G p - 1- IDI.

J . F. Fink et al.

56

Now let z E N ( x ) , and let F, denote the set of all edges from z to D. Then, since z @ D,no minimum dominating set of G - x - F, is contained in D ;thus

a(G - x - F,) > a(G - X ) = a(G) - 1, whence u(G - Ex - F,) > a(G).

From this last inequality, we see that lExl + lFzl 3 b ( G )S p - a(G) + 2.

Hence,

IEI > ( P or

-4

G ) + 2 ) - (P - 1 - IDl)

IFz[3 ID1 - a(G) + 3.

Now, let J be a minimum dominating set for G - X (and recall that 3 ID(- a(G) 3, we see that z is adjacent to at least IJI = a(G) - 1). Since two vertices of J . Let J1 be the vertices of J that are adjacent to z, and let J2 denote the set of vertices in D\J that are not adjacent to z. Then,

+

IFzl = lJll+

ID\(JUJ2)I

+ P I - IJI - 1521 = ID1- ( 4 G ) - 1)+ lJil = lJll

- 1521

so that

ID1 - (a(G) - 1)+ I J l l -

1521 3 ID1 - a(G)

+3

or lJll

3

lJ21

+ 2.

Now, let 5; be the set of vertices in J2 each of which is adjacent to exactly one vertex of Jl. Since lJll > lJ21 3 (5;1, there must be a vertex v that is adjacent to no vertex of J;. But then, K = (J\{v}) U { z } is a minimum dominating set for G - x , a contradiction since z $ D.Thus, b ( G )s p - a(G) 1. 0

+

We close with the following conjecture: Conjecture. If G is a nonempty graph, b(G)

A(G) + 1.

In [l],partial support for this conjecture is given. In particular, it is shown that if G is a graph with the property that a(G) a(G - v ) then b ( G ) A(G).

The bondage number of a graph

57

Reference [l] D. Bauer, F. Harary, J. Nieminen and C.L. Suffel, Domination alteration sets in graphs, Discrete Math. 47 (1983) 153-161. [2] M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs & Digraphs (Wadsworth, Belmont, CA, 1979). [3] F.S. Roberts, Graph Theory and Its Applications to Problems of Society (SIAM, Philadelphia, PA, 1978).

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Discrete Mathematics 86 (1990) 59-69 North-Holland

59

CHORDAL GRAPHS AND UPPER IRREDUNDANCE, UPPER DOMINATION AND INDEPENDENCE Michael S. JACOBSON* and Ken PETERS Department of Mathematics, University of Louisville, Louisville, KY 40.292, USA Received 2 December 1988

In this paper we consider the following parameters: IR(G), the upper irredundance number, which is the order of the largest maximal irredundant set, T ( G ) , the upper domination number, which is the order of the largest minimal dominating set and P(G), the independence number, which is the order of the largest maximal independent set. It is well known that for any graph G, p(G) s T ( G )s IR(G).

In this paper we show that these parameters are equal for all chordal graphs, and a class of graphs not containing a set of forbidden subgraphs.

1. Introduction Let G = (V, E) be a finite graph with no loops or multiple edges. If W G V ( G ) then we refer to the induced subgraph on W , denoted ( W), as the subgraph of G with vertex set W and all edges of E ( G ) with both end vertices in W. The open neighborhood of a vertex x is the set N ( x ) = {y E V ( G ) :( x , y) E E ( G ) } , and the closed neighborhood of x is the set N [ x ] U { x } . For a subset of vertices X E V ( G ) ,we define N ( X ) = Uxsx N ( x ) and N [ X ]= N ( X ) U X . A set of vertices D E V ( G ) is a dominating set of G if N [ D ]= V ( G ) , The domination number of G , denoted y ( G ) and the upper domination number of G , denoted T ( G ) , are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number, denoted i ( G ) , and the independence number, p ( G ) , are respectively the minimum and maximum cardinalities taken over all maximal sets of independent vertices of G. For x E X c V ( G ) , if N [ x ]- N [ X - { x } ]= 0, then x is said to be redundunt in X. A set X of vertices is irredundant if and only if no vertex in X is redundant. The lower and upper irredundance numbers, ir(G) and IR(G), are respectively the minimum and maximum cardinalities taken over all maximal irredundant sets of vertices of G. The modern study of domination in graphs was begun by Ore [18] and Berge [ 2 ] .For a survey of domination results, the reader is referred to [S], [ 5 ] , or [16]. * Research supported by O.N.R. Contract No. N00014-85-K-0694. 0012-365X/90/$03.50 0 1990-Ekevier Science Publishers B.V. (North-Holland)

60

M.S.Jacobsen, K . Peters

The concept of independent domination was first introduced by Cockayne and Hedetniemi [7], while irredundant sets in graphs were first studied by Cockayne, Hedetniemi and Miller [9]. Since then, results on irredundance have been presented by Bollobhs and Cockayne [3], Allan and Laskar [l], Cockayne, Favaron, Payan and Thomason [6], and a survey article by Hedetniemi, Laskar and Pfaff [13]. The six max-min parameters defined above, two each for domination, independence and irredundance, are all related by the following inequalities.

Theorem 1(Cockayne and Hedetniemi [7-81). For any graph G, ir(G) c y ( G ) i ( G ) P(G) T ( G ) IR(G). The questions that naturally arise are: Do there exist graphs for which these inequalities are strict, and if so, for any pair of the above parameters, are there necessary and sufficient conditions for equality of that pair? In answer to the first question, there are graphs for which these inequalities are strict. Slater gave the first example (Fig. 1) of a tree T for which ir( T) = 4, while y ( T ) = 5. Later, Cockayne, Favaron, Payan and Thomason [6] exhibit a graph which has unequal values of all six parameters. Regarding the second question, many results have been given that address the question of equality of some of these parameters. Most results of this type give sufficient conditions, usually in terms of forbidden subgraphs. However, forbidden subgraph characterizations for equality of parameters have been hard to obtain; in fact, it usually is impossible. As noted by Bollobhs and Cockayne [3], a necessary and sufficient forbidden subgraph list characterizing graphs G having ir(G) = y ( G ) is impossible to obtain. This is easy to see since the addition of a new vertex adjacent to all vertices of a graph G produces a graph G‘ containing G as an induced subgraph with i(G’) = y(G’)= ir(G’) = 1. This also indicates a similar problem for characterizing graphs G with y ( G ) = i ( G ) . We note here that the same is true for those graphs having P(G) = T ( G )= IR(G). For any graph G, the addition of an independent set of vertices to G, of size 2 JV(G)(,where each vertex in this set is adjacent to all the vertices of G, creates a graph G’ with P ( G ’ )= T ( G ’ )= IR(G’) = 2 IV(G)l, yet containing G as an induced subgraph. The first result involving forbidden subgraphs that implies equality of two of these parameters was the following presented by Allan and Laskar [l].

Fig. 1. A tree with ir(T) = 4 and y ( T ) = 5

Chordal graphs

G1

61

G2

Fig. 2. Forbidden subgraphs for Theorem 3.

Theorem 2 (Allan and Laskar [l]). If G is any graph without an induced subgraph isomorphic to K 1 , 3 , y ( G ) = i(C). Laskar and Pfaff [14] have also presented several sufficient conditions on G such that ir(G)= y ( G ) . We state several of their results in the following theorems. The first result pertains to chordal graphs. A graph G is chordal if any induced cycle of G (of length four or more) contains a chord, i.e., an edge joining two non-consecutive vertices of the cycle. A vertex x is a simplicial vertex if N ( x ) induces a complete subgraph. It is well known that any chordal graph contains at least one simplicia1 vertex. If both G and its complement are chordal, then G (and its complement) is referred to as a split graph.

Theorem 3 (Laskar and Pfaff [14-151). (i) Let G be a chordal graph. If G contains neither GI nor G2 of Fig. 2 as an induced subgraph, then ir(G) = y ( G ) . (ii) If G is the complement of a bipartite graph or a split graph and is connected, then ir(G) = y ( G ) = y,(G) = yc(G), where yt and yc are respectively the total and connected domination numbers of G. Theorem 4 (Laskar and Pfaff [14]). For any graph G that contains no induced K 1 , 3and no G3 (of Fig. 3) where the dotted edges of G3 are the only extra edges allowed, ir(G) = y ( G ) = i(G). Cockayne, Favaron, Payan and Thomason [6] give sufficient conditions for equality of some of the maximum parameters. In particular, they prove the

G3 Fig. 3. Forbidden subgraphs for Theorem 4.

M . S . Jacobsen, K . Peters

62

G4

Fig. 4. Forbidden subgraphs for Theorem 5

following:

Theorem 5 (Cockayne, Favaron, Payan and Thomason [6]). If G contains no subgraph isomorphic to the graph G4 (of Fig. 4), where the dotted edges of G4 are the only extra edges allowed, then IR(C) = I'(G). In the same paper they present several other results which d o not involve forbidden subgraph characterizations. One of these results deals with the following. Let e(G) denote the maximum number of pendant edges in a spanning subforest of G. A famous result relates E ( G )to the domination number.

Theorem 6 (Nieminen [17)). For any graph G with n vertices, y(G) + E ( G )= n. It is easily verified that the string of inequalities of Theorem 1 can now be extended as follows: ir(G) =S y(G) s i ( G )5 p(G) =S T ( G ) IR(G)

E(G).

Theorem 7 (Cockayne, Favoron, Payan and Thomason [6]). (i) If G is a graph of order n with no isolates and I R ( G ) = E ( G ) (i.e., y ( G ) + I R ( G ) = n ) , then P(G) = T ( G )= IR(G) = E ( G ) . (ii) ff G is bipartite, then p(G) = T ( G )= IR(G). In this paper w e will show that for any chordal graph G, P ( G ) = T ( G ) = IR(G). We also give a list of forbidden subgraphs that is sufficient for p(G) = T ( G )= IR(G).

2. Main results Before proceeding with the main results, we need to establish some further notation. Let I be an irredundant set with x E f such that x is adjacent to some other vertex of I . By the definition of irredundance, there must exist a vertex y E V ( G )- I such that N ( y ) f l I = { x } . For convenience we will refer to this set of vertices y E V - I , for such a vertex n, as B,. We will refer to B, as the set of private neighbors of x. Also, if I is a maximal irredundant subset of vertices with 111 = IR(G), we will refer to I as an IR-set of G. Likewise we will refer to a I--set or a p-set of G.

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63

The following lemma will be used extensively.

Lemma 8. Let I be an IR-set such that ( I ) contains as few edges as possible. If x is not an isolated vertex in ( I ) and y E B,, then there exists a vertex z E I which is not an isolated vertex in ( I ) , such that B, c N ( y ) .

Proof. Let I, x and y be stated as above. Consider the set I’ = Z - { x } U { y } . Note II’(= [I(= IR(G), thus if I’ was irredundant, then it would have to be a maximal irredundant set. Clearly, ( I r ) contains fewer edges than (I), since x is not an isolate in (I) and y is an isolate in (Z‘). Thus I’ can not be irredundant. Since y is an isolate in Z’, there must exist a vertex z # y which is redundant in I’. This is only possible if z is not an isolate in I and B, c N ( y ) . 0 With this lemma, we are now prepared to present the following result.

Theorem 9. Zf G is a chordal graph, B(G) = T ( G )= IR(G).

Proof. Since it is known that @ ( G )s T ( G )6 IR(G), We need only prove that B(G)>IR(G) to complete the proof. Let G be a chordal graph and Z an IR-set with (I) containing as few edges as possible. If Z is an independent set, the proof is complete, hence we assume that I is not independent. Let ( x l , y,) be an edge in (I) with w1 E B,, and z1E By,. By Lemma 8, there exist vertices x2, y2 E I with x2 adjacent to y2 such that B,, E N ( z , ) . Let w2 E B,,, let 2, E By2and continue applying the lemma. Since the graph is finite, at some point a vertex will be repeated but, up to that point, we will have the subgraph of Fig. 5. There are possible extra edges in the subgraph of Fig. 5 but, since each wi and z, are in B,, and By,, respectively for each i, there are no additional edges between vertices in I and vertices not in I, other than those shown. For a vertex that is finally repeated, there are four cases to consider. Case 1. Suppose Lemma 8 produces xk+*= xi for some i s k. Then B,. c_ N ( z k ) . x1

y1

x2

y2

Xk

‘k

k

w1 =2 Wk Fig. 5. Partial underlying structure of G.

=1

w2

M.S.Jacobsen, K . Peters

64

Let

C = {w,, xi, y,,

2,:

i 6j

sk}

This set C forms a cycle and since C is chordal, (C) is chordal and thus must contain a simplicial vertex. But in (C) every vertex is adjacent to two non-adjacent vertices, thus no simplicial vertex can exist. Case 2. Suppose Lemma 8 produces x k + l = Y , - ~ for some S k . Then By,_,c N ( z k ) , Let C = {zip1} U {wj, xi, yi, zi:i s j s k } .

The only candidate for a simplicial vertex in (C) would be zi-]since y i - l i$ C. For this to be true, ( w i , z k ) E E ( G ) . As above, by considering C' = C - {zipl}, a contradiction would result. Case 3. Suppose Lemma 8 produces yk = yi for some i < k . Then (xk, y i ) E E ( G ) . Let C = { y i , zi}U {w,, xi, y,, z,: i < j < k } U {wk,x k } .

Since ( C ) is chordal, a contradiction occurs as in the previous cases since no simplicial vertex can exist. Case 4. Suppose Lemma 8 produces yk = y i for some i < k . Then ( x k , x i ) E E ( G ) . Let C = { x i , yi, z i } U {w,, xi, y,,

2,:

i <j < k } U { w k ,x k } .

As in Case 2, the only candidate for a simplicial vertex is xi, but this would imply that ( x k , y i ) E E ( G ) and by letting C' = C - { x i } the previous case results. Exhausting all possibilities we may conclude that our original supposition that I was not independent is false, hence P(G) 2 IR(G). Thus, the result follows. 0 We have also obtained a forbidden subgraph list which also implies these parameters are equal.

Theorem 10. For any graph G that does not contain either K1,3,C4or the graph H of Fig. 6 as an induced subgraph, P(G) = T ( G )= IR(G).

Proof. Let G be a graph containing no induced subgraph isomorphic to K1,3,C, or H. As in the proceeding proof, let I be an IR-set containing as few edges as possible. Once again, if I is an independent set, the proof is complete, hence we assume that I is not independent. By Lemma 8 we can again form the subgraph of Fig. 5. As in the proof of Theorem 9, there are no additional edges between vertices in I and vertices not in I, other than the edges shown in Fig. 5. Also, since G contains no induced C,, (wi, zi) 4 E ( G ) and ( y i , xifl) c# E ( G ) for all values of i. Once again, there are four cases to consider.

Chordal graphs

65

H

Fig. 6. The graph H of Theorem 10.

Case 1. Suppose B,, E N ( z k ) for some i c k . Let

K = {zk, wi, x i , zi-,}. Note the case where i = 1 becomes somewhat complicated and we will postpone this until the end of the proof. Since (K) is not a K1.3 (zk,zi-l) E E ( G ) . Thus the vertices y ,z i - l , and zk form a triangle and we will claim for A = {xi-l,

Yi-19

zi-19

Wi,Xir

Yip

Zkr

yk, x k } ,

( A ) is isomorphic to the graph H. We show this is true by systematically eliminating the twelve possible additional edges in ( I ) . We use the subgraph of G shown in Fig. 7. The twelve possible additional edges in the induced subgraph (A) fall into three different types. The first is an edge between two vertices each of distance one from the triangle. The second is an edge between a vertex of distance one from the trainagle to a vertex of distance two from the triangle. And the third is an edge between two vertices each of distance two from the triangle. We will examine one edge of each type and the others of that type are eliminated in the exact same manner. xi)$ E ( G ) : { y i P l ,ziPl,wi,x i } would form a C4. (1) x i ) $ E ( G ) :{wi-lxi-l, y i - l , x i } would form a K1,3 (2) (3) (xi-1, yi) 6 E ( G ) :{wi-i,xi-1, yi-1, yi} would form a ki.3. Thus G contains a subgraph isomorphic to H, and thus we have a contradiction. Case 2. Suppose B+, G N ( z k ) for some i < k. Then (zk, wi)E E ( G ) or else {zk,zi-l,yi-l, wi} would induce a K 1 , 3 .Now, the vertices wi,zi-land zk form the X

i-1

Yi-1

X

i

Yi

X

k

k'

k'

W

Fig. 7. The subgraph of G isomorphic to H.

66

M . S . Jacobsen, K. Peters

same triangle as in Case 1 and the same set A leads to a subgraph isomorphic to H. Case 3. Suppose ( x k , y,) E E ( G ) for some i < k. Then (x,, x i ) E E ( C ) or else { y,, x i , zi, x k } would induce a K1,3.Thus the vertices x k , xi and y, form a triangle and we claim for

A = {zi-11wi,xi, Yij 4,W i t 1 9

xk,

”’,,

Z~-I},

( A ) is isomorphic to H. As in Case 1, we easily eliminate the twelve possible additional edges, this time in the subgraph induced by vertices not in I. We leave the details to the reader. Also note, if i = 1 we can apply Lemma 8 to x1 to find yo in (I) so that B y , z N ( w l ) . Pick Z ~ By,. E We need to show zo$ {z,, w,, z,-,, w k } in order to obtain distinct vertices to form the subgraph A. If zo = w2 then (wl, zl)E E ( G ) since there cannot be a K1,3 but then, if zo= w2 or zo = z,, {x,, y,, wl, z,} induces a C4. If z o = z k - , then (wl,w , ) E E ( G ) since again, there cannot be a K1,3but then, if zo = zk-l or zo = w,, {x,, wl, w,, x , } induces a C,. Thus G contains a subgraph isomorphic to H, and a contradiction results. Case 4. Suppose ( x k , x i ) E E ( G ) . Then ( x k , y,) E E ( G ) or else { x i , w,,y,, x k } would form a K1,3.This is exactly Case 3 and G would contain the same subgraph isomorphic to H. Finally, we return to Case 1 where i = 1. As in Case 3, we apply Lemma 8 to x, to find yo in ( I ) so that B, c N ( w , ) . Pick zo E Bye. If it happens that zo # z, then yo # y k (because there are no C4’s) so that {zo,w,,z k } induces a triangle. Then it follows that there is a vertex xo E I such that (xo, yo) E E ( G ) and xo # x k or else {zo, yo, ykrz,} would induce a C,. Now the remainder of the proof of Case 1 follows. So suppose zo = zk. Furthermore, suppose that (zo,zi) I# E ( G ) for 1 6 i < k and (zo, wi) $ E ( G ) for 1< i S k, otherwise we could choose zo # z., Now we can assume that ( I ) contains no additional edges other than those in Fig. 5 and the edge (w,,2,). If there would be additional edges, we can apply the appropriate one of the four cases to finish the proof. In this situation, let I’ = I - { y i : 16 i s k } U {zi:1 6 i 6 k}. The induced subgraph (Z’) has fewer edges than (I) but, each xi has a private neighbor (namely wi) and each zi has a private neighbor (namely itself) so that applying Lemma 8, there is a u E I’ such that B, E N ( z i ) for some i and note that u and {xi,yi, w,,z,: 1S i c k } . Pick t E B,. Thus (t, z,) E E ( G ) and also (t, E E ( G ) or else {t, u, zi, witl} induces a K,,3. Now, {t, zi, induces a triangle and since u is a vertex of distance one from this triangle, u is not adjacent to any of the vertices xi,y,, x i + l , yltl. But since u is irredundant in I’ there must be a vertex v E I’ such that ( u , v ) E E ( G ) . Now, the xifl,Y , + ~ } leads to the desired subgraph induced by {v, u, t, x i , y,, zi,w,+~, contradiction. Having exhausted all possibilities, we may again conclude that our original supposition that I was not independent is false and thus B(G) 3 IR(G). 17

Chordal graphs

(4

67

(b)

Fig. 8. No induced K,,,’s or H’s.

We conclude this section with examples that show that all three of the forbidden subgraphs of Theorem 10 are necessary, i.e., forbidding any two of the three does not imply that all three of the parameters are equal. In each example the verification of IR(G), T ( G ) and /3(G), though not trivial, is left to the reader. The graph G1 of Fig. 8(a) contains no subgraph isomorphic to K1.3 or to H (but contains C4’s) and has IR(Gl) = T(Gl) = 3 while /3(Gl) = 2. The graph G,, Fig. 8(b), satisfies the same conditions as GI and IR(G,) = 3 while T(G2)= /3(G2) = 2. The graph G3 in Fig. 9 contains no induced C4’s or H’s (but contains K 1 , 3 7 ~ ) and has IR(G3) = 7, T(G3)= 6, and /3(G3)= 5. The graph G4 of Fig. 10 contains no induced C4’s or K 1 . 3 ’ ~(but contains H’s) and IR(G,) = 9, r(G4) = 7 and /3(G4) = 6.

3. Conclusion Theorems 7 and 9 seem to present interesting algorithmic questions. For an arbitrary bipartite graph G, Dewdney [lo] shows that the problem of determining y(G) is NP-complete, while Hedetniemi, Laskar and Pfaff [19] show the same is

G3 :

Fig. 9. No induced C,’s or H’s.

M.S.Jacobsen, K . Peters

68

G4 :

Fig. 10. No induced C,’s or K,,,’s.

true for the problem of determining ir(G). For an arbitrary chordal graph G , the problem of determining y(G) and ir(G) are both NP-complete as shown by Booth [4]and Laskar and Pfaff [15]. The maximum independent set problem can be solved in polynomial time for the same two families of graphs (see [ll] and [12]). From this and Theorems 7 and 9, both problems of finding T ( G ) and IR(G), for G either bipartite or chordal, can be solved in polynomial time. Thus in some cases it is ‘hard’ to determine the minimum value of a parameter, while the corresponding max-min problem is relatively ‘easy’; This leads to the interesting question of why this occurs and in which other cases does this occur? We do not have answers to these questions, but feel they are worth investigating.

References [I] R.B. Allan and R.C. Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978) 73-76. [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [3] B. Bollobtis and E.J. Cockayne, Graph theoretic parameters concerning domination, independence and irredundance, J . Graph Theory 3 (1979) 241-249. [4] K.S. Booth, Dominating sets in chordal graphs, Report No. CS-80-34, Computer Science Department, University of Waterloo, Waterloo, Ont., 1980. (51 E.J. Cockayne, Domination of undirected graphs: a survey, in: Theory and Applications of Graphs, Lecture Notes in Mathematics 642 (Springer, Berlin, 1978) 141-147. [6] E.J. Cockayne, 0. Favoron, C. Payan and A. Thomason, Contributions to the theory of domination, independence and irredundance in graphs, Discrete Math. 33 (3) (1981) 249-258. [7] E.J. Cockayne and S.T. Hedetniemi, Independence graphs, in: Proceedings 5th S.E. Conference on Combinatorics, Graph Theory and Computing, Utilitas Math. (1974) 471-491. [8] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261. [9] E.J. Cockayne, S.T. Hedetniemi and D.J. Miller, Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull. 21 (4) (1978) 461-468. [lo] A.K. Dewdney, Fast Turing reductions between problems in NP; Chapter 4: Reductions between NP-complete problems, Report No. 71, Department of Computer Science, University of Western Ontario, London, Ont. 1983.

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[ll] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, (Freeman, New York, 1978). [12] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [13] S.T. Hedetniemi, R.C. Laskar and J. Pfaff, Irredundance in graphs: a survey, Congr. Numer. 48 (1985) 183-193. [14] R.C. Laskar and J. Pfaff, Domination and irredundance in graphs, Tech. Rept. 434, Department Mathematical Sciences, Clemson University, September 1983. [15] R.C. Laskar and J. Pfaff, Domination and irredundance in split graphs, Tech. Rept. 430, Department Mathematical Sciences, Clemson University, August 1983. [16] R.C. Laskar and H.B. Walikar, On domination related concepts in graph theory, Lecture Notes in Mathematics 885 (Springer, Berlin, 1980) 308-320. [17] J. Nieminen, Two bounds for the domination number of a graph, J. Inst. Math. Appl. 14 (1974) 183-187. [18] 0. Ore, Theory of Graphs, Amer. Math. SOC.Colloq. Publ. 38 (Amer. Math. SOC.,Providence, RI, 1962). [19] J. Pfaff, R.C. Laskar and S.T. Hedetniemi, NP-completeness of total and connected domination, and irredundance for bipartite graphs, Tech. Rept. 428, Department Mathematical Sciences, Clemson University, July 1983.

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Discrete Mathematics 86 (1990) 71-79 North-Holland

71

REGULAR TOTALLY DOMATICALLY FULL GRAPHS Bohdan ZELINKA Katedra Mathematiky VSST, Sokolsku 8,4 6 0 1 Liberec 1, Czechoslouakia Received 2 December 1988 The paper studies bipartite undirected graphs and directed graphs which are regular and totally domatically full.

1. Introduction The domatic number of a graph was defined by Cockayne and Hedetniemi [4] and the total domatic number by the same authors and Dawes [3]; the total domatic number was studied in [l].These concepts were defined for undirected graphs. The concept of the domatic number was transferred to directed graphs in (51. Here we shall transfer also the concept of the total domatic number to directed graphs; for this goal we shall first study bipartite undirected graphs. A11 considered graphs are finite without loops and multiple edges. Fundamental results concerning the domination in graphs can be found in Berge’s book [2]. A dominating (or totally dominating) set in an undirected graph G is a subset D of the vertex set V ( G ) of G with the property that for each vertex x E V ( G )- D (or x E V ( G ) respectively) there exists a vertex y E D adjacent to x . A domatic (or total domatic) partition of G is a partition of V ( G ) , all of whose classes are dominating (or totally dominating respectively) sets in G. The domatic (or total domatic) number d ( G ) (or d , ( C ) ) of G is the maximum number of classes of a domatic (or total domatic respectively) partition of G. The quoted authors have introduced some further related concepts. A graph G is called domatically full, if d ( G ) = md(G) + 1, where md(G) is the minimum degree of a vertex in G . (For any graph G the domatic number d ( G ) s md(G) + 1.) A uniquely domatic graph is a graph G in which there exists exactly one domatic partition with d ( G ) classes. Analogous concepts may be defined for the total domatic number. A graph G is called totally domatically full, if d,(G) = md(G). (For any graph d,(G)S md(G).) A uniquely total domatic graph is a graph G in which there exists exactly one total domatic partition with d , ( G ) classes. A total domatic partition of G with d , ( G ) classes will be called maximal. In the second section of this paper we shall study this concept for a particular case of undirected graphs, namely bipartite undirected graphs. In the third 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V. (North-Holland)

72

B . Zelinka

section we shall study them for directed graphs; we shall apply the results of the second section.

2. Bipartite undirected graphs In this section we shall consider bipartite undirected graphs. The vertex set of a graph G will be denoted by V(G) or shortly by V; the bipartition classes of a bipartite graph will be denoted by Vl and V,. Theorem 1. Let G be a bipartite undirected graph, let d , ( G ) = k. Then there exist at least k ! maximal total domatic partitions of G.

Proof. Let 9= {D,, . . . , D,} be a maximal total domatic partition of G. For each i = 1, . . . , k denote 0: = D, r3 V,, Df = Di n V,. We have 0: # 0 for each i = 1, . . . , k ; otherwise no vertex of V, would be adjacent to a vertex of Diand D, would not be a total dominating set of G. Analogously 0’ # 0 for i = 1, . . . , k. Hence 9‘= {Di,. . . , D:} is a partition of Vl and 9’={Df, . . . , D t } is a partition of V,. For each permutation p of the number set (1, . . . , k } we define the partition 9 ( p ) = { D , ( p ) , . . . , D , ( p ) } of V(G) such that D i ( p )= D: U D i ( i ) . Evidently 9 ( p ) is a maximal total domatic partition of G for each p and further 9 ( p J # 9 ( p , ) for p 1Z p , . As there are k ! permutations of (1, . . . ,k}, the assertion is proved. 0 Corollary 1. N o bipartite graph G with d , ( G ) 2 2 is uniquely totally domatic. We introduce a weaker concept than the uniquely totally domatic graph. A bipartite graph G will be called quasi-uniquely totally domatic, if for each $3’ are the same. (This is maximal total domatic partition 9 the partitions 9’, true for all partitions 9 ( p ) from the proof of Theorem 1.) This concept will be useful in the study of directed graphs. An interesting class of totally domatically full graphs is the class of regular ones. In such a graph G each vertex has degree d,(G). If 9is a maximal total domatic partition in G, then each vertex of G is adjacent to exactly one vertex from each class of 9. First we prove a theorem concerning undirected graphs which need not be bipartite. Theorem 2. Let G be a regular totally domatically full graph and let 9 be a maximal total domatic partition of G. Then all classes of 9 have the same cardinality.

Regular totally domatically full graphs

73

Proof. Let D1, D2 be two classes of 9. Each vertex of D1 is adjacent to exactly one vertex of D2 and each vertex of D2 is adjacent to exactly one vertex of D,. This yields a one-to-one correspondence between D, and D2 and thus ID,] = 1D21. As D1 and D2 were chosen arbitrarily, the assertion is true. 0 This implies an assertion on bipartite graphs.

Theorem 3. Let G be a regular totally domatically full bipartite graph, let 9be a maximal total domatic partition of G , let 9', g 2 have the same meaning as in the proof of Theorem 1. Then IVll = IV21 and all classes of 9'and 9 'have the same cardinality. Proof. As G is regular and bipartite, we have IVll = lV21. According to Theorem 2 all classes of 9 have the same cardinality p . Let D E 9,let D' = D n V,, D 2 = D n V2. Each vertex of D' is adjacent to exactly one vertex of D; as G is bipartite, this vertex is in D2. Similarly each vertex of D2 is adjacent to exactly one vertex of D' and thus there is a one-to-one correspondence between D' and D2. Hence ID'/= 1D21 = p / 2 . As D was chosen arbitrarily, this holds for all classes of 9. 0 Now we shall define an auxiliary concept. If G is a graph, then by H ( G ) we denote the graph with the vertex set V ( H ( G ) )= V ( G ) in which two vertices are adjacent if and only if they are connected in G by a path of length 2. The following theorem holds again for undirected graphs in general.

Theorem 4. Let G be a regular graph of degree k , let 9= { D l , . . . , Dk}be a partition of V ( G ) . The partition 9is a total domatic partition of G if and only if each Difor i = 1, . . . , k is an independent set in H ( G ) . Proof. Suppose that each Diis an independent set in H ( G ) . Let v be a vertex of G , let DiE 9. If there exist two distinct vertices of Diwhich are adjacent to v , then they are adjacent in H ( G ) , which is a contradiction with the independence of Di. As the degree of v is k, the vertex v is adjacent to exactly one vertex from Di.As v and Diwere chosen arbitrarily, this implies the assertion. 0 Now suppose that 9 is a total domatic partition of G. Consider DiE 9 and suppose that there exist vertices x , y of Diwhich are adjacent in H ( G ) . Then there exists a vertex z of G adjacent to both x and y in G. As z has degree k and there exists DiE 9 such that z is adjacent is adjacent to at least two vertices of Di, to no vertex of 0,. But then Diis not a total dominating set in G, which is a contradiction. Hence all classes of 9are independent sets in H ( G ) .

B. Zelinka

14

Corollary 2. Let G be a regular graph of degree k. Let there exist exactly one partition 9of V ( G ) into k classes, each of which is an independent set in H ( G ) . Then G is uniquely totally domatic. If G is bipartite, then in H ( G ) no vertex of V, is connected with any vertex of V,. Thus H ( G ) is the disjoint union of its subgraphs H 1 ( G ) ,H2(G)induced by V, and V, respectively.

Corollary 3. Let G be a bipartite regular graph of degree k and let there exist exactly one partition of Vl and exactly one partition of V, into k classes, each of which is an independent set in H ( G ) . Then G is quasi-uniquely totally domatic. We shall study an extremal case, when the number of edges of H ( G ) is as large as possible, i.e. H ( G ) is a disjoint union of two complete k-partite graphs. By %(k, m ) we denote the class of k-regular quasi-uniquely totally domatic bipartite graphs G such that d , ( G ) = k, the intersection of each class of any maximal total domatic partition 9 with V, and with V, has m vertices and two vertices of G belong to the same class of 9'(or of 9 ' )if and only if they both belong to Vl (or to V, respectively) and are not connected by a path of length 2 in G . By PG(k) we denote the finite projective geometry of order k, i.e. in which each point is incident to k 1 lines and each line is incident to k + 1 points. The geometry PG(k) has k2 k 1 points and also k2 k 1 lines.

+ + +

+ +

Theorem 5 . The class %(k, k) for a positive integer k is non-empty if and only if there exists a finite projective geometry PG(k). Proof. Let PG(k) exist. Choose a point p o of PG(k) and a line qo incident to p o . The points incident to qo and different from p a will be p , , . . . , p k ; the lines incident to p o and different from qo will be q , , . . . , qk. Let V, (or V,) be the set of all points (or lines) of PG(k) which are not incident to qo (or p o , respectively). Let G be the bipartite graph with the bipartition classes V,, V2 in which two vertices are adjacent if and only if they are incident in PG(k). The reader may verify himself that G E %(k, k). Each class of 9'is the set of all points incident to one of the lines q l , . . . , qk except p o , each class of 9 ' is the set of all lines incident to one of the points p l , . . . ,p k except qo. Now suppose that %(k, k) it 0 and let G E %(k, k). Then we can consider $3' and 9 ' . We shall construct PG(k). The set of points of PG(k) will be V, U { p o ,p l , . . . ,P k } , its set of lines will be V2U (qo, q,, . . . , qk}, where Po, p l , . . . ,P k , qo, q l , . . . , qk are distinct elements not belonging to V, U V,. Every point from Vl is incident to all lines from V, which are adjacent to it in G and further it is incident to qi if and only if it is in 0:. Every point p i for 1< i s k is incident to all lines from Df and to qo. The point p o is incident to the lines qo, q , , . . . , qk. To any two of the vertices p o , p l , . . . , P k there exists exactly one

Regular totally domatically full graphs

75

line incident with both of them (their joining line), namely qo. To any two vertices of Vl belonging to the same class 0: the joining line is qi, for two vertices of Vl not belonging to the same class of 9’ it is the line of V, adjacent to both of them in G. For the point p o and a point x E Dl c V, the joining line is qi. For a point p i , where 1< i < k and a point x E V, it is the line from D fto which x is adjacent in G. Evidently the joining line is always unique. Analogously we can find intersection points of all pairs of lines. Therefore the geometry thus constructed is PG(k). A graph thus constructed from PG(2) is a circuit of length 8. The graph obtained in this way from PG(3) has the following matrix of adjacency between Vl and V,:

1 0 0 0 1 0 0 0 1

0 1 0 0 0 1 1 0 0

0 0 1 1 0 0 0 1 0

0 1 0 1 0 0 0 0 1

0 0 1 0 1 0 1 0 0

1 0 0 0 0 1 0 1 0

0 0 1 0 0 1 0 0 1

1 0 0 1 0 0 1 0 0

0 1 0 0 1 0 0 1 0

Corollary 4. Let k be a power of a prime number. Then %(k, k ) # 0.

Theorem 6. The class %(k, 1)# 0 for each positive integer k , the class %(k, 2 ) # 0 for each integer k z=2.

Proof. For each positive integer k the complete bipartite graph Kk,kbelongs to %(k, 1). The circuit of length 8 is in %(2, 2). Consider k 3 3 . Let V,= { x ~ ., . . , x k , x i , . . . , x ; } , V, = {yl, . . . ,yk, y ; , . . . ,y ; } be the bipartition classes of a graph G and let the edge set of G consist of edges x i y l ! ,x ] y i for i = 1, . . . , k and x i y j , xl!y,! for any pair of different numbers i, j from the numbers 1, . . . , k . The graph G is regular of degree k. For each i = 1, . . . , k the vertices xi,xf have distance greater than 2. If i # j , then x i , xi are both adjacent to y,, for any h different from both i and j ; analogously x ] , xi’ are both adjacent to y l for such a number h. The vertices x i , x i are adjacent both to y ! . Thus the family 9’ = {{xl, x i } , . . . , { x k , x ; } } is a partition of V, into classes with the property that two vertices of Vl belong to the same class if and only if they are not adjacent in H ( G ) . An analogous assertion holds for 9’= {{yl, y ; } , . . . , { y k , y ; } } . Therefore G E %(k, 2). 0

B. Zefinka

76

-

r

0 1 1 1 0 -0

1 0 1 0 1 0

1 1 0 0 0 1

1 0 0 0 1 1

0 1 0 1 0 1

0 0 1 1 1 0 _

Theorem 7. For k < m the class %(k, m ) is empty. Proof. Suppose that there exists a graph G E % ( k , rn) for k < m . Let D:,0: be two classes of 9'. In H ( G ) the subgraph induced by D : U D : is a complete bipartite graph and has m 2 edges; therefore there are at least m2 paths of length 2 going from 0 : into Dl in G. But, on the other hand, each vertex of V2is adjacent to exactly one vertex of D: and to exactly one vertex of 0:in G and thus it is an inner vertex of exactly one path of length 2 connecting a vertex of 0 : with a vertex of 0:. As IV21 = km < m2,this is a contradiction. 0

Theorem 8. For each m there exists a quasi-uniquely totally domatic graph G with d , ( G ) = 2 in which all classes of 9'and 9 'have cardinality m . Proof. This graph is a circuit of length 4m.

3. Directed graphs

Directed graphs considered here are finite without loops and pairs of vertices joined by two or more equally directed edges. Let G be a directed graph. A subset D of the vertex set V ( G ) of G is called dominating (or totally dominating), if to each vertex x E V ( G )- D (or to each x E V ( G ) respectively) there exist vertices y , z of D such that there exist edges from x to y and from z to x. Outgoing from this concept, we may define the domatic number and the total domatic number (and further related concepts) of a directed graph quite analogously as in the case of an undirected graph. Note that the total domatic number of a directed graph is well defined only for directed graphs without sources and sinks (analogously as for undirected graphs without isolated vertices). In order to transfer the results on bipartite undirected graphs to directed graphs, we introduce an auxiliary concept. Let G be a directed graph with the vertex set V ( G ) . Consider two disjoint sets V,, V2of the same cardinality as V ( G ) . Choose a one-to-one mapping f , of V ( G )

Regular totally domatically full graphs

77

onto V, and a one-to-one mappingf, of V ( G )onto V2. By B ( G ) we denote the bipartite undirected graph with the bipartition classes V,, V, in which a vertex fi(u)E V, is adjacent to fi(u) E V2if and only if an edge goes from u to u in G .

Theorem 9. Let G be a directed graph, let D be a subset of V ( C ) . Then D is a dominating set in G if and only if fi(D) Uf2(D)is a dominating set in B(G). Theorem 10. Let G be a directed graph, let D be a subset of V ( G ) . Then D is a totally dominating set in G if and only i f f i ( D ) U f , ( D )is a totally dominating set in B(G). Proofs are left to the reader.

Corollary 5. Let G be a directed graph. Then d ( B ( G ) )z=d ( G ) and d , ( B ( G ) )3 dt(G). Corollary 6. Let G be a directed graph, let B ( G ) be uniquely domatic and d ( B ( C ) ) = d ( G ) . Then G is uniquely domatic. Corollary 7. Let G be a directed graph, let B ( G ) be quasi-uniquely totally domatic and d , ( B ( G ) ) = d,(G). Then G is uniquely totally domatic. This yields two constructions of uniquely domatic and uniquely totally domatic directed graphs.

Construction 1. Let Go be an undirected bipartite graph which is uniquely domatic and regular of degree k = d(Go) - 1. Let V,, V, be bipartition classes of Go, let 9 be the (unique) maximal domatic partition of Go. Choose a one-to-one mapping g of Vl onto V, which maps each class of 9onto itself. Direct all edges of Go from Vl to V2.Identify x with g ( x ) for each x E V,. The obtained graph will be denoted by G. Construction 2. Let Go be an undirected bipartite graph which is quasi-uniquely totally domatic and regular of degree k = d,(Go). Let V,, V2 be the bipartition classes of Go, let 9 be a maximal total domatic partition of Go. Choose a one-to-one mapping g of V, onto V2 which maps each class of 9onto itself and each vertex onto a vertex non-adjacent to it. Direct all edges of Go from V,to V,. Identify x with g ( x ) for each x E V,. The obtained graph will be denoted by G.

Theorem 11. The graph G obtained by Construction 1 (or Construction 2 ) is a uniquely domatic (or uniquely totally domatic, respectively) directed graph. Proof. Evidently Go= H ( G ) and thus the assertion follows from Corollaries 6 and7. 0

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Corollary 8. Let k , m be positive integers such that either k = m and it is a power of a prime number, or k 3 2, m = 2. Then there exists a regular totally domatically full directed graph G such that d,(G) = k , the graph G is uniquely totally domatic and each class of the maximal total domatic partition of G has m vertices. At the end we shall present some fundamental assertions on total domatic numbers of directed graphs.

Theorem 12. Let G be a directed graph with n vertices. Then d,(G) n/2. Proof. Let D be a totally dominating set in G. As each vertex of D must be adjacent to a vertex of D and there are no loops, JDI2 2. This implies the assertion. 0 Theorem 13. Let G be a directed graph with n vertices with the property that any pair of vertices is joined by at most one edge. Then d,(G) S n / 3 . Proof. Let D be a totally dominating set in G. Each vertex of D is joined by edges with at least two vertices of D ; one of these edges comes into it, the other goes out. Thus the subgraph of G induced by D contains a circuit and ID1 3 3. This implies the assertion. 0 Theorem 14. Let k , n be positive integers, n 3 3k. Then there exists a tournament T with n vertices such that d,( T ) = k. Proof. The vertex set of Twill be the union of two disjoint sets X , Y. The set X is the set of vertices x(i, j) for 1 G i s k , 1 S j S 3. If IZ = 3k, then Y = 0; else it is the set of vertices y ( i ) for 1S i < n - 3k. For each i the edges go from x ( i , 1) to x ( i , 2), from x ( i , 2) to x(i, 3 ) and from x(i, 3 ) to x(i, 1). Let i , < i2;if j , = j 2 , then an edge goes from x ( i , , j , ) to x ( i 2 , j 2 ) else it goes inversely. Further edges go from y ( i l ) to y ( i 2 ) for i , < i 2 . Finally, for i = 1, . . . , k edges go from each vertex of Y to x(i, 1) and from x ( i , 2) and x ( i , 3 ) to each vertex of Y. Thus the tournament T is given. If n = 3k, then d , ( T ) k according to Theorem 13. If n > 3k, then there exists the vertex y ( n - 3 k ) E Y which has the out-degree k and thus also d , ( T ) k. Denote Di= {x(i,l), x ( i , 2 ) , x ( i , 3 ) ) for i = 1, . . . , k. Then 9= { D l , . . . , Dkp1, DkU Y} is evidently a total domatic partition of T and thus d , ( T ) = k. 0

4. Problems (1) For which numbers k, m , is the class %(k, m ) non-empty? ( 2 ) For which positive integers k does there exist an undirected graph G with k 2

Regular totally domatically full graphs

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vertices which is regular of degree k and in which there exists a partition 9 = { D l , . . . , Dk}of the vertex set of G into classes of equal cardinalities with the property that two vertices x , y are connected by a path of length 2 in G if and only if they belong to different classes of 9? Such a graph would be a non-bipartite analogy of the graphs from %(k, k) and would be uniquely totally domatic. It is possible to look also for further results analogous to those from Section 2 for graphs which are not bipartite in general.

References [l] R.B. Allan, R. Laskar and S.T. Hedetniemi, A note on total domination, Discrete Math. 49 (1984) 7-13. [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [3] E.J. Cockayne, R. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-215. [4] E.J. Cockayne, S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261. [5] B. Zelinka, Semidomatic numbers of directed graphs, Math. Slovaca 34 (1984) 371-374.

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Discrete Mathematics 86 (1990) 81-87 North-Holland

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DOMATICALLY CRITICAL AND DOMATICALLY FULL GRAPHS Douglas F. RALL Department of Mathematics, Furman University, Greenville, SC 2%13, USA Received 2 December 1988 A subset, D, of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in D or adjacent to some vertex in D . The maximum cardinality of a partition of the vertex set of G into dominating sets is the domatic number of G, denoted d(G). G is said to be domatically critical if the removal of any edge of G decreases the domatic number, and G is domatically full if d ( G ) assumes the known lower bound of 6(G) + 1. An example is given to settle a conjecture of B. Zelinka concerning the structure of a domatically critical graph. We also prove that a domatically critical graph G is domatically full if d ( G ) 3 and provide. examples to show this does not extend to the cases d ( C )> 3.

1. Introduction Consider a finite graph G with vertex set V ( G ) and edge set E ( G ) , which has neither loops nor multiple edges. For x E V ( G ) , N ( x ) will denote the set of vertices in V ( G )each of which is adjacent to x. If G and H are two graphs having no vertices in common, then G H denotes the join of G and H. G + H has vertex set V ( G )U V ( H ) and edge set E ( G ) U E ( H ) U {xy I x E V ( G ) and y E V ( H ) } . See [l]for any undefined terms. A subset D of V ( G )is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D.A partition of V ( C ) into dominating sets of G is a D-partition. The concept of dominating sets in graphs has been studied extensively in recent years. Many of the resulting papers have involved the domination number of G , A(G), which equals the minimum number of vertices in a dominating set of G. See [2-41 for a survey of some of these results. Since the paper of Cockayne and Hedetniemi [3] much attention has been given to the maximum number of pairwise disjoint dominating sets of a graph. Specifically, the domatic number, d ( G ) , is the maximum number of classes in a D-partition of G. For any given D-partition, P, with d ( G ) classes each vertex of G must be adjacent to a vertex of every dominating set in P other than its own. This requires each vertex to have degree at least d ( G ) - 1. Hence we have the following result as found in [3].

+

Proposition 1. Zf G is any graph, then d ( G )c 6 ( G )+ 1. 0012-365X/90/$03.50 @ 1990- Elsevier Science Publishers B.V. (North-Holland)

82

D.F. Rail

When 6 ( G ) = 0 or 6 ( G ) = 1 the above upper bound is sharp. While it was known that the bound was not sharp for 6(G)> 1, in [ 6 ] , Zelinka constructs graphs of arbitrarily large minimum degree but having domatic number 2. A graph G for which d ( C ) = 6(G) 1 is said to be domatically full. The following proposition along with Proposition 1 imply that any graph G with 6(G) S 1 is domatically full.

+

Proposition 2 (Ore [ 5 ] ) . If D is a minimal dominating set in a graph G with 6(G) 3 1, then V ( G )- D contains a minimal dominating set. Thus d ( G ) = 1 if and only if G has an isolated vertex. Thus if we add an endvertex to any graph without isolated vertices we obtain a graph of domatic number 2. A t the other extreme Zelinka's result yields graphs with domatic number 2 and large minimum degree. It appears then that the interesting open question posed by Cockayne and Hedetniemi in [3], namely to characterize those graphs of domatic number 2, will indeed be difficult to resolve. The domatic number of a subgraph need not be related in any consistent way to that of the original graph. For example, d ( K n U K,) = 1 while d ( K , ) = n, and d ( K , ) = m but d ( T ) = 2 for any nontrivial subtree T of K,. However, if H is a spanning subgraph of G, then d ( H ) s d ( G ) . If d ( G ) = k 2 and d(G - e ) < k for every edge e of G we call G domatically k-critical, or critical for brevity. In this paper we present some results concerning critical graphs and the relationships between critical and domatically full graphs.

2. Domatically critical graphs A mentioned in the introduction d ( H ) s d ( G ) whenever H is a spanning subgraph of G. In particular we easily obtain the following bounds for the domatic number upon removal of a single edge.

Proposition 3. d ( G ) - 1 c d ( C - e ) c d ( G )for any e E E ( G ) . Now it follows that G is domatically k-critical if and only if d(G - e ) = k - 1 for every edge e E E(G). One reason for studying domatically k-critical graphs is stated in the following.

Proposition 4. If G is a graph with d ( G ) = k, then G contains a spanning subgraph H which is domatically k-critical. Proof. One can simply remove edges (one at a time) from G which do not decrease the domatic number until a spanning subgraph G' of G is obtained which is domatically k-critical. 0

x

Domatically critical and domatically full graphs

'ar G

A G1

83

G2

Fig. 1 . A graph and two of its critical spanning subgraphs.

Of course the k-critical spanning subgraph so obtained is not unique nor is its domination number necessarily the same as that of the original graph. In Fig. 1, G1 and G2 are two different domatically 3-critical spanning subgraphs of G, y(G) = 1, y(G,) = 2, y(G2)= 1. It is also easy to show that one may have to remove an arbitrarily large proportion of the original edge set to arrive at a critical spanning subgraph.

Proposition 5 . Let G be a domatically k-critical graph with a D-partition P = { V l , V,, . . , , v k } . For each i, is an independent set.

Proof. If there is a j , 1s j 6 k for which induces a subgraph which contains an edge e, then P is also a D-partition of G - e, which is a contradiction to G being domatically k-critical. 0 Cockayne [2] proposed the study of domatically critical graphs and Zelinka [7] provided a necessary condition on a graph G if it is domatically critical.

Theorem 6 (Zelinka). Let G be a domaticaliy k-critical graph with D-partition { V l , V2,. . . , v k } . For any two distinct i, j the subgraph induced by vi U V, is a bipartite graph with parts and each of whose connected components is a star with at least two vertices. Zelinka also conjectured that the necessary condition was sufficient.

Conjecture 7 (Zelinka). Every graph G which has a D-partition of order d ( G ) having the property described in Theorem 6 is critical. If d ( G ) = 2 then the above conjecture is correct since G is then the disjoint union of stars. Every edge e in such a graph in incident with a vertex of degree one. G - e has an isolated vertex and so by Proposition 2 d ( G - e ) = 1. Thus G is 2-critical.

D . F. Rall

84

Fig. 2. H , counterexample to conjecture 7.

For d ( G ) = 3 we have a counterexample to the conjecture which is shown in Fig. 2. d ( H ) = 3 with D-partition {V,, V,, V,}. For i # j , V , U induces a bipartite subgraph whose components are all stars, and yet H is not 3-critical since H-(3, 6) has { (3, 4, 6}, ( 2 , 8, 9}, { 1, 5, 7)) as a D-partition. Counterexamples to the conjecture for d ( G ) = n 3 4 can be constructed by forming the join G = H + K,,-3. In fact concerning the join of a critical graph to a complete graph we have the following result.

Lemma 8. Suppose G is a graph with d ( G ) = k. Then G is domatically k-critical and only if G + K,, is domatically (k + n)-criticalfor every positive integer n.

if

Proof. Assume G is k-critical. Let H = G + x . By Proposition 4.2 of [3], d ( H ) = k + 1. Consider an arbitrary edge e of H. If e E E ( G ) , then d ( H - e) = d ( ( G - e ) + x ) = d ( G - e ) + 1 = k - 1+ 1 = k. Suppose that e E E ( H ) but e @ E ( G ) , say e = ( x , u), and assume d ( H - e ) = k + 1. Let P = {Vl, V,, . . . , V,+l} be a D-partition of H - e with x E V,,, and u E V,, r # 1. If r = k + 1, set v = u. Otherwise there exists a vertex v # x in V,,, since u must be dominated by V,,, in H - e. But then W, = V, U (V,+, - { x } ) is a dominating set of G and W, is not independent since u is dominated by V, in H - e. P' = { W , , V,, . . . , V,} is a D-partition of the k-critical graph G contrary to Proposition 5. Thus, d ( H - e ) = k and H is domatically (k 1)-critical. The general conclusion regarding G + K,, follows by induction. If G has domatic number k but is not critical, then G K, cannot be critical since each vertex of K,, forms a dominating set of G + K,, and an edge whose removal from G does not lower the domatic number can also be deleted from G + K, leaving the domatic number unchanged. 0

+

+

Starting with a graph Gkwhich is known to be domatically k-critical we can

Domatically critical and domatically full graphs

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repeatedly apply Lemma 8 to obtain a sequence of graphs Gk, Gk+l,Gk+2,. . . where G, (n 3 k) is n-critical. If G2 = K 2 then this sequence is the sequence of complete graphs. The wheel, W,, occurs as the second term of a sequence which begins with C3n,a 3-critical graph; n copies of K4 with one vertex from each identified results from letting G, be n disjoint copies of K 3 .

3. Relationships between critical and full If G is a graph each of whose edges is incident with a vertex of degree one, then G is 2-critical. In fact, if d ( H ) = n 2 2 and each edge of H is incident with a vertex of degree n - 1, then the deletion of any edge of H leaves a graph H’ with 6 ( H ‘ )= n - 2 and so by Proposition 1, d ( H ’ )= n - 1 and H is n-critical. For example, such a graph H can be constructed either by the method described in the proof of Theorem 2 in Zelinka [7] or by joining a vertex to a regular domatically full graph having domatic number n - 1. We now investigate a modified converse to the preceding. In particular, if G is a domatically n-critical graph, is G necessarily domatically full? The structure of a 2-critical graph G - each component is a nontrivial star - allows one to see that G is domatically full. It is also true for n = 3 but is false for n 2 4, as the following shows. Theorem 9. Every domatically 3-critical graph i s domatically full. However, for each n 3 4 there exists a graph which is n-critical but which is not domatically full.

Proof. Let G be a domatically 3-critical graph with D-partition P = {V,, V,, V,}. Assume G is not domatically full; that is 6(G) 2 3 . We handle first the case 6(G) = 3. Let u be a vertex of G with deg(u) = 3. Assume without loss of generality that u E V,, N ( u ) = { u l , u,, u,} with u1 E V,, u2, u, E V,. Since G is 3-critical, P is not a D-partition of either G - (u, u2) or G - (u, u,), and so N ( u 2 )c { u } U V,, N ( 4 ) c { u } u v2. Let w,,w, E N(u2)n V2 and w,, w4 E N(u,) n V2. If { w l , w2} n { w , , w4} # 0 then the subgraph induced by V,U V, has a component which is not a star, contradicting Theorem 6. Thus V2has at least 4 vertices. Each wi, 1=zi s 4, has a unique neighbor in V3 by Theorem 6 and so must have at least 2 neighbors in v,:x , , x2 E N(w,) n v,;x,, x4 E N(w*) n vl; x S , x6 E N ( w ~n) v,;x 7 , x s E N ( w ~n) V,. As above the sets {xl, x , } , {x,, x4}, { x 5 , X g } , and { x 7 , x 8 } are pairwise disjoint and so V, contains at least 8 vertices. Continuing in this manner we see V, contains at least 16 vertices, V, has at least 32 vertices, and so on. Since G is finite this is clearly impossible, and so 6(G) cannot be 3. The proof showing 6(G) cannot be 4 or larger proceeds similarly starting with any vertex of minimum degree. Thus 6(G) = 2 and G is domatically full if it is 3-critical. 0

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86

"4

"3

Fig. 3. A 4-critical graph K , 6 ( K ) = 4.

The graph K of Fig. 3 is 4-critical but is not domatically full. Since K is regular of degree 4 it is easy to see that any dominating set of K must have at least 3 vertices, so d ( G )s = 4 and the indicated partition {Vl, V,, V,, V,} is a D-partition so d ( K ) = 4. To prove that K is critical, we found (by computer) all dominating sets of cardinality 3. In addition to V,, V,, V,, V, are { 2 , 5 , 7 } , {4,8, lo}, {3,6,11} and {1,9,12}. It is now straightforward to check that d ( K - e) = 3 for every edge e of K. By Lemma 8, K K n - , is n-critical but not domatically full. We have also constructed a 5-regular graph which is domatically 5-critical and suspect that for any k 3 4 there exists a k-regular graph which is k-critical.

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References [l]

M. Behzad, G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs (Pnndle, Weber and

Schmidt, Boston, MA, 1979). [2] E.J. Cockayne, Domination of undirected graphs-a survey, in: Y. Alavi and D . R . Lick, eds., Theory and Applications of Graphs (Springer, Berlin, 1978) 141-147. [3] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261.

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[4] R. Laskar and H.B. Walikar, On domination related concepts in graph theory, Tech. Rept. 352, Dept. Math. Sciences, Clemson Univ., Clemson, SC, 1980. [5] 0. Ore, Theory of Graphs, Amer. Math. SOC.Colloq. Publ. 38 (Amer. Math. SOC.,Providence, RI, 1962). [6J B. Zelinka, Domatic number and degrees of vertices of a graph, Math. Slovaca 33 (1983) 145-147. [7] B. Zelinka, Domatically critical graphs, Czech. Math. J. 30 (1980) 486-489.

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Discrete Mathematics 86 (1990) 89-97 North-Holland

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ON GENERALISED MINIMAL DOMINATION PARAMETERS FOR PATHS B. BOLLOBAS Trinity College, Cambridge, LIK

E.J. COCKAYNE University of Victoria, B. C . , Canada

C. M. MYNHARDT University of South Africa, Pretoria, South Africa Received 2 December 1988 A subset X of vertices of a graph is a k-minimal P-set if X has property P, but the removal of any 1 vertices from X,where 1 < k, followed by the addition of any ( I - 1) vertices destroys the property P. We note that 1-minimality is the usual minimality concept. In this paper we determine G(Pn),the largest cardinality of a k-minimal dominating set of the n-vertex path P,. We also prove for any n-vertex graph G, T,(G)y(G) n and finally a 'Gallai-type' theorem for k-minimal parameters is established.

1. Introduction The concept of minimality may be generalised as follows. Let S be a set and P a property enjoyed by soi,le of the subsets of S. A subset of S with (without) property P is called a P-set (P-set). A subset X of S is called a k-minimal P-set if X has the property P, but for all 1 satisfying 1 S 1 c k, all 1-subsets U of X and all (1 - 1)-subsets R of S, ( X - U ) U R is a P-set. We note that 1-minimality is the usual concept of minimality. In this paper the set S will be the vertex set V of a graph and a subset X E V has property P if and only if X is dominating. This specialization of the above defines a k-minimal dominating set of a graph. We define &(G) to be the largest cardinality of a k-minimal dominating set of G. Let y(G) ( T ( G ) )be the smallest (largest) cardinality of a minimal dominating set of G. The following inequalities are obvious for any graph C:

y ( G )C

.

* *

c &(C)c . .. C &(G) C &(G) = T ( G ) .

In this work, we first strengthen the theorem of Jaeger and Payan [6] which asserts that the product of domination numbers of a graph and its complement is at most the number of vertices in the graph. The examination of their proof, in fact, motivated the new extended definitions of minimality given here. The principal result of the paper is the exact determination of G(Pn). This calculation is surprisingly complex although the evaluations of y(P,,) and I'(P,) are trivial. 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V. (North-Holland)

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It is clear that many other parameters of graphs and more general structures, which are defined in terms of minimality may also be similarly generalized and that the maximality concept may also be extended. In [ 2 ] , Cockayne, MacGillivray and Mynhardt compute &(Pa) and Pk(Cn)where Pk(G) is the smallest cardinality of a k-maximal independent set of vertices of G. The final result of this paper is a generalization of a theorem of Gallai 131 concerning certain k-maximal and k-minimal parameters. For an excellent bibliography of the study of domination in graphs, the reader is referred to [ 5 ] . Extensions of Gallai’s Theorem are given in [l].

2. The results 2.1. The Jaeger- Payan generalisation

In [6] Jaeger and Payan proved the following Nordhaus-Goddum type result for the domination number.

Theorem 1 (Jaegar and Payan). For any n vertex graph G, y ( G ) y ( G )

n.

We show here that their proof may be adapted to prove the stronger result:

Theorem 2. For any n-vertex graph G, T , ( G ) y ( G ) n. Proof. The result is trivial for &(G) = 1, hence we assume T,(G)= t 2 2. Let X = {xl,. . . ,x , } be a largest 2-minimal dominating set. Since X is dominating, there exists a partition of V ( G ) into classes V,, . . . , V,, such that for each i = 1, . . . , t, xiE V , and x, is adjacent to all other vertices in Let B be such a partition such that the number of vertices which are adjacent to all other vertices in their class, is maximum. We show that each class of $9’ is a dominating set of For suppose V, (say) does not dominate Then there exists a vertex x which is in V, (say) and is adjacent in G to all vertices of V,. Vertex x is not adjacent to all vertices of V,, for otherwise ( X - { x l , xz}) U {x} dominates G, contrary to 2-minimality. Therefore x # x 2 and we now consider the partition B’ = V ; ,. . . , V : where V ;= V,U { x } , V ’- V2 - { x } , V l = V , for 2 < j -
v.

c.

c.

,-

c

n

IV,l>

= i=l

2 y ( c ) = &(G)y(C). i=l

The domatic number d ( C ) of G is the largest order of a partition of V ( G ) into dominating sets of G. The following deduction from the proof of Theorem 2 is obvious.

On generalised minimal domination parameters for paths

91

Corollary 1. For any graph G , &(G) G d ( G ) . 2.2. The calculation of &(P,) Define an 1-subset Q of a dominating set X of a graph G to be stable (unstable) if and only if there does not exist (there exists) an ( 1 - 1)-subset R of V - X such that (X - Q) U R is dominating. Notice that a dominating set X is k-minimal if for each l s l s k , all 1-subsets of X are stable. Let P, have the (left to right) vertex sequence v l , . . . , v,. The simple proof of the following proposition is omitted.

Proposition 1. Zf X is a (k - 1)-minimal but not k-minimal dominating set of P,, then any unstable k-set in X consists of consecutive vertices of X , i.e. if X = {v,,, vm2,. . . , v,,} where m , 6 m26 . . * G m,, then any unstable k-set in X may be written as {urn,+,, . .., for some i. In what follows P, (i, m ) where 1s i 6 m S n, will denote the subgraph of P, induced by the vertex subset {v;,v ~ +. .~. ,, v,}. Our first Lemmas l(a) and (b) give necessary conditions for consecutive vertices of P, to appear in a k-minimal dominating set.

Lemma l(a). For any k, n 2 1, if X is a k-minimal dominating set of P,, such that {vi-,,v;} E X for some i, then exactly one of the following holds: (i) n < i 3k, in which case n = i + 3r + 1, where O s r s k - 1 and V ( p n ( i , n ) ) n x = { v i + 3 1 ; = 1~,,. . . , r } ; (ii) n 2 i + 3k and V(Pn(i,i + 3 k )) n X = { v ~ + ~1 j, = 0, . . . , k}.

+

Proof. By induction on k. Let k = 1 and let X be a 1-minimal (i.e. minimal) dominating set of P, with { v ~ - ~v ,i } E X for some i. Suppose n < i + 3. If n = i, then X is not minimal, hence n 3 i 1. If n = i 2 then vi+2= v, is dominated, therefore v ; +or ~ u ; + is ~ in X . In either case X is not minimal so the case n = i 2 cannot occur. Therefore n = i 1 and (i) holds. If n 2 i 3, it is easily seen that v i + ,or ui+* in X contradicts minimality and hence v ; +must ~ be in X so that v i t 2 is dominated. Hence the assertion (ii) holds. Now suppose that the result is true for k - 1 where k 2 2 and let X be a k-minimal dominating set of P, with { v ~ - ~vi} , GX. Case 1. Let n < i + 3(k - 1). Since X is (k - l)-minimal, by the induction hypothesis, condition (i) holds. Case 2. Let i 3(k - 1)S n. By the induction hypothesis,

+

+

+

+

+

+

V(P,(i, i + 3(k - 1))) n X If n = i

1;

= { v ; + ~ ,= 0, . . .

, k - l}.

+ 3(k - l), then X ' = ( X - {vi+3jI j = 0, . . . , k - 1)) U { v ~ +1 j ~=~0, + ..~ . ,k - 2 )

(1)

B . Boiiobris et ni.

92

+

dominates P,, which contradicts the k-minimality of X . If n = i 3(k - 1) + 2, then since v, is dominated, v, or v,-, is in X. Again X ' dominates and we conclude that either n = i + 3 ( k - 1)+ 1 in which case (i) is satisfied, or n 2 i 3k. In this case, it follows from (1) and the 1-minimality of X , that neither u ~ + ~ ( ~nor - ~Vi+3(k-1)+2 ) + ~ is in X . Therefore vi+3kE X to dominate its predecessor and (ii) holds. This completes the proof of Lemma l(a). 0

+

Lemma l(a) has a 'dual' form concerning the vertices to the left of a consecutive pair v,-,, v, in a k-minimal dominating set. The proof is similar to that of Lemma l(a) and is omitted.

Lemma l(b). For any k, n 2 1, if X is a k-minimal dominating set of P, such that { v ~ -vi} ~ ,c X for some i , then exactly one of the following holds: (i) i - 1 - 3k < 1 (i.e. i < 2 + 3k), in which case 1 = i - 1 - 3r - 1 (i.e. i = 3r) for some r, 1d r S k, and

v ( P , ( 1, i - 1)) n X (ii) i - 1 - 3k 2 1 (i.e. i

V ( P ,( i

-

= { vi-

3 3k

I O ~j

cr

- I} ;

+ 2) and

1 - 3k, i - 1 ) ) fX l = { ~ i - l - - 3 , I 0 G j < k}.

By a &-set we mean a k-minimal dominating set of largest cardinality.

Theorem 3. For any k, n 3 1, P, has a &-set which is independent. Proof. Let n and k be such that the statement is false. Since the result follows easily if k = 1, it is clear that k 3 2. Let X be a &-set in P,, such that P , ( X ) has as few edges as possible and let i be the largest integer such that I J - ~ v, , E X. By the ~ v , +IS~ an end vertex of choice of and Lemma 1, respectively, neither v , + nor P, and { v , + ~~ ,, + ~ } f l X =hence 0; n 3 i + 3 and V , + ~ E XConsider . the set X' = (X - {v,}) U { u l t l } which clearly dominates P,. By the choice of X, X' 1s not k-minimal. Hence there is a smallest integer 1 s k for which there exists an I-set Q c X ' such that (XI - Q) U R dominates P, for some ( I - 1)-set R E V ( P , ) . Suppose u , + E~ Q. By the choice of I, u , + 4~ R. But then (X - ((Q - { u , + ~ U }) {v,})) U R = (XI - Q) U R which dominates P,, contradicting the k-minimality of X . Hence v Z t l 4Q. By the choice of I and by Proposition 1, either Q E {vl, 212, . . . , V , - I } or Q c { v , + 3 ,~ ~ . . .~ , vn}. 4 If , Q c {v1, v2, . . . , v l - l } , then since (XI - Q) U R dominates P,, it is clear that (X - Q) U R dominates P,. Hence Q G { v , + ~ vtt4, , . . . , v,}. If v , +4~Q then v1+3 E (X' - Q) U R in which case (X - Q) U R dominates P,. Hence, again by Proposition 1 and Lemma 1, Q = { v , , ~I,j = 1, 2, . . . , I } . By the choice of i , ~ , + 3 1 + 1 Hence at least 31 - 3 1 vertices of P are not dominated by X' - Q and since Y ( P ~ [ - ~=+[ ~( 3)( I - 1) + 1)/3], no (I - 1)-set

ex'.

+

On generalised minimal domination parameters for paths

93

R E V(P,,) exists such that (X’ - Q ) U R dominates P,,. This contradiction completes the proof. 0 We now state and prove the principal result of the paper. Theorem 4. For all k 3 1 and n 3 2, ( k + 1)n i f n = 31 1(mod3k 1) for some 1 E (0, 1, . . . , k - l}, T,(Pn)= otherwise.

+

+

Proof of lower bound. Write n as n = h + r, where h = m(3k integer m, and 0 < r s 3k. Let

+ 1) for

X = { v i eV(P,) I i = 1(mod3k + 1) or i =3f (mod3k + l), 1

some

~k}. 1

+

Clearly, X dominates P,, unless r = 31 2 for some 1, 1s 1 s k - 1. Define

y={

(x-{?Jh+3jl 1Sjsl})u{?Jh+3j+l l l s j S 1 } if r = 31 2 for some 1, 1c 1 k - 1, X otherwise.

+

Then Y dominates P,, for all n. Moreover, we show that Y is k-minimal unless r =31 for some 1, 1c l 6 k - 1. In order to see this, suppose firstly that r = 31 for some 1, 1S 1S k - 1. Let Q = { u ~ +U~{?J,,+~~ } I 1S j S I}. Then lQl= 1 1 s k and exactly 31 vertices of P,, the vertices of P,,(h 1, n ) , are not dominated by Y - Q . But y(P3[)= [31/31 = 1 and hence there exists an 1-set R such that (Y - Q ) U R dominates P,,. In particular, R = { v ~ +I 1~s j~s-l}.~ Now suppose r # 31 for any 1, 1S 1s k - 1. Let S be any subset of Y consisting of k consecutive vertices of Y. Note that at least 3(k - 1) 1 consecutive vertices of P,, are not dominated by Y - S and since = [(3(k - 1) + 1)/3] = k, no set T with fewer than k vertices exists such that (Y - S) U T dominates P,,. Hence S is stable and therefore any subset of Y consisting of fewer than k consecutive vertices of P,, is also stable. Now let s be the smallest integer such that Y is not s-stable. By Proposition 1, any unstable s-subset of Y consists of s consecutive vertices of Y . Since any set of k or fewer consecutive vertices of Y is stable, s > k and Y is k-minimal as asserted. In view of the above, define

+

+

+

(Y-Q)UR

.=(Y

ifr=31forsomel, l s l s k - 1 , otherwise,

where Q and R are as above. Z is a k-minimal dominating set of P,, for all n ; hence &(P,,) 3 IZI. Moreover,

lZl= (k + 1)m + ) Z n V ( P , ( h + 1, .) I

(2)

~

B. Bollobris et al.

94

(where n = h

+ r with h = m(3k + 1) and 0 s r s 3k), and f

)zn V ( P , ( h +

f

i ++

k+l 1, n ) ) l = 1 1 (I 1

if if if if

r = 31 for some 1, 1s I s k - 1, r=3k, r = 31 2 for some I, 0 s 1 s k - 1, (3) r =31+ 1 for some 1, O s l s k - 1.

+

It is easily verified from (2) and (3), that in the first three cases of (3), IZI = [(k l)(n + 1)/(3k + l)] and in the fourth case 121= [(k + l)n/(3k + 1)1. This completes the proof of the lower bound.

+

Proof of upper bound. We first prove the following lemma. Lemma 2. If X is an independent & set of P,, then among any 3k vertices of P,, at most k + 1 are in X .

+ 1 consecutive

Proof. Suppose there are 3k + 1 consecutive vertices S = {vi, . . . , vi+3k} of P,, such that lSnXl k + 2. Let j (I respectively) be the smallest (largest) integer such that vj E S f l X (vlE S f l X) and let S’ = { v ~ +v j~+,3., . . , v l - J . Then IS’fl XI = (S n XI - 2 z=k while IS’I s 3k - 3. But y(P3k-3)= [(3k - 3)/3] = k - 1 and therefore there exists a set R with IRI S k - 1 which dominates S ’ . Clearly (X- (S’ f l X)) U R dominates P,,, contradicting the k-minimality of X. Hence the lemma is proved. 0 To continue with the proof of the upper bound, we again put n = h + r with h = r n ( 3 k + 1) for some integer m, and O s r s 3 k . Suppose, contrary to the result, (k

Gc(Pn)

+ l)n

’

if r =31+ 1 for some I,

OGI

Gk

-

1,

otherwise, i.e.,

(k (k (k (k

+ 1)m + I + 1 + 1)m + 1 + 2 + 1)m + I + 2 + 1)m + k + 2

if r = 31 for some 1, 1G I c k - 1, if r =31+ 1 for some 1 , O s l s k - 1, if I = 31 + 2 for some 1, O s l s k - 1, if r = 3k.

By Theorem 3, P,, has an independent r-set X. Let X’ = X n V ( P n (1, h ) ) . By Lemma 2, IX‘/s (k + l)m, which implies

I + 1 if r = 31 for some I , 1< 1 s k IX-X’IZ

1+ 2

- 1,

if r =31+ 1 for some 1, O s l s k - 1, if r = 3 1 + 2 for some 1 , O s l < k - 1, k + 2 if r = 3 k .

I +2

On generalised minimal domination parameters for paths

95

The last case, i.e. ( X - X ' ( 2 k + 2 for r = 3k, contradicts Lemma 2 applied to P,, (h, n ) . Hence three cases remain. Case 1. Let r = 31 + 1 for some I, O S I S k - 1 and consider the 3(k - I ) vertices of the subgraph P,(h - 3(k - I ) + 1, h ) of P,. At least 3(k - I ) - 2 of these are not dominated by X - V(P,,(h - 3(k - 2) 1, h ) ) and therefore

+

But then

+ 1, .))I s k - I + I + 2 = k + 2 while IV(P,,(h - 3(k - I) + 1, .))I = 3k + 1, contradicting Lemma 2. I X n V(P,,(h - 3(k - I )

Case 2. Let r = 31 where 1 S f S k - 1. If vh E X ' or vhp1E X ' , then X - X' is unstable since y(P3J = [31/31 = I < I + 1. Hence { u ~ -v~h }, fl X ' = 0 and therefore vh-* E X ' , vh+lE X - X ' . Consider the 3(k - I) + 1 vertices of P,(h - 3(k I), h). At least 3(k -I) - 1 of these are not dominated by X - V(P,,(h - 3(k I), h ) ) and therefore

If IX'I < ( k

+ l)m, then IX - X'I 2 I + 2 so that

Ix n v(P,,(h - 3(k - I ) , n ) ) l > k - I + I + 2 = k + 2

+

while lV(P,,(h - 3(k - I), n))l = 3k 1, contradicting Lemma 2. Hence IX'I (k 1)m. Since {vhPl, vh} f Xl ' = 0,

+

=

I X 'n v(p,( 1, h - 2))1= (k + i)m. Consider the subgraphs P,( 1, 3k - 1) and P,,(3k, h - 2) of P,,. Clearly, P,( 3k, h - 2) has order (3k l)(m - 1) so that

+

IX'n V ( P f l ( 3 kh, - 2))l s ( k + l ) ( m - 1) by Lemma 2. Therefore IX' n V(P,,( 1, 3k - 1))l

3k

+ 1.

Let j be the largest integer such that vi E X ' n V ( P , (1, 3k - 1 ) ) = X" and consider P,( 1,j - 2). Clearly X " f V(P,( l 1, j - 2 ) ) = Q satisfies IQI a k, while P,,(l, j - 2) has order at most 3k - 3. Since y(P3kp3)= k - 1, There exists a set R containing k - 1 vertices which dominates P,, (1, j - 2). But then (X - Q') U R dominates P,,, where Q' is any k-subset of Q, contradicting the k-rninimality of X. Case 3. Let r = 31 2 where 0 Sf S k - 1. If {vh-*, vhpl,v,,} n X # 0 , then at most 31 + 3 vertices remain to be dominated by Q = X - X ' which has at least I + 2 vertices. But Y ( P ~ ( +=~f )+ 1; hence there exists an (I 1)-element subset R of V(P,,) such that ( X - Q) U R dominates P,,, contradicting the k-minimality of X if 2 S k -2. Hence in this case, { v ~ - vhpl, ~ , u h } n X ' = 0 implying that vhpl is

+

+

B. Boliobh et al.

96

not dominated which is impossible. If I = k - 1, then by Lemma 2, {vhp1,u h } n X’ = 0 and IX-X’I = I + 2 = k 1, so that V ~ - ~ Eand X ’IX’(= (k l)m. As in Case 2, a contradiction of the k-minimality of X can now be obtained. 0

+

+

2.3. A generalization of Gallai’s Theorem

The point covering number a ( G ) of a graph G (i.e. smallest number of vertices which cover all the edges) and the independence number P(C) (i.e. largest cardinality of an independent set of vertices) are related by the well-known result of Gallai [3].

Theorem 5 (Gallai). For any n vertex graph, a ( G )+ P(G) = n. In order to generalize this result we need three definitions. Let P be a property associated with the subsets of a set S. The subset X of S is a k-maximal P-set if X is a P-set but the addition of any 1 elements to X where 1 S k, followed by the removal of any 1 - 1 elements, yields a P-set. Let Y G S be a Q-set if and only if it intersects every P-set (i.e. it is a transversal of the family of P-sets. Finally property P is hereditary if each subset of a P-set is also a P-set.

Theorem 6. Let P be a hereditary property. Then X is a k-maximal P-set if and only if S - X is a k-minimal Q-set. Proof. Let X be a k-maximal P-set and Y = S - X. Y is a transversal of the P-sets for otherwise there is a P-set entirely contained in X contrary to the hereditary property. Suppose Y is not a k-minimal Q-set. Then for some I-subset T of Y where 1 S k and an (I - 1)-subset U of S - Y, ( Y - T ) U U is a Q-set. Consider the set (X U T) - U. It is not a P-set since it does not intersect the Q-set ( Y - T) U U . Hence (X U T) - U is a P-set which contradicts the kmaximality of X. Therefore Y is a k-minimal Q-set. The proof of the converse is similar and omitted. 0 If mk(S,P) and &(S, P) denote the smallest cardinality of a k-minimal Q-set and the largest cardinality of a k-maximal P-set, we have immediately:

Corollary 2. If P is a hereditary property on the subsets of S, then

a!@, P ) + P

k G

P ) = IS(.

In the special case where k = 1, S = V ( G ) and P-sets are independent sets of vertices, Corollary 2 reduces to Gallai’s Theorem.

On generalbed minimal domination parameters for paths

97

Finally we note that Hedetniemi [4] has also obtained some generalisations of Gallai’s Theorem. Most of these may be deduced from Corollary 2 by taking k = 1 and P to be a suitable hereditary property associated with the subsets of the set S which is either the vertex set or edge set of a graph.

Acknowledgements Research support from the Canadian Natural Sciences and Engineering Research Council Grant A7544 and the University of South Africa is gratefully acknowledged.

References [l] E.J. Cockayne, S.T. Hedetniemi and R. Laskar, Gallai theorems for graphs, hypergraphs and set systems, in: Proceedings of 1st Japan Conference on Graph Theory, Hakone, 1986, to appear. [2] E.J. Cockayne, G. MacGillivray and C.M. Mynhardt, Generalized independence parameters for paths, in preparation. [3] T. Gallai, Uber extreme Punkt-und-Kantenmengen, Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2 (1959) 133-138. [4] S.T. Hedetniemi, Hereditary properties of graphs, J . Combin. Theory Ser. B 14 (1973) 94-99. [5] S.T. Hedetniemi and R. Laskar, A bibliography of domination in graphs, private communication. [6] F. Jaeger and C. Payan, Relations du type Nordhaus-Gaddum pour le nombre d’absorption d’un graphe simple, C.R. Acad. Sci. Paris Sir. I Math. Series A, 274 (1972) 728-730.

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Part 111. New Models

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Discrete Mathematics 86 (1990) 101-116 North-Holland

101

DOMINATING CLIQUES IN GRAPHS Margaret B. COZZENS Mathematics Department, Northeastern University, Boston, MA 02115, USA

Laura L. KELLEHER Massachusetts Maritime Academy, USA

Received 2 December 1988 A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding a dominating clique. A forbidden subgraph characterization is given for a class of graphs that have a connected dominating set of size three.

Introduction A set of vertices in a simple, undirected graph is a dominating set if every vertex in the graph which is not in the dominating set is adjacent to one or more vertices in the dominating set. The domination number of a graph G is the minimum number of vertices in a dominating set. Several types of dominating sets have been investigated, including independent dominating sets, total dominating sets, and connected dominating sets, each of which has a corresponding domination number. Dominating sets have applications in a variety of fields, including communication theory and political science. For more background on dominating sets see [3, 5 , 151 and other articles in this issue. For arbitrary graphs, the problem of finding the size of a minimum dominating set in the graph is an NP-complete problem [9]. The dominating set problem remains NP-complete even for some specific classes of graphs, including chordal graphs [ 2 ] , split graphs and bipartite graphs [7, 11. The problem remains NP-complete for some types of graphs even when the type of domination is extended. The total dominating set problem is NP-complete for bipartite graphs [17]. The connected domination problem has been shown to be NP-complete for arbitrary graphs [9] and for bipartite graphs [17]. However, there are classes of graphs for which there exist linear algorithms to locate a minimum cardinality dominating set. Cockayne, Goodman and Hedetniemi [4] presented such an algorithm for trees and this has been generalized by Natarajan and White [16] to weighted trees. Booth and Johnson [ 2 ] , in investigating chordal graphs, have presented a linear algorithm for locating a minimum dominating set in directed path graphs (which include interval graphs), 0012-365X/90/$03.50 0 1990-Elsevier Science Publishers B.V. (North-Holland)

102

M.B.Cozzens, L.L. Kelleher

given an appropriate path representation. Pfaff, Laskar and Hedetniemi [18] have presented a linear algorithm for the total domination problem in series-parallel graphs. A complete subgraph or a clique is an induced subgraph such that there is an edge between each pair of vertices in the subgraph. In this paper, characterizations of classes of graphs that contain dominating sets that induce a complete subgraph are given in terms of forbidden subgraphs. For a certain class of graphs, a polynomial time algorithm is given for finding a dominating set that induces a complete subgraph. Dominating sets that induce a complete subgraph have a great diversity of applications. In setting up the communications links in a network one might want a strong core group that can communicate with each other member of the core group and so that everyone outside the group could communicate with someone within the core group. A group of forest fire sentries that could see various sections of a forest might also be positioned in such a way that each could see the others in order to use triangulation to locate the site of a fire. In addition, the properties of dominating sets are useful in identifying structural properties of a social network [13, 141 and in computing the threshold dimension of certain classes of graphs [6].

Clique dominated graphs A ciique dominating set or a dominating clique is a dominating set that induces a complete subgraph. A clique dominated graph is a graph that contains a dominating clique. The smallest size dominating clique possible in a graph would be a single vertex. Wolk [19] presents a forbidden subgraph characterization of a class of graphs which have a dominating clique of size one. He called such a dominating clique a central vertex or central point. In the following theorem and throughout this paper the notation P,, denotes the path on n distinct vertices and C,, denotes the cycle on n vertices.

Theorem 1 (Wolk [19]). If G is afinite connected graph with no induced P4 or C4, then G has a dominating vertex. This theorem can be extended to get forbidden subgraph conditions sufficient to imply the existence of a dominating set that induces a complete subgraph, a dominating clique. This is presented in the next theorem.

Theorem 2. If G is a finite graph that is connected and has no induced Ps or C s , then G has a dominating clique.

Dominating cliques in graphs

103

Proof. By induction on n, the number of vertices in G. (i) The proposition is clearly true for n = 1. (ii) Assume that any finite connected graph with n vertices, n 2 1, that has no induced Ps or Cs has a dominating clique. Let G be a finite graph with n + 1 vertices, n 3 1, that is connected and has no induced Ps or Cs. Let v be a vertex of G that is not a cutpoint. Such a vertex exists [ll]. Let G' be the subgraph of G induced by all vertices of G except v. Since G' is a finite graph with n vertices that is connected and has no induced Ps or Cs it has, by the induction hypothesis, a dominating set that induces a clique. Let K' be a dominating set of G' that induces a clique. In G, if v is adjacent to any vertex in K', then K' will also be a dominating set of G that induces a clique. Suppose that in G, v is not adjacent to any vertex in K'. Since G is connected, v must be adjacent to some vertex of G. Let x be any vertex of G that is adjacent to v. Let K = { x } U ( N ( x ) n K'). It will be shown that K is a dominating set of G that induces a clique. By construction K induces a clique. Suppose K is not a dominating set of G. Then there must be a vertex u that is not adjacent to any vertex in K. However, since K' is a dominating set of G', u must be adjacent to some vertex in K'. Let a be a vertex in K' that is adjacent to u. Let b be a vertex in K other than x . Such a vertex exists since x itself is not in K' but x must be adjacent to some vertex in K'. See Fig. 1. If u is not adjacent to v then v-x-b-a-u is an induced Ps, a contradiction to the assumption that G has no induced Ps. If u is adjacent to v then v-x-b-a-u is an induced Cs, a contradiction to the assumption that G has no induced Cs. Therefore, G has a dominating set that induces a clique. 0 It should be clear that the converse of Theorem 2 is not true. For example, the graph in Fig. 2 has a dominating clique of size one and an induced Ps. The following theorem establishes a relationship between the forbidden subgraph conditions sufficient for a graph to contain a dominating clique and the size of a dominating clique in the graph. The notation Kn+pdenotes the complete

Fig. 1 .

M . B . Cozzens, L.L. Kelleher

104

t

Fig. 2.

Fig. 3.

graph on n vertices with n pendants, one at each vertex of the complete graph. K3+,, is shown in Fig. 3.

Theorem 3. If G is a finite graph that is connected and has no induced Ps, Cs or K ( k + l ) + pk, 2 2, then G has a dominating clique of size s k .

Proof. By Theorem 2, G has a dominating clique. Let K be a minimum dominating clique of G, and let m = the size of k. If m k , then K is a dominating clique of G of size s k . Suppose m > k. Since K is a minimum dominating clique, each vertex, x i , 1s i d m, in K must be adjacent to at least one vertex, y,, that is not in K and that is not adjacent to any other vertex in K. Let this set of vertices yi be called S . Case 1. At least k + 1 of the vertices in S form an independent set. Then these k + 1 vertices in S together with their neighbors in K form an induced K(k+l)+p,a contradiction. Case 2. S does not contain k + 1 independent vertices but S does not induce a complete subgraph. Let y , and y z be vertices in S that are not adjacent to each other and let y3 be a vertex in S that is not adjacent to both y , and y 2 . By symmetry it is sufficient to consider that y , is not adjacent to y , . Then y3-x3-xz-yz-y, is an induced Ps or C5,a contradiction. See Fig. 4. Case 3. S induces a complete subgraph. Since { x l , y , } is not a dominating edge of the graph, there must be a vertex, say z, that is not adjacent to either x 1 or y,. However, since K is a dominating clique, z must be adjacent to some vertex, say x 2 , in K. Then z-x2-x,-yI-y3 is an induced Ps or C s , a contradiction. See Fig. 5. Therefore, G has a dominating clique of size S k .

t

ys

YZ

Fig. 4.

Fig. S .

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We can now establish conditions under which a graph must have a dominating clique of size two, a dominating edge.

Corollary 3.1. If G is a finite graph with two or more vertices that is connected and has no induced P5, C5 or K3+pthen G has a dominating edge. Proof. By Theorem 3, with k = 2, G has a dominating clique of size one or two. Since any edge containing a dominating vertex is a dominating edge, G has a dominating edge. 0 A bipartite graph is a graph whose vertex set can be partitioned into two subsets V, and V2 such that every edge of G joins a vertex from V, with a vertex from V2. A split graph is a graph such that there is a partition of the vertex set into a complete graph and an independent set. There is no restriction on edges between vertices of the complete graph and the independent set. The following corollaries relate Theorem 3, and particularly Corollary 3.1, to these well-known classes of graphs.

Corollary 3.2. If G is a connected bipartite graph that does not contain an induced P5 then G has a dominating edge. Proof. Since all of the cycles of a bipartite graph are even [ l l ] , a bipartite graph cannot have an induced C5 or K3+p. Therefore, by Corollary 3.1, G has a dominating edge. 0 Corollary 3.3. If G is a connected split graph that does not contain an induced K3+pthen G has a dominating edge. Proof. Since a split graph contains no induced 2K2, C, or C5 [S], by Corollary 3.1, a connected split graph that does not contain an induced K3+p has a dominating edge. 0

Parameters The diameter of a connected graph is the maximum possible distance between any two vertices of the graph. The diameter of any clique dominated graph is less than or equal to three. For graphs that have a dominating clique a parameter similar to those previously defined for various types of dominating sets can be associated with the size of a minimum dominating clique. For a graph G, P(G) denotes the domination number of the graph, the size of a minimum dominating set, and i ( G ) denotes the cardinality of a minimum independent dominating set of the graph. The connected domination number of a graph G is denoted by

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P,(G) and the total domination number of a graph that has a total dominating set is denoted by P,(G). If a dominating clique exists in a graph G, let Pk(G) denote the clique domination number, the cardinality of a minimum dominating clique of the graph G. Some elementary properties of the clique domination number can now be presented.

Property 1. If G is a clique dominated graph then P(G)=sP,(G) S Pk(G). Proof. Since every dominating clique is a connected dominating set it follows that the size of the smallest connected dominating set is less than or equal to the size of the smallest dominating clique. The size of the smallest dominating set is, in turn, less than or equal to the size of the smallest connected dominating set. 0 Property 2. If G is a clique dominated graph with p vertices and maximum degree < p - 1, then P ( G ) P,(G) P,(G) d P k ( G ) . Proof. Since any nonsingleton connected dominating set is a total dominating set, the inequality in Property 1 can be extended for graphs that contain a total dominating set. 0 Property 3. If G is a connected graph that has no induced P4 or C4 then P(G) = i(G) = Pk(G) = 1. Proof. By Theorem 1, G has a dominating vertex. Property 4. If G is a connected graph that has no induced Ps, Cs or K3+p then P(G) = PdG) d 2. Proof. By Corollary 3.1, G has a dominating edge.

0

Connected split graphs clearly have a dominating clique. The complete graph of the partition of the vertices is a dominating clique. However, this clique may not be a minimum dominating clique. The following property relates the domination number, the independent domination number and the clique domination number of a connected split graph.

Property 5. If G is a connected split graph then P ( G ) = Pk(G) 6 i ( G ) . Proof. For any graph C, P(G) i ( G ) and for any clique dominated graph C, P(G) S Pk(G). Let D be a minimum dominating set of a connected split graph G whose vertex set can be partitioned into clique K and independent set S. Let P ( G )= m . To show that P ( G )= Pk(G), a dominating clique of size m must be found. If D is a dominating clique then clearly G has a dominating clique of size

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m. If D is not a dominating clique, form a half dominating clique of the connected split graph by replacing each x E D such that x E S by y E N ( x ) . Such a y exists since G is connected. Since each neighbor of a member of S must be a member of K , and since D is a minimum dominating set, the set formed in this way must be a dominating clique of size m. Therefore, B (G ) = &(G) G i ( G ) . 0

A polynomial algorithm

Determining whether or not a graph has a central vertex can be accomplished easily by checking the degree of each vertex or by checking each vertex to see if the closed neighborhood of the vertex, the union of the vertex and its neighborhood, is the vertex set of the entire graph. The proof by induction that established that a finite connected graph with no induced Ps or C, has a dominating clique suggests an algorithm for finding a dominating clique in such graphs. This algorithm will be shown to run in O ( n 3 )time where n is the number of vertices in the graph. This algorithm finds a dominating clique (which may be a minimum or maximum dominating clique, or neither a minimum nor maximum dominating clique) in a finite connected graph which does not contain an induced P, or Cs. If the algorithm is run on a finite connected graph which does contain an induced P, or C, then the algorithm will either terminate, saying an induced P5 or C, exists, or find a dominating clique anyway. Thus, it is not necessary to first check if the graph has an induced P, or C5. In the following algorithm, K represents the set that is currently under consideration as a dominating clique of the graph, T is the set of vertices that have already been considered, and W is the set of vertices that have not yet been considered. The notation A ( K ) denotes the set of all vertices adjacent to the present clique K , while A ( T ) denotes the set of all vertices adjacent to the set T, When a vertex of the graph is considered, the present clique K is tested to see if it dominates this vertex. If K does dominate this vertex, then another vertex is chosen. If K does not dominate this vertex, a new clique K is formed which will dominate that vertex. In this case, the clique being replaced is called K 1 and the neighbors of this set are represented by A ( K 1 ) . If the algorithm continues until all vertices of the graph have been considered and dominated by K , then K is returned as a dominating clique. If at any stage a new clique is formed which fails to dominate a vertex that had been dominated by the previous clique, then the algorithm terminates because an induced P, or C, was found.

Algorithm DC Dominating clique of a connected graph. Input: The adjacency lists, A ( v ) , v E V , of a connected graph G = ( V , E ) .

M.B .

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Cozzens,

L. L. Kelleher

Output: A set of vertices K that induces a dominating clique in G or “The graph contains an induced P5or Cs.”

begin K +0; T 4; W+V; choose v E W ; K+{v); WtW-{v}; T+{v); A(K) +-A(v); A(T) - 4 ~ ) ; flag +-0; while W # 0 and flag = 0 begin choose II E A ( T )n W ; ifvEA(K)then K t K else begin choose x E A ( v )n T ; K +- { x } U ( N ( x ) f l K) end; T +T U {v}; w tw - {v};

A(K1) +A(K);

i f A ( K l ) $ A ( K ) then f l a g t l end; if flag = 1 then return “The graph contains an induced Ps or Cs.” else return K end. Theorem 7. Algorithm DC is correct and runs in O(n’) time. Proof. If G does not contain an induced Ps or C s then, by the proof of Theorem 2, the set K formed in Algorithm DC is a dominating clique. If G does contain an induced P5 or C , then it may be that each K that is formed dominates all previously dominated vertices, as well as the latest chosen vertex, so that the last

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K

Kl

-

* V

W

Fig. 6 .

K is a dominating clique of the graph. Suppose this is not so. Then a K is formed by Algorithm DC that does not dominate some vertex, say v, that was dominated by the previous K. Let x be a vertex of the previous K that dominated v. Let w be the latest chosen vertex and let z be a vertex in the new K that dominates w. Since z was dominated by the previous K, by construction, the new K must contain some vertex, say y, that is adjacent to z and that was in the previous K. See Fig. 6. Since x and y were in the previous K, x is adjacent to y. Since v is not dominated by the new K, v cannot be adjacent to z and v cannot be adjacent to y. If x were adjacent to z then, by the construction of the new K, x would be in the new K. Thus, x cannot be adjacent to z. If x or y were adjacent to w then the previous K would have dominated w. So x is not adjacent to w and y is not adjacent to w. Thus, if v is not adjacent to w then v-x-y-z-w is an induced P5 and if 21 is adjacent to w then v-x-y-z-w is an induced C5. The message returned by Algorithm DC is therefore correct. Since adding vertex number k to the set of chosen vertices requires O(k2) steps, the algorithm runs in O ( n Z+ ( n + 1)2 + . . . + 12) = O ( n 3 )time. 0 To illustrate Algorithm DC, consider the graph in Fig. 7. This is a connected graph with no induced P5or C5. Initially K = 0, T = 0, and W = {a, b, c, d, e, f }. Choose, for example, v =a. Then K = { a } , T = { a } , and W = {b, c, d , e , f } . A ( K ) = A ( T ) = { b , c } . Flag=O. e

A a

c

d

Fig. 7.

f

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W is not the empty set until all vertices have been chosen and flag remains equal to one unless an induced P, or C, is found. Since W is not empty and flag = 0, choose a vertex adjacent to the one originally chosen. Suppose vertex b is chosen next. Since b is already dominated by K , K remains the same. At this stage, K = { a } , T = { a , b } , and W = { c , d , e , f } . A ( T ) = { a , b, c , d, e } . A ( K 1 ) = { b , c}. A ( K ) = { b , c } . Since A ( K 1 ) G A ( K ) , flag remains = 0. Again, since W is not empty and flag = 0, choose a vertex that has not yet been considered and is adjacent to a vertex that has been chosen. Suppose vertex c is chosen next. Since c is also already dominated by K , K still remains the same. Now, K = { a } , T = { a , b , c } , and W = { d , e , f } . A ( T ) = { a , b , c , d , e } . A ( K 1 ) = { b , c } . A ( K ) = { b , c}. Since A ( K 1 ) G A ( K ) , flag remains = 0. Neither condition to terminate the while loop has been met. Choose another vertex that has not yet been considered and is adjacent to a vertex that has been chosen. Suppose vertex d is chosen next. Vertex d has not been dominated by K , so a new K must be formed. Choose a vertex that is adjacent to d and that has already been considered. Vertices b and c both meet these conditions. Suppose vertex b is chosen. A new K is formed by uniting vertex b with the neighbors of vertex b that are in the previous K. A t this stage, K = { a , b } , T = { a , 6 , c, d } , and W = { e , f } . A ( T ) = { a , b , c , d , e , f } . A ( K l ) = { b , c } . A ( K ) = { a , b , c , d , e } . Since A ( K 1 ) G A ( K ) , flag remains = 0. All of the vertices of the graph have not yet been considered and an induced P5 or C, has not been found. Choose a vertex that has not yet been chosen. Suppose vertex e is chosen. Since e is already dominated by the present set K , this set remains the same. Now, K = { a , b } , T = { a , b, c , d , e } , and W = {f}. A ( T ) = { a , 6 , c, d , e , f } . A ( K 1 ) = { a , b , c, d, e } . A ( K ) = { a , b , c, d , e } . Since A ( K 1 ) c A ( K ) , flag remains = 0. W is still not empty and flag = 0. Since it is the only vertex that has not been chosen, vertex f must now be chosen. Vertex f is not already dominated by K , so a new K must be formed. Choose a vertex adjacent to f that has already been considered. Suppose this is vertex d. The new K is formed by uniting vertex d and the neighbors of vertex d that are in the previous K. Now, K = { b , d } , T={a,b,c,d,e,f}, and W=0. A(T)={a,b,c,d,e,f}. A(K1)= { a , b , c, d , e } . A ( K ) = { a , b, c , d , e , f } . Since A ( K 1 ) c A ( K ) , flag remains = 0. Since W is now empty, the while loop is terminated. Since flag=O, the set K = { b , d } is returned as a dominating clique of the graph. This set is a dominating clique of the graph in Fig. 7. Many choices were made arbitrarily in implementing the algorithm. A different dominating clique, such as { b , e } or {c, d } , may have been returned by the algorithm.

Threshold dimension The class of clique dominated graphs as well as the class of connected graphs that do not have an induced P5or C, are not perfect graphs since the complement

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of C7 is in both of these classes of graphs. However, Algorithm DC is a polynomial algorithm to find a dominating clique of a finite connected graph with no induced Ps or C5.It may be possible to use this dominating clique to find other parameters for these graphs. Since this class of graphs contains the class of connected split graphs, any polynomial algorithms which are found for clique dominated graphs apply to the class of connected split graphs. Also, the computation of any parameters which are known to be NP-complete for connected split graphs will be NP-complete for clique dominated graphs. For example, a threshold graph is a graph that has no induced P4, C4, or 2K2. The threshold dimension of a graph is the minimum number of partial subgraphs of a graph that are threshold graphs and that cover the edges of the graph. Yannakakis [20] proved that determining if the threshold dimension of an arbitrary graph is less than or equal to k , for fixed k 2 3, is NP-complete. As we now show, determining if the threshold dimension of a connected split graph is less than or equal to k, for fixed k 3 3, is NP-complete. Therefore, determining if the threshold dimension of a clique dominated graph is less than or equal to k, for fixed k 3 3, is NP-complete. A chain graph is a bipartite graph that has no induced 2K2. (A graph with no induced 2K2 is said to be nonseparable [lo].) The chain dimension of a graph is the minimum number of chain subgraphs that cover the edges of the graph.

Theorem 5 . It is NP-complete to determine if the threshold dimension of a split graph is less than or equal to k , for fixed k 2 3. Proof. Let G be a split graph whose vertex set can be partitioned into clique K and independent set S. Form the bipartite graph B ( G ) by removing the edges of the clique K from G. Similarly, any bipartite graph whose vertices can be partitioned into sets V, and V, can be transformed into a split graph by adding the edges to make either V, or V2, but not both, a clique. Since any vertices that induce P4 in the split graph G must induce 2K2 in the bipartite graph B ( G ) , the threshold dimension of G is equal to the chain dimension of B ( G ) . Since it is NP-complete to determine if the chain dimension of a bipartite graph is less than or equal to k , for fixed k z=3 [18], it is NP-complete to determine if the threshold dimension of a split graph is less than or equal to k, for fixed k 2 3. 0

Corollary 5.1. It is NP-complete to determine if the threshold dimension of a connected split graph is less than or equal to k , for fixed k 3 3. Proof. This follows from Theorem 5 and Corollary 3.1 since the threshold dimension of a graph is the sum of the threshold dimensions of the components of the graph. 0

Corollary 5.2. It is NP-complete to determine if the threshold dimension of a clique dominated graph is less than or equal to k , for fixed k 2 3.

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Proof. This follows from Corollary 5.1 by restricting the problem to allow only instances that are connected split graphs. 0 However, as the following theorem shows, it is possible to determine in polynomial time if the threshold dimension of a connected split graph is s 2 . Lemma 1. Let G be a connected split graph whose vertices can be partitioned into clique K and independent set S . Let B ( G ) be the bipartite graph formed from G by removing the edges of K . Then the threshold dimension of G is s 2 if and only if the chain dimension of B ( G ) is S 2 .

Proof. If the threshold dimension of G is S 2 , then G = C, U G2 where GI and G2 are threshold graphs. G1 and G2 are split graphs whose vertex sets can be partitioned into a clique that is a subset of K and an independent set that is a subset of S. Thus, GI can be partitioned into K 1 c K and S, E S. Similarly, G2 can be partitioned into K , c K and S, E S. Form the bipartite graph B(G,) by removing the edges of K 1 from G I .Form the bipartite graph B(G2)by removing the edges of K 2 from G2. Since G, and G2 are threshold graphs they contain no induced P, or 2 K Z .This implies that bipartite graphs B(Gl) and B(G,) contain no 2K2. Therefore, B(G,) and B(G2)are chain graphs. Since B(G,) U B(CJ covers the edges of B ( G ) , the chain dimension of B ( G ) s 2. If the chain dimension of B ( G ) 2, then B ( G ) = GI U G2 where G, and G2 are chain graphs. Form the graphs G, U K and G2U K . The fact that the chain graphs GI and G2 have no induced 2 K , implies that G, U K and G2 U K have no induced P,. Since, by construction, GI U K and G2U K are split graphs, they have no induced C , or 2K2. Therefore, they are threshold graphs. Since their union covers the edges of G, the threshold dimension of G S 2. 0 Theorem 6. There is a polynomial algorithm to determine if the threshold dimension of a connected split graph is s 2 .

Proof. Let G be a connected split graph whose vertex set can be partitioned into clique K and independent set S. Form the bipartite graph B ( G ) by removing the edges of K from G. By results of Yannakakis [20] and Ibaraki and Peled [12] there is a polynomial algorithm to determine if the chain dimension of a bipartite graph is ~ 2 By. Lemma 1, this same algorithm will determine if the threshold dimension of G is ~2 by determining if the chain dimension of B ( G ) is s 2 . A polynomial algorithm exists for determining the threshold dimension of split graphs with no K 3 + p ,and, more generally, for graphs with a dominating edge such that each induced subgraph has a dominating edge [6].

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Graphs that have a dominating K 3 or P3 Dominating vertices and dominating edges are the only connected dominating sets of size one and size two. However, connected dominating sets of size three can be either cliques or paths. In this section, a forbidden subgraph characterization for a graph to have a connected dominating set of size three is given. Let A l , A2, A3, and A, be the graphs shown in Fig. 8. Let A = {p63 c6, K4+p,A l , A*, A3, A4).

Theorem 7. If G is a finite, connected graph with three or more vertices that has none of the graphs in A as an induced subgraph, then G has a connected dominating set of size three. Proof. By induction on n, the number of vertices in G. (i) The proposition is clearly true for n = 3. (ii) Assume that any finite connected graph with n vertices, n 2 3, that has none of the graphs in A as an induced subgraph has a connected dominating set of size three. Let G be a finite connected graph with n + 1 vertices, n 3 3, that has none of the graphs in A as an induced subgraph. Let v be a vertex of G that is not a cutpoint. Such a vertex exists [ll].Let G' be the subgraph of G induced by all vertices of G except v. Since G' is a finite connected graph with n vertices that has none of the graphs in A as an induced subgraph it has, by the induction hypothesis, a connected dominating set of size three. Let D = { a , 6 , c } be a connected dominating set of size three of G'. In G, if v is adjacent to any vertex in D, then D will also be a connected dominating set of size three of G. Suppose that, in G, v is not adjacent to any vertex in D. Since G is connected, v must be adjacent to some vertex of G. Let x be any vertex of G that is adjacent

A3

A4

Fig. 8.

M.B . Cozrens, L. L. Kelleher

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(ii)

a

C

a

(iii)

VXHV

X

V

(V)

(vi)

(vii)

(viii)

Fig. 9.

to v. Since D is a dominating set of G', x must be adjacent to some vertex in D. Up to symmetry, the set {v, x , a, b, c } induces one of eight possible subgraphs. See Fig. 9. Now, either x and two of the vertices of D form a dominating P3 or K , of G, or G has one of the subgraphs (not necessarily induced) shown in Fig. 9. Since the subgraphs formed by {v, x, a, b, c } are induced subgraphs, if G has no edges between the pendants of the subgraphs in Fig. 9 then there is a contradiction to the assumption that C has none of the graphs in A as an induced subgraph. Suppose G has at least one of the edges between the pendants shown in Fig. 9. Suppose further than G contains the subgraph shown in Fig. 9(i). If G has exactly one edge between the pendants then G has an induced P,. If G has exactly two nonsymmetric edges between the pendants then G has an induced C,. If G has two symmetric edges between the pendants or all three edges between the pendants then either {v, x, a } is a dominating P3 of G or there is a vertex not adjacent to any vertex in {v, x, a } but adjacent to a and/or c. This would imply that G has an induced P, or C,. See Fig. 10. In a similar way. consideration of the edges between the pendants of each of

AAA Fig. 10.

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the subgraphs in Fig. 9 can be shown to lead to the conclusion that either G has a dominating P3 or K 3 or there is a contradiction to the assumption that G has none of the graphs in A as in induced subgraph. Therefore, G has a connected dominating set of size three. As noted earlier for Theorem 2, the converse of Theorem 7 is However, it is possible to extend the forbidden subgraph relationships in this paper to if and only if statements and thus to give forbidden characterizations of these classes of graphs. An example of this type of is given in the following corollary.

not true. described subgraph extension

Corollary 7.1. Let G be a finite connected graph with three or more vertices. Every connected induced subgraph of G with three or more vertices has a connected dominating set of size three if and only if G has none of the graphs in A as an induced subgraph. Proof. This follows easily from Theorem 7 and the fact that none of the graphs in A has a connected dominating set of size three.

Open problems This paper has given forbidden subgraph characterizations of graphs with a dominating clique or a connected dominating set of size three. Several problems related to this area remain open. These problems include: Is there a forbidden subgraph characterization of graphs that have a connected dominating set of size four? Is there a polynomial algorithm to locate a minimum dominating clique in a clique dominated graph or in a connected graph with no induced P5 or Cs? What other parameters are computable in polynomial time for clique dominated graphs or for connected split graphs? Are there forbidden subgraph characterizations of classes of graphs that contain other specific types of dominating sets?

References [l] A. Bertossi, Dominating sets for split and bipartite graphs, Inform. Process. Lett. 19 (1984) 37-40. [2] K.S. Booth and J.H. Johnson, Dominating sets in chordal graphs, SIAM J. Comput. 11 (1982) 191- 199. [3] E.J. Cockayne, R.M.Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219. [4] E.J. Cockayne, S. Goodman and S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975) 41-44. [S] E.J. Cockayne and S.T. Hedetniemi, Toward a theory of domination in graphs, Networks 7 (1977) 247-261.

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[6] M.B. Cozens, Algorithms to compute the threshold dimension of some clique dominated graphs, submitted. (71 A.K. Dewdney, Reductions between NP-complete problems, in Fast Turing reductions between problems in NP, Report 71, Univ. Western Ontario, Dept. of Computer Science, 1981. [8] S. Foldes and P. Hammer, Split Graphs, in: Proceedings Eighth S.E. Conference on Combinatorics, Graph Theory and Computing (Utilitas Mathematica, Winnipeg, 1977) 311-315. [9] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1978). [lo] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [ l l ] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [12] T. Ibaraki and U.N. Peled, Sufficient conditions for graphs to have threshold number 2, Ann. Discrete Math., to appear. [I31 L.L. Kelleher, Domination in graphs and its application to social network theory, Ph.D. Thesis, Northeastern University, 1985. [14] L.L. Kelleher and M.B. Cozens, Dominating sets in social network graphs, Math. Social Sci., to appear. [15] R. Laskar and S.T. Hedetniemi, Connected domination in graphs, Tech. Report 414, Clemson Univ., Dept. Mathematical Sci., 1983. [16] K.S. Natarajan and L.J. White, Optimum domination in weighted trees, Inform. Process. Lett. 7 (1978) 261-265. [17] J . Pfaff, R. Laskar and S.T. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Tech. Report 428, Clemson Univ., Dept. Mathematical Sci., 1983. [18] J. Pfaff, R. Laskar and S.T. Hedetniemi, Linear algorithms for independent domination and total domination in series-parallel graphs, Congr. Numer. 45 (1985) 71-82. [19] E.S. Wolk, A note on “The comparability graph of a tree,” Proc. Amer. Math. SOC.16 (1965) 17-20. [20] M. Yannakakis, The complexity of the partial order dimension problem, SIAM J. Algebraic Discrete Methods 3 (3) (1982) 351-358.

Discrete Mathematics 86 (1990) 117-126 North-Holland

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COVERING ALL CLIQUES OF A GRAPH* Zsolt TUZA Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13-17, Hungary

Received 28 August 1986 The following conjecture of T . Gallai is proved: If G is a chordal graph on n vertices, such that all its maximal complete subgraphs have order at least 3, then there is a vertex set of cardinality < n / 3 which meets all maximal complete subgraphs of G. Further related results are given.

1. Introduction Let G = ( V , E) be a graph with vertex set V and edge set E. A vertex set Y c V is called a clique if JYI3 2 and the graph G I induced by Y is a complete subgraph maximal under inclusion. (Thus, isolated vertices are not considered to be cliques here.) In this paper we present theorems of the following type: (* )

Let 3 be a given class of graphs, and suppose a G E 3 is such that every edge of G is contained in a complete subgraph of order k. Then there is a vertex set of at most n / k elements that meets all cliques of G (where n = IV(G)l is the number of vertices in G ) .

Professor Gallai (private communication) asked if ( 9 ) is true for the class of chordal graphs (i.e., graphs having no cycle of length >3 as an induced subgraph). He conjectured that the answer is affirmative for k = 2, 3. The case k = 2 was settled by Aigner and Andreae [l]. Later Andreae [2] gave further sufficient conditions on %, implying ( * ) for k = 2. Here we prove Gallai’s conjecture for k = 3 (Theorem 3). Our method provides a short proof for k = 2 also; moreover, the present approach is suitable for characterizing all chordal graphs for which the upper bound of n / 2 is sharp (Theorem 2). Restricting 3 to the class of split-graphs, we verify ( * ) for k = 4 (Theorem 9) and show that it does not hold for any k 2 5; i.e., for all k 3 5 there exist some split-graphs satisfying the suppositions of ( * ), in which the cliques cannot be covered by n / k vertices (Proposition 10). On the other hand, for the class of so-called strongly chordal graphs ( * ) is proved for all k and n (Theorem 7), and for interval graphs a linear-time * Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences. 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V.(North-Holland)

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algorithm exhibiting an at most (n/k)-element covering set is presented (Algorithm 8). There are various interesting related problems which remain open. The simplest one for chordal graphs is the following.

Problem 1. Is ( * ) true when k = 4 and 3 is the class of chordal graphs?

2. Results

For a graph G, define the clique hypergraph of G as the set system Xc = &(G) = {H c V ( C ) :H is a clique of G}. We say that a vertex set T is a clique-transversal set if T fl H # 0 for all H E Xc, and define the clique-transversal number tc = t c ( G ) as the minimum cardinality of a clique-transversal set. Note that t c ( G )= t(Xc(G))where ~ ( 2 is the ) transversal number of a set-system X, in the sense of Berge [3]. As Erdos, Gallai and the author have observed [ 5 ] , t c ( G ) can be very close to n = IV(G)l, namely tC = n - o ( n ) can hold. (The asymptotic behavior of n - max tC, however, is not known.) It is reasonable to ask how drastically tC decreases when some assumptions are put on the graph G. From this point of view, Gallai proposed to study the class of chordal (=triangulated = rigid circuit) graphs satisfying the following property.

(Pl) Every edge of G is contained in a clique of cardinality s k . One can assume another very similar property. (P2) All cliques of G have cardinality 2 k . Though (P2) is slightly stronger than (Pl), in many cases the difference is not significant. As a matter of fact, in the classes of graphs studied below (Pl) and (P2) happen to be equivalent. Moreover, trivially, ( P l ) e ( P 2 ) for k = 3, without any further restriction on the graphs. Somehow (P2) looks easier to handle because then our aim can be formulated as to prove t ( X ) =sn / k for a hypergraph of lower rank 2 k . (The lower rank of X is min(lH1: H E Re>.) In most proofs, the following algorithmic approach will be used in order to obtain a clique-transversal set of small cardinality. Algorithm (&). Let & = X C and V o = U H E q H . If Y t P l and V,-l have been obtained, pick a vertex ti E VPl, set %$ = { H E2lP1: ti @ H} and V , = U H EH. K Repeat this procedure until V;. = 0. Then, obviously, the set T = {tl, t Z ,. . . } is a clique-transversal set.

lvl

Clearly, if =S [V,Pll - k holds in every step of (A"), then T has cardinality S a l k , so that t c ( G )< n / k follows immediately. Hence, the essential question

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is to tell how to pick ti, and to show that a suitable choice of ti yields Iv(6 ly-ll - k whenever Z0. We have to stress the following important aspect of (&): In each step, the algorithm handles as a set system (hypergraph), without considering the graph induced by The idea that the graph G itself is used only for deriving some properties of Xc, will yield significant simplification in finding upper bounds on zc(G).

vp1

v-l.

2.1. Chordal graphs One of the most important properties of a chordal graph G (following from results of Dirac [4]) is that its vertices have a simplicia1 order (=perfect elimination ordering) v l , v 2 , .. . , vn, i.e., for all i (1 6 i s n ) the vertex set r: = {vj: i < JG n , vivj E E ( G ) } induces a complete subgraph of G, where E ( G ) is the edge set of G. As a consequence, Xc(G) consists of the sets r: U {vi} which are maximal under inclusion. Moreover, every vertex can be the first element of at most one clique. (The order v l , . . . , v, defines an order on any subset of V ( G ) ,so we can define the first, second, . . . ,last element of a clique.) From this observation we can deduce ( * ) for chordal graphs if k 6 3. For k = 2, the theorem of Aigner and Andreae [l] can be proved in a much stronger form. Namely, all extremal graphs can be characterized. For a graph G = (V, E), call { e l , . . . ,en,2}= EM c E a perfect matching if the edges of EM are pairwise disjoint and their union is V. A cut-edge of a graph G is an edge whose deletion increases the number of connected components of G. Using these notions, we can formulate the following result. (The second part of the theorem is valid without assuming that the graph is chordal.)

Theorem 2. (a) Let G be a chordal graph on n vertices. Then zc(G) s n / 2 , and equality holds if and only if G contains a perfect matching EM such that every ei E EM is a cut-edge of G. (b) Let G be an arbitrary graph on n vertices. If G contains a perfect matching all of whose edges are cut-edges, then zc(G) = n / 2 . Before proving Theorem 2, let us show the validity of ( * ) for k

= 3.

Theorem 3. If every edge of a chordal graph G is contained in a triangle, then z,(G) s n / 3 . Proof. Applying (&), we are going to find a clique-transversal set {tl, t2, . . . } of at most n / 3 vertices. Suppose that vl, . . . , v, is a simplicia1 order of the vertex set of G. Suppose further that Xi-._, and have been obtained by (A,,) (starting with % = Xc and V,,= V). As a first guess for ti, choose vj E &-l with minimum j among all vertices being a second element of some H E Xi-l. If there is another H’ E Xi-] containing vj, then vj is the first or second element of H ’ (for J is

v.-l

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minimum). In the latter case, put ti = vi. Then vi $ V,, and & does not contain the first element of H and H' either, i.e., /&I =slV,-ll - 3. Suppose now H is the only set in q.-._,, having vj as its second element, and let ti be the third (or any but the first or second) element of H. Since the first element of H does not belong to V,, it is enough to show that vj $ &, because then IKj s - 3 follows immediately. Now vi $ V , is obvious in the case when there is no other H' E Xi-,such that vj E H ' Z H . On the other hand, if vi E H' E Xi-, then vi must be the first element of H', i.e., H and H' are the only members of Xi-l containing v,. Now vjtjE E ( G ) , for vj and ti belong to the same clique H, so that ti E H' by the definition of a simplicia1 order. Thus, vj $ V,, since ff,H'$%-. 0

lv-ll

Now Theorem 2 can be shown by an argument which uses some ideas of the previous proof also.

v-l

Proof of Theorem 2. First we show tc(G) < n / 2 . In (Ao), let ti= v,,E be the vertex which is the second element of at least one H E Xi-,, such that the subscript j , is minimum. Then vit $ V,, as well as the first element of H , that is, IV,ls lV,.-ll - 2 in every step, implying t,-(G) s n / 2 . Next, suppose that G is an arbitrary graph on n vertices, containing a perfect matching EM all of whose edges are cut-edges of G . Since each e, E EM has to contain at least one element of any clique-transversal set, t,-(G) 3 n / 2 follows immediately. Thus, the assumption in (a) is sufficient for t c ( G ) = n / 2 when G is chordal. Moreover, if every e, E EM is a cut-edge, then G contains a vertex of degree 1. Indeed, for all e i E E M , let Xi and Xz! denote the vertex sets of the two components after the deletion of e, (Xi U X,! = V ( G ) , X , f l X,! = 0 for 1s i s n), and assume IX,l SIX,!l. (If G is not connected, the partition X,lJXt! is not unique. Then let Xibe the smallest component occurring after the deletion of e, in the component containing e,, and set Xz!= V(G)\X,.) For all j # i, e, c X , or e, c X l holds. Since every vertex of Xiis incident to some member of E M , there is an ej c Xiwhenever lXil > 1. Then Xi5 X , and, consequently, if X , is minimal under inclusion, then lXil = 1, and ei has an end-vertex of degree 1. As a consequence, the other vertex of e, is a suitable choice for t l when (&) is applied, and IV,l =s IVo(- 2 follows. Since each cut-edge of G remains a cut-edge in G\e, also, a similar argument can be repeated after every step, so that a vertex with a 1-degree neighbor can be found. Thus, the procedure exhibits an (n/2)-element vertex set that meets all cliques of G . This proves (b). Finally, we show that if a chordal graph G does not contain a perfect matching of cut-edges, then t c ( G ) < n / 2 . We proceed in (&) as follows. Whenever there = {v, v'} E such that v' is not contained in any other H' E X,-,, is an HiW1 then we put t, = v. If such an H j - , exists in every step ( O < i S n/2), then a perfect matching EM has been found at the end of the procedure. Observe that

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each e E EM is a cut-edge of G. Otherwise, there is a minimum-length cycle C containing e . Since G is chordal, ICI = 3 follows, implying that e is contained in a triangle of G. This contradicts the fact that each clique (and, in particular, each Hi-,) is maximal under inclusion. Suppose that, after a certain number of steps, some XiPl has no vertices of degree 1 , incident to a 2-element clique HiP1E 2’Z-,. Then the first vertex of XiPl (in the simplicia1 order) is contained in an H E 2tPl such that /HI L 3. Thus, the procedure described in the proof of Theorem 3 yields IKI s lK-,l - 3 in this step of (&). Since ll$l s ly-ll - 2 holds in all steps of (&), as proved above, the strict inequality t,-(G) < n12 follows. We do not know if a result similar to Theorems 2 and 3 holds for k = 4. For k 3 5, however, the answer is negative: As we will show in Section 2.3, ( * ) is not true for k 3 5 even in the far more special class of split-graphs. On the other hand, there is another interesting subclass of chordal graphs (containing all interval graphs) in which ( * ) holds for all k, as shown in the next section.

2.2. Strongly chordal graphs For s 3 3, an s-trampoline T“ is a graph on 2s vertices x,, . . . ,x,, yl, . . . ,ys, with the edge set {xixi: l s i < j 6 s } U { x i y i : l s i 6 s } U { x i y i + , : l 6 i s s } , where subscript addition is taken mods. Fig. 1 exhibits T 3 and T s . A graph is called strongly chordal if it is chordal and contains no trampoline as an induced subgraph. This class of graphs has been extensively investigated by Farber [6] who proved that the above definition is equivalent to assuming the existence of a strong elimination order v l , . . . , v, of the vertex set, i.e., a simplicial order in which any three edges viv,, vivk, vjvmE E ( G ) imply vkv, E E ( G ) whenever i < m and j < k. For this class of graphs, the equivalence of ( P l ) and (P2) is implied by the following statement.

Lemma 4. Let G be a chordal graph not containing T 3 as an induced subgraph. Zf G satisfies (Pl) then (P2) also holds in G. This statement will be deduced from a structural property of hypergraphs.

Lemma 5 . Let % be a set system with underlying set X , 1 x13 3. Zf X I$ X but all pairs x, E X are contained in some H E X , then there exist xl, x2, x 3 E X and H,, H2, H3 E %such that xiE Hi if and only i f i =j or j + 1 (mod3). X I

Proof. The statement is trivial for 1 x1= 3, so we proceed by induction on 1x1.If there is an x E X such that X \ {x} I$ X then setting X’ = {H\ {x}: H E X } and X ’ = X \ { x } we obtain a set system satisfying the assumptions on a smaller underlying set. Otherwise, for any three elements x,, x2, x3 E X , the sets X\{xi} E X ( i = 1 , 2, 3) form the required structure. 0

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V Fig. 1. The 3-trampoline T 3 and 5-trampoline T 5

Proof of Lemma 4. Let H be a clique and assume to the contrary that 3 6 IHI < k. Then (Pl) says that the set system X = {H' n H : H' E X,-(G), H' # H}, with underlying set X = H , satisfies the suppositions of Lemma 5. Indeed, Thus, for each each edge xx' is contained in some clique H' of cardinality a/?. x x ' , we have H' # H, and also H' # H since H is maximal under inclusion. Choose xi E H and HiE X with the properties guaranteed by Lemma 5, and let HI!E Xcbe a clique such that HI!fl H = Hi(i = 1, 2, 3). Since each HI! is maximal under inclusion, there can be chosen yi E Hi\H for which yixitl 6 E ( G ) (i = 1, 2, 3; subscript addition mod3). On the other hand, y i x j , yixi-l E E ( G ) , since G I , is a complete subgraph for 1S i S 3 . Thus, the yi are distinct and we have found T 3 as a subgraph. Since any induced T 3 is forbidden, there must be an edge within { y l , y 2 , y 3 } ; say, y , y 2 E E ( G ) . In this case, however, the cycle y , y 2 x 2 x 3is not chordal-a contradiction. 0

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Fig. 2 .

Note that (Pl) does not imply (P2) in every chordal graph. The smallest counterexample (with maximum cliques of cardinality 4) is shown in Fig. 2. We shall also need the following property of strongly chordal graphs for deducing ( * ).

Lemma 6. Let v l , . . . , v, be a strong elmination order of a strongly chordal graph G, and let Hi be the maximal clique having v i as its first element. Then the last element of Hi is contained by all cliques H, meeting Hi, whosefirst element v, satisfies m > i. Proof. Let v k be the last element of Hi and let H, E &, vj E H, fl Hi. Now we have viv,, vivk, vjvmE E ( G ) (H, and H, are cliques), moreover i < m and j < k (since v k is the last vertex in Hi). Hence, v,vk E E ( G ) , so that vk E r:, since m < j < k . Consequently, vk E H,, for a strong elimination order is simplicia1 also. 0 Theorem 7. Let G be a strongly chordal graph on n vertices. If every edge of G is contained in a clique of cardinality a k , then t,-(G) n Jk. Proof. By Lemma 4, X,-(G) has lower rank at least k. Let v l , . . . , v,, be a strong elimination order of the vertices, and apply (&) in the following way. Having obtained Xi-, and V,-l, let v be the first vertex of %,-,. Denote by H the that contains v , and choose ti as the last element of H. (unique) clique H E Xi-, Then Lemma 6 implies V, c V,-l\H. Since [HI 2 k in every step, t,-(G)< n / k follows. 0 Theorem 7 implies the validity of ( * ) for all k in any interval graph. Below we give a linear-time algorithm for the case when the interval representation of a

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graph is known. (The proof of the algorithm is omitted, it can be deduced from the proof of Theorem 7.)

Algorithm 8. (Given a collection 4 of n intervals, any pair of them incident to a clique of order s k in the intersection graph G,, the algorithm exhibits at most n / k of them, such that {Zl,Z,, . . . } meets all cliques of G9.) Put 4o= 4. Having obtained .Yj, for 9, # 0 let p i be the leftmost point of the real line, which is a right endpoint of some interval of 9,and is contained by at Let Z, E 4; be the interval whose right endpoint is the least k intervals of 9;. rightmost one among all intervals containing p i . Let 4,+1 consist of the intervals of LJi, having at least one point to the right of p i . Then i := i 1, and stop when 9; = 0.

+

We just mention that, if the intervals are ordered from left to right according to their right endpoint, then a strong elimination order is obtained for the corresponding interval graph. Note that the upper bound tC s n / k is sharp. The interval graph C, belonging to the family 9 = { [ i , i + k - 11: 1S i s n } has tc(G,) = Ln/k]. (This G, is k-fold connected.)

2.3. Split-graphs

A split-graph G = (P, Q, E) has edge set E and vertex set P U Q, where P is independent and Q is a clique. Trivially, every split-graph is chordal, so that ( * ) is true for k = 2 and 3 in any split-graph, as we have seen in Section 2.1. In this section we prove the analogous result for k = 4 and show that it cannot be extended for any larger k.

Theorem 9. If in a split-graph G on n vertices every edge is contained by a clique of order 3 4 then t,-(G) n/4. Clearly, Xc(G)= {Q} U { I ' ( p )U { p } : p E P } for any split-graph G, where T ( p ) denotes the set of vertices adjacent to p . Hence, (Pl)=(P2), and Theorem 9 is equivalent to the following result. (The degree of a vertex is the number of edges containing it.)

Theorem 9'. Zf X = (V, 8)is a hypergraph of IVI with lower rank 3 3 , then t ( X ) ( n + m ) / 4 .

=n

+

vertices and 181= m edges

Proof. In most cases, we proceed by induction on n m . The statement is trivial for n = m = 0 (and also for n 6 2, because then m = z ( X ) = 0). Without loss of generality, we can assume that X is 3-uniform (i.e., JEl= 3 for all E E 8). If there is a vertex v of degree 3 3 , then deleting v and the edges containing it, we obtain a hypergraph X' with n ' 6 n - 1 vertices and m' 6 m - 3 edges. Thus,

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t(%) 6 t(%') + 16 (n' + m')/4 1S (n + m)/4 follows by the induction hypothesis. Suppose v E V is a vertex of degree 1 and let v E E, E 8. If E, is disjoint from all E E 8\{Eu}, deleting v we obtain a hypergraph X' with n' = n - 3 vertices and m' = m - 1 edges, i.e., n + m decreases by 4. On the other hand, if v' E E, n E' for some E' E 8\{E,}, then deleting v' we obtain n' 6 n - 2 and m' = m - 2, proving the statement by induction again. Suppose there are two edges E, E' E 8 with an at least 2-element intersection which does not meet any other edge of X.Then the deletion of any v E E n E' yieldsn'6n-2, m ' = m - 2 . Let X be a hypergraph for which none of the above three reductions can be applied, i.e., from now on suppose that X = (V, 8) is a 3-uniform hypergraph, regular of degree 2, such that JEn E'J6 1 for all E, E' E 8. Then n/3 = m/2. Putting n = 3t and m = 2t, we have to show t( X ) < 5t/4. Let X* = ( V * , %*) be the dual of X , i-e., for each v E V take an edge E ( v ) E 8* and for each E E 8 take a vertex v ( E ) E V*, where v(E)E E ( v ) in X* if and only if v E E in X (cf. e.g. [3]). Then X* is a 3-regular graph (i.e., 2-uniform) on the 2t vertices, and we have to prove that its vertex set can be covered by at most 5t/4 edges. By a theorem of Gallai [7], this is equivalent to finding at least 3t/4 pairwise disjoint edges in X * . Let {el, . . . , eq} c 8* be a maximum family of pairwise disjoint edges, and suppose to the contrary that q < 3t/4. Then V' = V*\(el U * - U eq) is an independent vertex set of cardinality >t/2. Since X* is 3-regular, at least 3t/2 edges join V' with el U . . U eq, so that there is an ei, say eq, incident to at least three edges joining eq = xy with V'. Since both x and y have degree 3, one of those three edges, say ei, is incident to y, the other two (eq+l and ei+l) are incident to x . These edges are disjoint from el U * U eqPl, and some of eq+, and ehtl must be disjoint from eh. Thus, eq can be replaced by eh and one of eq+l and eq+l,contradicting the maximality of q. 0

-

Finally, it remains to show that ( * ) is no longer true for k 3 5, in the class of split-graphs.

Proposition 10. For every k 2 5 there exists a split-graph G = (P, Q , E ) such that every clique of G has at least k vertices, and tc(G) > IV (G) I lk.

Proof. (Construction.) Let l Q l = k + 1, IPI = [(k + 1)/2J =s, Q = (41,. . . , q k + l } ,P = { p l , . . . ,ps}. For i = 1,2, . . . , [(k + 1)/2], let ei = ( q Z i - 1 , q z j } and, if k is even, let e, = { q k ,q k + l } .Now the edge set E will consist of all pairs 4.9. ' I (16 i < j 6 k + 1) and piqi (1 6 i 6s, 1s j 6 k 1, j $ el). Then, obviously, t c ( G ) = 2 > (3k 4)/2k 3 IP U Q l / k , for k 2 5. On the other hand, every pi E P has degree k - 1, so that G satisfies the requirements. 0

+

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In connection with Theorem 9', the following problem arises.

Problem 11. For k 2 4, find the smallest constant c k such that z ( X ) 6 ck(m + n ) for every k-uniform hypergraph X with n vertices and m edges. Theorem 1.1 of [8] implies that c k S (log k)/k-more precisely, z( X ) S n / k + (m log k ) / k always holds. On the other hand, a variant of the construction described in the proof of Proposition 10 yields a lower bound approximately 2/(k - 1 + We also note that the inequality q 2 3t/4 proved in the last paragraph of the proof of Theorem 9' is not sharp. The best possible estimate is q 2 7 t / 8 , and also the structures satisfying q = 7 t / 8 can be characterized (see the particular case d = 3 of Theorem 1 (1.6) in [9]).

rn).

Note added in Proof. Concerning Problem 11, Lai and Chang [12] proved that c4 = 2/9, i.e., the 4-uniform hypergraph with 6 vertices and 3 edges of empty intersection (cf. the proof of Proposition 10) gives the smallest ratio of z ( X ) / ( n m). Moreover, Alon [lo] showed that ck = (1 + o(l))(log k ) / k as k tends to infinity. In connection with (*), some classes of graphs with z c ( G ) S n / 2 are characteized in [111.

+

Acknowledgements

I am grateful to professor Gallai and the referee for their remarks on the first version of the paper. References [l] M. Aigner and T. Andreae, Vertex-sets that meet all maximal cliques of a graph, manuscript, 1986. [2] T. Andreae, Covering the maximal cliques of a graph with at most half of its vertices, manuscript, 1987. [3] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [4] G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76. [5] P. Erdos, T. Gallai and Zs. Tuza, Covering the cliques of a graph with vertices, Discrete Math., to appear. [6] M. Farber, .Characterizations of strongly chordal graphs, Discrete Math. 43 (1983) 173-189. [7] T. Gallai, Uber extreme Punkt- und Kantenmengen, Ann. Univ. Sci. L. Eotvos Sect. Math. 2 (1959) 133-138. [S] Zs. Tuza, Extremal Problems on Graphs and Hypergraphs, Thesis (Hungarian Academy of Sciences, Budapest, 1983). [9] Zs. Tuza, Matchings and coverings in regular uniform hypergraphs, to appear. [lo] N. Alon, Transversal numbers of uniform hypergraphs, Graphs and Combinatorics 6 (1990) 1-4. [ l l ] T. Andreae, M. Schughart, and Zs. Tuza, Clique-transversal sets of line graphs and complements of line graphs, Discrete Math., to appear. [12] Feng-Chu Lai and G.J. Chang, An upper bound for the transversal numbers of 4-uniform hypergraphs, J . Combin. Theory Ser. B, to appear.

Discrete Mathematics 86 (1990) 127-136 North-Holland

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FACTOR DOMINATION IN GRAPHS Robert C. BRIGHAM Departments of Mathematics and Computer Science, University of Central Florida, Orlando, FL32816, USA

Ronald D. DUITON Department of Computer Science, University of Central Florida, Orlando, FL 32816, USA Received 2 December 1988 Given a factoring of a graph, the factor domination number yr is the smallest number of nodes which dominate all factors. General results, mainly involving bounds on yr for factoring of arbitrary graphs, are presented, and some of these are generalizations of well known relationships. The special case of two-factoring K, into a graph G and its complement G receives special emphasis.

1. Introduction All graphs considered will have no loops or multiple edges. A graph H = (V, E) has a t-factoring into factors G1, G,, . . . , G, if each graph Gi= (V;, Ei)has node set V; = V and the collection {El, E,, . . . , E,} forms a partition of E. Standard graphical invariants may be associated with H and the factors Gi. In this and the following section we shall employ unsubscripted symbols to designate invariants of H and the subscript i to designate invariants of Gi. Thus y represents the domination number of H and yi the domination number of Gi. This paper is concerned with a domination concept defined for a graph and a specific factoring of that graph.

Definition 1. Let H = (V, E) be a graph having the t-factoring G1, G,, . . . , G,. Then (i) D f s V is a factor dominating set if Of is a dominating set for each Gi, l s i s t , and (ii) the factor domination number yf(Gl, G,, . . . , G,) is the size of a smallest factor dominating set. We will write yf with no arguments when the graph H and its associated factoring are understood and the interpretation of the definition when t = 1 is that yp= y. The following observations are straightforward.

Observation 1. Let H be a graph with factors G1, G,, . . . , G,. Then

0012-365X/90/%03.5001990- Elsevier Science Publishers B.V.(North-Holland)

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Observation 2. The decision problem of determining whether a graph and associated factoring have a factor domination set of size k or less is NP-complete. Proof. Restrict t = 1. The problem reduces to DOMINATING SET [2]. 0 It is interesting to speculate on possible applications of factor domination numbers. One can envision a communication network (the graph H) composed of t subnetworks (the factors Gi). The reasons for several subnetworks might include security, redundancy, or the desire to limit recipients of certain classes of messages. The factor domination number of such a structure represents the minimum number of “master” stations required so that a message issued simultaneously from all masters reaches all desired recipients after traveling over only one communication link, no matter which subnetworks are active. Equivalently, it represents the minimum number of links which must be constructed between the network and a single control center so that messages from the center can reach any desired recipient over any subnetwork using at most two communication links. In this paper, we mainly consider bounds for y f in terms of other invariants. Section 2 deals with general results involving factorings of arbitrary graphs into an arbitrary number of factors. Section 3 studies the special case in which H = K p and t = 2 , i.e., when the complete graph is factored into a graph G and its complement 5. Although not treated here, other extensions of domination concepts to graph factorings are possible. For example, there are natural extensions of connected domination number, irredundance number and domatic number; and many results parallel the standard ones for graphs.

2. General factorings

Throughout this section we assume that the graph H is factored into GI, G,, . . . , G,, and that the node set is { u l , v,, . . . , v,}. The next observation shows that the problem of finding yf can be reduced to determining the ordinary domination number of a specially constructed graph.

Observation 3. There is a graph H ’ , constructible from H a n d its factors, such that Y(H’)= Yf. Proof. Construct H‘ on p ( t + 1) nodes from disjoint copies of H, GI, G2, . . . , G,. Additional edges connect vi of H to vi of Gj and to all nodes of G, which are adjacent to uj, 1 6j < p , 1=si =st. It follows that some minimum dominating set of H’ is contained in the copy of H and it is clear that the same nodes form a factor dominating set of ff and its factorization; thus y f ~ ( f f ’ ) .

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The reverse inequality is obtained by observing that the nodes of any factor dominating set, when interpreted as nodes of H in H’, form a dominating set of H‘. We now concentrate on finding bounds for yf. The first result is a generalization of Observation 1 and its proof is straightforward.

Observation 1 follows immediately by repeated application of Theorem 1. The next two theorems present upper bounds for yf. Here di is the minimum degree of Cj and a. is the node covering number of H.

Theorem 2. y f < p - miqGiGr{Si}. Proof. Let Of be any set of p - minl,iGt { S j } nodes. Clearly Of is a factor dominating set. 0

Theorem 3. Let I be the set of nodes in H which are isolated in at least one Gj. Then yf =S a. + IZ(. Proof. Let X be a minimum node cover of H. Suppose v is a non-isolated node of Ciwith incident edge e. Then e is also an edge of H so X dominates v in Gj. It follows that X U I is a factor dominating set. An immediate consequence of Theorem 3 is that yf G p - y + ( I ( , since y < p o = p - ao. Next we present some lower bounds for yf. The maximum degree for H is represented by A. First notice that t and A are related by t . minl-isf {Si} A, providing an upper bound on t when minISjG,(Si} is positive. On the other hand when min,,iGf { Si} = 0, t may exceed A or even p .

Theorem 4. If t G A, then yf 3 t, else yf = p . Proof. Let Df be a minimum factor dominating set. If t S A, any node v in H - Df must have in H at least t edges to Df so it can be dominated in each Gi; hence lDfl 3 t. If t > A, no such node v can exist and H - Df must be empty, i.e., Yf=P.

Theorem 5. If t S A, then yf 2 y + t - 2, else yf = p .

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130

Proof. When t > A the argument in the proof of Theorem 4 shows yf = p. Thus assume t s A and let Df be a minimum factor dominating set. If Df = V ( H ) , then y f = p * y + A 3 y + t > y + t - 2 a n d the result holds. If Df#V(H), let v e H Df.As in the proof of Theorem 4, is adjacent to at least t nodes of Of. Let X E D,be a set of t - 1 nodes contained in the neighborhood of v in H. In H every node of H - (Of U {v}) is adjacent to at least one node of Of - X. Thus ( D f - X ) U {v} dominates H so y e IDf -XI + 1= yf - ( t - 1)+ 1. 0 The final lower bound generalizes the well known result y a p / ( A

+ 1) [3].

Theorem 6. y, >p t / ( A + t).

Proof. If yf = p , the result holds. Thus we may assume t < A and yf < p . Let D, be a minimum factor dominating set. Each node of H - Of # 0 has at least t edges to Df for a total of at least (p - yf)t such edges. Thus

Solving for yf yields the result.

0

+

We conclude this section with an extension of the equality y E = p found in E is the maximum number of end edges in a spanning forest of H. This is an illustration of the generalization of domination concepts and proofs to factorizations.

[4]where

Definition 2. The invariant Ef is the cardinality of the largest set of nodes X such 1s i S t, there is a spanning forest in which X is independent and that in each Gi, each node of X has degree one. Notice that the definition is equivalent to that of

Theorem 7. yf

E

when t

= 1.

+ Ef = p .

Proof. If yf = p , then E~ = 0 and the result holds. Thus we may assume t s A and y f < p . Let Df be a minimum factor dominating set. In each Ciarbitrarily select for each node of Gj - Df one edge between it and Df. The subgraph of Githus formed is a union of stars centered on the nodes of Of and is a spanning forest of C j . In each of these spanning forests the nodes of Ci- Of are independent and have degree one so it follows that ~ ~ ->yf. pNow suppose X is a set of Ef nodes satisfying the rquirements of Definition 2. Then the nodes of H - X form a factor dominating set and yf s p - Ef. 0

Factor domination in graphs

131

3. Two-factors of Kp In this section we restrict attention to 2-factors of the complete graph Kz Thus GI = G and G2= ??. To simplify notation we employ y = y ( G ) and 7 = y(G). The same convention will apply to other graphical invarants of G and G. We begin by noting that Observation 1 may be rephrased for this special case as max{ y, 7 )s

Yf Y + 7.

r}

Equality with max{ y, is achieved by several classes of graphs and we list a few with their easily computed values. Here Kp is the complete graph, C, is the cycle, W, is the wheel, and G* is the complete r-partite graph K,,,, *,..., n,.

(9 YXK,, KJ

= P a

(iv) yf(G*, G*) = r. A more general class of graphs for which yf = max{ y,

7 }is given in the following.

Theorem. If either G or G is disconnected, yf = max{ y,

U}.

Proof. Assume G is disconnected. Any dominating set of G must contain at least one node from each of its components and such a set clearly dominates E. 0 We will see other conditions which guarantee yf = max{ y, Y}, but we first give a result which shows there are graphs for which yf may have any of the values between max{ y, 7}and y + 7. Verification of Theorem 9 is not difficult, but it is tedious and not particularly instructive. Theorem 9. For any integers m, n , and k such that 2 exist graphs G for which y = m, 7 = n, and yf = k.

m
k

m

+ n, there

are connected. Thus We now consider graphs for which both G and diameters d and ;iboth exist and are greater than or equal to two. Theorem 10. If G and are connected, then ( 9 Yf= max{y, 71 ford +dz=7, (ii) yf s max(3, y, 7} 1 for d + 2 = 6, (iii) yfs max{ y, 7 }+ 2 for d + = 5, for d = d = 2. yf s min{ S, + 1 (iv)

+

s}

R.C. Brigham, R . D . Dutton

132

Proof. Without loss of generality assume d 3 d 2 2 and that nodes x and y have distance d in G. Let X denote the set of nodes containing x and its neighbors in G and similarly define Y for y. We verify each result in turn. (i) Since d > 3 implies d < 3 [l], we may assume d 3.5 and d = 2 . Every dominating set of G contains a node from X and a node from Y. These two nodes have no common neighbor in G and thus dominate It follows that Yf = Y = max{y, 3. (ii) Either d = 2 = 3 or d = 4 and d = 2. Since two distance three nodes in any graph dominate the complement graph, we have when d = d = 3 that y = 7 = 2 and yf G 4. When d = 4 and 2 = 2 the argument presented in (i) suffices except, to dominate G, we may also need to add node x to a minimum dominating set of G. (iii) Here d = 3, d = 2 and nodes x and y dominate 5. Thus any minimum dominating set of G along with { x , y > will dominate both G and G. (iv) In a diameter two graph any node and its neighbors dominate both the graph and its complement. 0

z.

In a similar vein, a result concerning the radii r and F can be derived.

Theorem 11. If G and

G are connected and max{r, 3 3 3, then

yf = max{ y, 7).

Proof. Assume without loss of generality that r 2 3 and yf > max{y, U}. Then every minimum dominating set of G has a common neighbor from which it follows that r S 2, a contradiction. We now examine bounds for yf involving other invariants of G and E. The first of these concerns the maximum degrees A and d.

Lemma 1. If yf > y, then yf =sA

+ 1.

Proof. Let D be a set of y nodes which dominates G and let X be a maximum set of nodes of G - D each of which is adjacent to every node in D and all other nodes of X. Notice that D U X dominates both G and G. The set X has at least yf- y nodes. Otherwise D U X would be a factor dominating set having less than y (yf - y ) = yr nodes. The degree of every node in the nonempty set X is at least y (yf- y - 1) = yf- 1, i.e., A 2 yf - 1. 0

+

+

Applying Lemma 1 to both G and 5 produces the following result.

Theorem 12. Either yf = max{ y, 7}or y f Smin{A,

d}+ 1.

Another conclusion which follows from the proof of Lemma 1 is that the size of the largest clique o in G is at least yf - y + 1.

Factor domination in graphs

Theorem W. yf s min{ w

133

+ y, W + 7 )- 1.

Corollary 1. If G is triangle free then y s yr G y

+ 1.

We can be more specific for acyclic graphs. Let T' be the set of trees on two or more nodes with radius one, i.e., star graphs, or radius two with a node of degree at least two and all of whose neighbors have degree at least three.

Theorem 14. If G is a tree, then y+l $GET', Yf=(y otherwise.

Proof. Corollary 1 shows y s yfs y + 1 and it is clear that yf = y + 1 when G is in T'. Suppose G is not in T'. Then either G is K1 (in which case yf = y = l), has r 5 3 (yf = max{ y, = y by Theorem ll),or r = 2 and every node of degree two or more has a neighbor of degree one or two. Considering the last case, let x be a node of distance at most two from every other node in G and which has at least one neighbor of degree one or two. Let Nl be the set of neighbors of x of degree one. If Nl is not empty, let D = ( N ( x ) - N , ) U { x } . Otherwise let y E N ( x ) have degree 2 and be adjacent to z Z x , and set D = ( N ( x ) - y ) U (2). In either case the set D dominates both G and G and no smaller set can dominate G. Thus Yf = Y. 0

u}

We now present an upper bound for yf in terms of the minimum degrees of G and G.

Theorem 15. if6=&2, max{6,s) 1 otherwise.

+

Proof. Suppose yf > max{6,s)f 1. Let x be a node of minimum degree in G. Then x and its neighbors dominate G but do not dominate G. Thus there is a node y of distance 3 from x which means x and y dominate G and 7 S 2. A similar shows that y S 2 . It follows that y f S 4 so by the supposition argument using max{ 6, S}S 2. If 6, say, is zero, has diameter at most two. Such graphs have yr6 8 1 and contradict our supposition. If 6 = 8 = 2 then the conclusion holds, since yf = 4 in this case. Finally, it is easy to show y f s 3 for all graphs with min{b, S>= I.

c

+

Notice that Theorem 2 implies yfSp-min{6, strengthen this.

s}.With Theorem 15 we can

134

R. C. Brigham, R. D. Dutton

Corollary 2.

p-A+l if A = d s p - 3 , p - min{ A, d} otherwise. The edge independence numbers of G and G also provide bounds on yr. We will use two preliminary lemmas to obtain the general result.

p1and p nodes, k of which are isolated. Then f o r any maximum independent set of edges X the nodes of C may be labeled and partitioned into sets A = {ai 1 s i s PI}, B = {b, 1 1s i s pl}, C = {ci I 1 S i S p - 2p1- k} and D = { d , 1 1 s i G k} where ( 1 ) {a;, bi} is an edge of X f o r 1 s i s PI, ( 2 ) di is an isolated node of G for 1 i k , and (3) each ai is adjacent to at most one node in C and if ai is adjacent to cj then bi is also adjacent to cj and to no other node of C.

Lemma 2. Let G be a graph with edge independence number

I

Proof. For a set of p1independent edges label and partition the nodes of G into sets A, B, C , and D in such a way that (1) and (2) are satisfied. Now for i = 1, 2, . . . , PI if a; is adjacent to some c, and bi is not adjacent to that cj interchange the labels of ai and bi as well as their appearances in the sets A and B. Observe that (1) and (2) remain valid. If an {a;, bi} pair was not interchanged in the above process, the neighbors of aj in C form a subset of the neighbors of bi in C. Suppose a, has at least one neighbor cj in C, and bi is adjacent to c, as well as to cj. Then we could replace, in the original set of PI independent edges, edge {ai, bi} with the two edges {a;, c j } and { b;,c,} and obtain an independent set of edges of size p1+ 1. Therefore all such {ai, bi} pairs individually satisfy (3). Suppose instead that the pair {a;, b;} was interchanged. Then, after the interchange, bi (under the new labeling) is adjacent to some cj and a, is not. I f a, is adjacent to any other C node x , then { x , a;} and {b;,c j } could be used as above to form an independent set of p1+ 1 edges, again contradicting the maximality of Bl, Thus we may assume ai has no neighbors in C and therefore {a;, bi} satisfies (3). In this construction, the sets C and D induce an empty subgraph on p - 28, = a1- P1+ k nodes where a,= p - B1 - k is the edge covering number. Since no node in C is isolated, each must be adjacent to at least one node in B. Thus the nodes in B and D dominate the graph G, i.e., y S P1+ k. Further, if D is not empty every dominating set of G must contain the nodes in D and any of these will also dominate G. Thus, yf = y =sPI + k when k > 0, i.e., 6 = 0. Lemma 3. If

PI G a1- 2, then y f < max{p, + 1, p1+ k } .

Fucror domination in graphs

135

Proof. From the preceding remarks we need only consider graphs for which k = 0. Using the structure in Lemma 2, we have that D is empty and the set C contains at least two nodes. First assume no a, node has a neighbor in set C . Then the set B dominates G, and B along with any ci node will dominate E. Thus y f S PI + 1. Now suppose at least one ai node is adjacent to some cj node. Consider the set X = ( B - {bi})U { c j ,c,} where ai,and thus bi, is adjacent to cj and m Zj.First, notice that by ( 3 ) of Lemma 2 neither ai nor bi can be adjacent to any other node in C. Therefore every node in C - { c j } must have a neighbor in B - { b i } and thus X dominates G. Since C is an independent set of nodes, b, is not adjacent to c, and no node in A can have two neighbors in C , we have in E that { c j , c,} dominates C U A U { b i } . Thus, the set X also dominates G, i.e., YfG (XI = p, + 1. 0 Theorem 16. If 6, 8> 0, then yfS min{/3,,

PI}+ 1.

Proof. When 6 > 0, p = a1+ PI and then G ( p - 2 ) / 2 is equivalent to PI s a,- 2. Thus 6, and rnin{&, s ( p - 2)/2 gives, by Lemma 3, y r s min{p,, 1. The only remaining case involves graphs with 6, 0 and PI = [ p / 2 ] . Using the notation of Lemma 2 independently for G and ??,we have D and 5 empty and IC( = S 1. Sets A U C and B U C both dominate G and AUC and 3 dominate ??. If any one of these four sets of size at most PI 1 dominates both G and G, the result holds. If none do, then in each A , B, and B there is a node which dominates B U C , A U C, B U c , and X U C respectively, i.e., we must have y = y = 2 and thus y f G 4 . Now suppose the conclusion is false. Then [ p / 2 ] 1= PI + 1 = + 1 < yfS 4. Since 6 > 0 and y = 2 , we conclude that PI = P 1 = 2 , y f = 4 , and p s 5 . From Theorem 2, yrs p - min{ 6,s).Thus p = 5 , min{ 6, = 1 and max{A, d}= 3. We cannot have A =d =3, since then yr = 4 > 3 = p - A + 1 and contradicts Corollary 2. The alternative, A # also gives a contradiction since Corollary 2 now implies min{A, d} s p - yf= 1. This means either G or G must have an isolated node = 1. since p is odd which violates min(6,

=PI

PI} +

+ A,

s>O Uc

PI}

s>

Icl

+

PI

s}

z,

s}

4. Summary

The concept of a factor domination number yf has been defined for an arbitrary t-factoring of any graph H . Determining yf in general was shown to be a difficult problem, and our efforts concentrated on finding reasonable bounds. While it has not been demonstrated here, it can be shown that almost all of the bounds presented are sharp. The most fruitful directions of research seem to concern graphs with a small number of highly structured factors. We have chosen in Section 3 to concentrate on graphs H = K , with t = 2 . Our results certainly are

136

R. C. Brigham, R. D . Dutton

not complete, e.g., it is possible to relate yf to the node independence and chromatic numbers of G and It may also be interesting to study restrictions on the 2-factors G and G, e.g., where G is self-complementary or planar. Many other cases await further research.

c.

References [l] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (North-Holland, Amsterdam, 1976) 14. [2] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [3] R. Laskar and H.B. Walikar, On domination related concepts in graph theory, in: Lecture Notes in Mathematics 885 (Springer, Berlin, 1980) 308-320. [4]J. Nieminen, Two bounds for the domination number of a graph, J . Inst. Math. Applics. 14 (1974) 183- 187.

Discrete Mathematics 86 (1990) 137-142 North-Holland

137

THE LEAST POINT COVERING AND DOMINATION NUMBERS OF A GRAPH E. SAMPATHKUMAR Mysore University, M y s o r e - S 7 W , India Received 2 December 1988 A set S c V of a graph G = (V, E) is a total point cover (t.p.c.) if S is a point cover containing all isolates of G, if any. The number a,(G) is the minimum cardinality of a t.p.c. A t.p.c. S is a least point cover (1.p.c.) if a t ( ( S ) ) sa , ( ( S , ) ) for any t.p.c. S,, where (S) is the subgraph induced by S. The least point covering number a , ( G )of G is the minimum cardinality of a 1.p.c. A dominating set D of G is a least dominating set (1.d.s.) if y ( ( D ) )s y ( ( D , ) ) for any dominating set D , ( y denotes domination number). The least domination number y,(G) of G is the minimum cardinality of a 1.d.s. If yt is the total domination number, we prove among other things: (i) yI s yt, and (ii) for a tree, yIs a,. Conjectures. For any graph G of order p 3 2, (1) y I s a,,(2) ylC 3p/5, if G is connected.

We consider graphs G = (V, E) which are finite, undirected, loopless and have no multiple lines. Any definitions not given here can be found in [4]. A set f)c V is a dominating set if every point in V - D is adjacent to some point in D. The domination number y ( G ) of G is the minimum cardinality of a dominating set. For a survey of results on domination see [3, 61. A set S c V is a total point cover (t.p.c.) of G if S is a point cover containing all isolated points of G, if any. The total point covering number a,(G) of G is the minimum cardinality of a t.p.c. Clearly, the point covering number a o ( G ) S a t ( G )and equality holds if G has no isolates. For a set S c V , let (S) be the subgraph of G induced by S. A t.p.c. is a least point cover (1.p.c.) if a , ( ( S ) ) s at((&)) for any t.p.c. S,. The least point covering number a,(G) of G is the minimum cardinality of a 1.p.c. The p-point covering number a p ( G )of G is defined as follows: Let S be a 1.p.c. Then a,(G) = a , ( ( S ) ) . A dominating set D is a least dominating set (1.d.s.) if

for any dominating set D 1 . The least domination number y,(G) of G is the minimum cardinality of a 1.d.s. The d-domination number yd(G) of G is defined as follows: yd(G)= y ( ( D ) ) where

D is a 1.d.s.

An &,-set is a minimum t.p.c. and an &,-set is a minimum 1.p.c. Similarly we define a y-set and a yl-set. 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V.(North-Holland)

E. Sampathkumar

138

Fig. 1.

For C 5 , yI = 3 while y = 2. In Fig. 1, { 1 , 5 , 7 , 9 } is a y-set, { 1 , 4 , 7 , 9 } is a yl-set, { 1 , 3 , 5 , 7 , 9 } is an a,-set and { 1 , 3 , 5 , 6 , 7 , 9 } is an a,-set. Thus for G , y = y, = 4, at = 5, and aI= 6. . be the smallest integer not less than r. For a real number r > 0, let 11

Proof. We prove only (i) and (v) by induction on n. First we make the following observations. Suppose G,, = P,, or C,,. Then

(A)

al(Gn+l) = a,(G,,)+ 1 when II = 0, 2 or 3 mod 4, when n = 1mod 4,

= a,(G,,)

and

(B)

y,(Gn+l)= yl(Gn)+ 1 when n = 1,2, 3 or 4 mod 5, when n = 0 mod 5.

= Yl(Gn)

(i) The result is true when n = 3. Suppose it is true for some n or 3 mod 4, then by observation (A), aI(Pn+l) = a,(Pn)

-

If n = 0 , 2

+1

= ( n - 1) -

=n

3 3.

[ ( n - 1)/4]

+1

(by inductive hypothesis)

[n/41.

If n = 1 mod 4, then

a l ( P n + , )= al(Pn)= ( n - 1) - [(n - 1)/41 = n - [n/41 The proof of (iii) is similar and we omit it. (v) The result is true when n = 3. Suppose it is true for some n

3 3.

If

The least point covering and domination numbers of a graph

139

n = 1 , 2 , 3 or 4 mod 5, then by observation (B),

YI(Gn+J = YdGJ + 1 = n -2[n/Sl + 1 = ( n 1) - 2 [(n+ 1)/51. If n=O mod5,

+

YdGn+I) = Y d G n ) - 1 = n - 2[n/51 - 1 = (n + 1) - 2[(n + 1)/5].

Cl

Proposition 2. For any graph G (i)

aps at s a,,

(ii)

yd s y 6 y,.

Proof. (i) Let S be a 1.p.c. Since V is a t.p.c. we have by the definition of ap(G), ap(G) = at((S)) s a t ( ( V ) )= at(G). The other inequality follows since any 1.p.c. is a t.p.c. Likewise, (2) follows.

0

Proposition 3. For any graph G (i)

Y

(ii)

yd s ap.

(3)

at,

(4)

Proof. (i) is true since any t.p.c. is a dominating set. (ii) Let D be a 1.d.s. and D1be a 1.p.c. Then D1is a dominating set and by ( 3 ) , Yd=Y((D))s

r((Dl))sat((Dl>)=

&p.

0

Proposition 4. For any tree T, YI

a,.

Proof. By induction on the number of points p in T. The result holds when p = 2 or 3. Suppose p 2 3 and (5) holds for any tree with p - 1 points. Let T be a tree with p points. We consider two cases. Case 1. There exists a point u in T adjacent to at least two pendant points. Suppose v is a pendant point adjacent to u. Clearly, y l ( T )= yl(T - v ) and a,(T ) = a,(T - v). Since y,( T - v ) s a,(T - v ) by hypothesis, (5) holds. Case 2. The exists no point u as in Case 1. Certainly there exists a pendant line uv with deg v = 1 and deg u = 2. It is easy to see that y l ( T )S yt(T - v ) + 1 and a , ( T )= mI(T - v ) + 1, and (5) holds. 0 We see from Proposition 1 that the inequality (5) holds for cycles. Also, it is verified for many other graphs and we are led to believe that it may be true in general.

E. Sampathkumar

140

Conjecture 1. For any graph C YI =s f f l . We now investigate the relationship of yI with total and connected domination numbers. The total domination number y t ( G ) of G is the minimum cardinality of a dominating set D such that ( D ) has no isolates. (See [l,21.) The connected domination number y c ( G ) of G is the minimum cardinality of a dominating set D such that ( D ) is connected (See [5,8].)

Proposition 5. For any graph G without isolates YI s Yt.

(6)

Proof. Among the minimum total dominating sets of G, choose a set D such that y ( ( D ) ) is least. We claim that D is a 1.d.s. For, let D,be any dominating set. If ( D l ) has no isolates, then D, is a t.d.s. and hence y ( ( D ) ) s y ( ( D l ) ) , by our choice of D. On the other hand, suppose ( D l ) has k isolates, say u,, i = 1 , 2 , . . . , k. For each ui,let vi be a point in V - D 1adjacent to ui.(Note that the 21,’s may not all be distinct). Clearly, D2 = D1U {v,, v2, . . . , vk} is a t.d.s. and Y ( ( D ) ) s Y ( ( D 2 ) )s Y((D1)). This proves that D is a 1.d.s. Since ID1 = y t , (6) holds.

0

For a p point graph G without isolates, Cockayne, Dowes and Hedetniemi [2] have shown that yt s 2 ~ 1 3 Hence . by (6), y , =s 2pl3

(7)

We strongly feel that this bound can be improved. Conjecture 2. For a connected graph with p 3 2 points yI s 3pl5. We observe that from Proposition 1, Conjecture 2 holds for paths and cycles. Also, it has been verified for many other graphs. In fact, the bound is suggested by the formula for yI in Proposition 1. Let A and 6 respectively denote the maximum and minimum degrees of a graph G of order p . We now relate yIwith yc. If G is connected, then YI =s Yc. For, (8) holds when p = l or A ( G ) = p - 1 , p 3 2 . If A ( G ) < p - l , yt s yc, and (8) follows from (6).

(8)

then

The least point covering and domination numbers of a graph

141

It is known that y c < p - A (see [5]). Hence, from (8), one can deduce Y1SP-A

(9)

for any graph G. Let be the complement of G and establish Gp

y1+

rl= y l ( c ) . Using (9),

one can easily

+ 1.

(10)

rl

If G is a tree with p b 3 points, we note that = 2, and if e is the number of pendant points of G, then yc = p - e (see [8]). With the help of (8), we see that for a tree yI

+

s p -e

+ 2.

(11)

Let k ( C ) be the connectivity of G . Since k ( G ) s 6 ( G ) , using (9) we can deduce the following: Let y = yI or yc. If G is connected, then

y sp

- k(G).

Likewise, since yt < p - A

(12)

+ 1 when G has no isolates (see [2]), one can deduce

sP -k(G)

Yt

(13)

for a connected graph G # K,. We now obtain a necessary and sufficient condition for y = yI = yc. Let eF( E ~ be ) the maximum number of pendant lines in any spanning forest (tree) of G.

Proposition 7. If G is connected, then:

+ eF= p

(i)

y

(ii)

yc

(Nieminen [7]),

+ E~ = p (Laskar and Hedetniemi [ 5 ] ) , +E ~ Z P ,

(iii)

YI

(iv)

yl+

ET

sp.

Clearly, (16) and (17) follow from (2), (8), (14) and (15). Corollary 7.1. For a connected graph C , y = y I= yc if, and only if,

Acknowledgement Thanks are due to the referees for their valuable comments.

EF =

eT.

142

E. Sampathkumar

References [1] R.B. Allan, R. Laskar and S.T. Hedetniemi, A note on total domination, Discrete Math. 49 (1984) 7-13. (21 E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219 [3] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247-261. [4] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [5] R. Laskar and S.T. Hedetniemi, Connected domination in graphs, in: B. Bollabas, ed., Graph Theory and Combinatorics (Academic Press, London, 1984) 209-218. (61 R. Laskar and H.B. Walikar, On domination related concepts in graph theory, in: Combinatorics and Graph Theory (Calcutta, 1980), Lecture notes in Mathematics 885 (Springer, Berlin, 1981) 308-320. [7] J . Nieminen, Two bounds for the domination number of a graph, J. Inst. Math. Applics. 14 (1974) 183- 187. [8] E. Sampathkumar and H.B. Walikar, The connected domination number of a graph, J . Math. Phys. Sci. 13 (6) (1979) 607-613.

Part IV. Algorithmic

This Page Intentionally Left Blank

Discrete Mathematics 86 (1990) 145-164 North-Holland

145

DOMINATING SETS IN PERFECT GRAPHS Derek G. CORNEIL Department of Computer Science, University of Toronto, Toronto, Ont., Canada MSS 1Al

Lorna K . STEWART Department of Computing Science, University of Alberta, Edmonton, Alta., Canada T6G 2E7 Received 2 December 1988

In this paper, we review the complexity of the minimum cardinality dominating set problem and some of its variations on several families of perfect graphs. We describe the techniques which are used to attain these complexity results, with emphasis on the dynamic programming approach to the design of algorithms.

1. Introduction Since their introduction by Claude Berge in the early 1960s [2], perfect graphs have attracted considerable attention, and many interesting families of graphs have been shown to be contained in the perfect graphs. Perfect graphs are graphs in which the maximum clique size is equal to the chromatic number for every induced subgraph. One of the problems which has been widely studied in relation to these graphs is that of finding a minimum dominating set. In this paper, we review the results in this area and attempt to give the reader an understmding of the techniques which have been used. Many of these methods involve dynamic programming; we illustrate this approach by developing solutions to the dominating set problem for the family of perfect graphs known as 1-CUBS, and the total dominating set problem for permutation graphs. These algorithms are described in a ‘tutorial’ manner in an attempt to give the reader a detailed understanding of the problem solving techniques.

2. Complexity summary For a graph G(V, E), a dominating set S is a subset of the vertices such that every vertex in V - S is adjacent to some vertex in S. Throughout this paper, d ( G ) will denote the size of a minimum dominating set in a graph G. A dominating set S is independent if the vertices of S are pairwise non-adjacent, total if the subgraph induced by S has no isolated vertices, connected if the vertices of S induce a connected subgraph, a dominating cycle if the subgraph 0012-365X/90/$03.50 01990- Elsevier Science Publishers B.V. (North-Holland)

D.G. Corneil, L.K. Siewart

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Graph family

Domination

Independent

Connected

Total

Cycle

Clique

Comparability Bipartite Trees Permutation Cographs 2-CUBS 1-CUBS Chordal Split k-trees (fixed k) k-trees (arb k) Strongly chordal Undirected path Directed path Interval

“221 “221 p[151 P[10][26) P[201

“201 “201 p[71 P[10][26] Pi261

“381 “381 P[lI P[10][18] PPOI

“381 “381 ~1361 P[*][lO], [ l l ] PWI

”161 “171 P[eI P[16] P[16]

P[12] P[12] P[eI P[12] P[20]

“*I

“*I

“*I

”*I

”*I

P[*l “8”l “41[201 P[21] “211 P[25] “91 P[91 P[91[251

P[*I “351 “351 P[lI

P[*l “361 “351 P[lI

P[*l “171 “171

P[*l “121 “121

~ 4 1 ~241 P[lI PI241 ~ 4 1 ~ ~ 4 pi241 w41

~1421 1 “341 ~1421 p[421

“361 P[5][32][40]

P[341 “61[341 P[341 PP41 P[33] P[12]

induced by S has a Hamiltonian cycle, and a dominating clique if the subgraph induced by S is a complete graph. The minimum dominating set problem is that of finding a minimum cardinality dominating set. Variations of this problem which have been studied include finding the minimum cardinality dominating set which is independent, total, connected, a cycle or a clique. We note in passing that considerable work has been done on the weighted cases of these problems as well. Since all of these problems are known to be NP-hard for general graphs [27], much research has been focussed on finding polynomial time solutions for certain families of graphs. Table 1 summarizes the complexity status of the six minimum cardinality dominating set problems for several families of perfect graphs. The definitions of the various families referred to can be found in the glossary at the end of this paper and in many of the references. The graph theory terminology used is standard and may be found in [28]. The relationships among these and other families of perfect graphs are described in many sources, including [28] and [30]. Some of the straightforward containment relations are as follows: cographs c permutation c comparability c perfectly orderable, perfectly orderable c strongly perfect c perfect, trees c bipartite c comparability, bipartite c parity c Meyniel c strongly perfect, split c chordal c weakly chordal, trees c k-trees (fixed k ) c k-trees (variable k ) c chordal, interval c directed path c undirected path c chordal, interval c directed path c strongly chordal c chordal, cographs c 1-CUBS c 2-CUBS c CUBs, chordal c CUBs, chordal c Meyniel.

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Notice that the results of Table 1 imply a number of other polynomial and NP-hardness results because of the containment relationships among the various perfect graphs families. For example, we see that the dominating set problem is NP-hard for perfect graphs as well as for perfectly orderable graphs and strongly perfect graphs, since the comparability graphs are contained in each of these families [3, 141. The problem can also be seen to be NP-hard for Meyniel graphs and parity graphs since bipartite graphs are contained in these families [13, 371. In addition, the NP-hardness of the dominating set problem on weakly chordal graphs follows from the result on chordal graphs [29]. Finally, we see that a polynomial time algorithm for domination on interval graphs follows from the algorithm for strongly chordal graphs and from the algorithm for directed path graphs, since interval graphs are contained in both of these classes of graphs [9, 251.

3. Techniques In this section we examine the techniques most commonly used for establishing the complexity status of various domination problems on perfect graphs. Polynomial algorithms will be demonstrated for the domination problem on 1-CUBS and the total domination problem on permutation graphs. First we show that the domination problem (i.e. given a graph G and integer k is d(G) =S k ? ) is NP-complete for k-CUBS (k 3 2). 3.1. NP-completeness techniques The most commonly used transformation is from the h-vertex cover problem namely given a graph G(V, E) and a positive integer h S IVl does there exist a subset V’ c V s.t. IV’(=S h and each edge in E has at least one endpoint in V’? This problem was shown to be NP-complete by Karp [31]. As an illustration of a transformation from the h-vertex cover problem to the domination problem, consider the class of k-CUBS ( k 3 2 ) (see [19]). This proof is identical to that used to establish the NP-completeness of the domination problem on split graphs [201.

Theorem 3.1. The domination problem is NP-complete for k-CUBS (k 2 2). Proof. Clearly the problem belongs to NP. We reduce the h-vertex cover problem to this restricted domination problem as follows. Given graph G ( V , E) and integer h we construct a 2-CUB G‘(V’,E’) by bonding using a K,,, and (El copies of K,. (Clearly these complete graphs are 2-CUBS.) For each edge (i, j ) E E a new K , is 2-bonded to the edge ( i , j) in K,,,. We will show that G has a vertex cover of h vertices if and only if G’ has a dominating set of h vertices. Let A be a vertex cover of G. The same set of vertices chosen in the K,,, in G’ is

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clearly a dominating set in G'. Conversely let B be a dominating set in G'. Any vertex which does not belong to KIVlcan be replaced by a neighbouring vertex in Klvl without destroying the domination property. This new dominating set B' c KIV,also forms a vertex cover of G. 0 As an immediate corollary of this theorem we have:

Corollary 3.2. The connected, total, cycle and clique domination problems on k-CUBS ( k 3 2) are NP-complete. 3,2. Polynomial techniques We now turn to the techniques used to show that various classes of perfect graphs have polynomial time domination algorithms. Many classes of perfect graphs have tree representations (sometimes unique). Dynamic programming on such a tree is then used to solve the domination problem. In these algorithms a minimum dominating set for the subgraph G, represented by node x in the tree is determined from minimum dominating sets for the subgraphs represented by the children of x. For some families of graphs, this straightforward dynamic programming is not sufficient and a more sophisticated variation is needed. This could involve the storage of all minimum dominating sets for G, or other information such as a minimum dominating set with specific properties (such as the inclusion or exclusion of a particular node). Furthermore in some cases there could be an exponential number of minimum dominating sets for G,. Sometimes a polynomial sized representative set suffices. An example of this is presented in

WI. We now illustrate these techniques by solving two new dominating set problems, namely the dominating set problem on 1-CUBS and the total domination problem on permutation graphs. In these presentations, we follow a tutorial approach and illustrate both the false attempts and the solution. 3.2.1. Polynominal algorithm for the domination problem on 1-CUBS

I-CUBS are an extension of cographs and thus it seems possible that an algorithm similar to one for cographs will work for I-CUBS. The essential difference arises in the treatment of 1-bonding. A parse tree for a CUB is a tree which illustrates a possible composition of the graph using the operations of complement, union and bonding. Assuming that there is a 1-CUB recognition algorithm which will produce a parse tree for a 1-CUB (such an algorithm will be presented later), we focus on the part of the domination algorithm dealing with graph G where G is the 1-bonding of graphs G, and G2 at vertex x . For this analysis we will try to determine the domination number, d ( G) from information which could have been recursively calculated for GI and G2. (In a subsequent algorithm an actual minimum dominating set will be produced.) The first

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observation is that either d ( G ) = d(Gl) + d(G2) or d ( G )= d(Gl) + d(G2) - 1. An attempt to determine which case holds leads to the following lemma. In this lemma d ( G , x ) refers to the smallest domination set of G where x must be in the domination set.

Lemma 3.3. Zf G is formed by the 1-bonding of graphs G1 and G2 at vertex x , then

i

4 G l ) + 4G2) - 1

d ( G )=

ifd(G1) = d(G*,x ) A d(G2) = 4G2, x ) ,

d(G*)+ M"(G2),

d(G2 - x ) )

4G2) + M"(Gl),

d(G1 -XI)

4 G l ) +4 G 2 )

+ d(G1, x ) A d(G2) = d(G*,x ) , ifd(G1) = d(G1, x ) A d(G2) + d(G2, x ) , ifd(G1)

$d(Gl)

+ 4 G ,X I A d(G2) f 4 G 2 , x ) .

In order for this lemma to be useful in a polynomial time algorithm for calculating the domination number for a 1-CUB we have to calculate the following information for each subgraph H determined by a vertex in the parsing tree of the given 1-CUB:

d(H);

d(H,Y), VYEH;

Wf-y),

VYEH.

Although it may be possible to calculate d ( H , y ) efficiently there appear to be difficulties with d ( H - y ) . It seems that one may need to calculate d ( H y - z ) V z E H , z Z y . This in turn would require the exclusion of all possible triples etc. Thus we see that an exponential amount of work may be required. The obvious difficulty with this general approach is that we have not exploited the structure of 1-CUBS. We now present a polynomial time algorithm which does exploit this structure. This algorithm will use a tree representation of the 1-CUB which is constructed in the following polynomial time 1-CUB recognition algorithm.

Algorithm 3.1. 1-CUB(G) (1-CUB recognition algorithm). Input: G. Output: a 1-CUB parse tree for G if G is a 1-CUB, "NO" otherwise. (1) If G = {x}, then output

(2) If G = A , U A2, then output

R

l-CUB(A1) 1-CUB(A2).

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(3) If G = A , UA,, then output

l-CUB(A1) l-CUB(A2).

(4) If G is formed by the 1-bonding of A, and A, at vertex x , then output

(5) If G has no cutpoints, output ‘NO’ and halt. Otherwise let A l , A,, . . . ,A, be the blocks of G, that is, the induced subgraphs of G which have no cutpoints and which are maximal with respect to this property. Output n

T

l-CUB(A1) 1-CUB(A2) l-CUB(Ak)

where T is the rooted tree whose nodes are Ai (1 s i s k ) and T represents the tree structure among the blocks of G. As an example of the above algorithm consider the graph in Fig. 1. The 1-CUB tree produced by the algorithm is presented in Fig. 2. Examination of Algorithm 3.1 yields the following facts about the 1-CUB tree TG .

Fact 3.4. If the root of TG is a B, node with children A, and A2 then: (i) there exists a uertex y E A l s.t.(x,y ) cf E G , (ii) there exists a vertex z E A2 s.f. (x,z) @ E G , (iii) there exists a uertex w € A I or A , s.t.(x, w ) E EG. Fact 3.5. I f the root of TG is a B T node with children A , , . . . , A, then each A , is rooted at a node or a B, node. 1

2

10

9

Fig. 1. G.

4

6

8

7

Dominating sets in perfect graphs

T

=

l1.2.3.9.10

I

13.4.5.8

-

O

1

15.6.7

151

I

Fig. 2. 1-CUB tree representation of G .

We now present Algorithm 3.2 which will calculate a minimum dominating set of 1-CUB G.

Algorithm 3.2. Domination (1-CUB domination algorithm). Input: T,, 1-CUB tree representing G (produced by Algorithm 3.1). Output D a minimum dominating set of G. (1) If G = { x } set D to { x } and halt. (2) If TG is rooted at a U node with children A, and A 2 , use Algorithm 3.2 to calculate D , , a minimum dominating set of A, and D2, a minimum dominating set of A2. Set D to D1U D , and halt. node with children A l and Az then look for x , a (3) If TG is rooted at a universal vertex in G, that is, a vertex which is adjacent to all other vertices of G. If such an x exists set D to { x } and halt. Otherwise choose x € A I , y € A 2 ,set D to { x , y} and halt. (4) If TG is rooted at a Exnode with children A , and A2 then look for y (Zx) a universal vertex in G in which case set D = {y} and halt. Otherwise choose y € A I and z E A 2 , set D to {y, z } and halt. (y and z are chosen so that either y or z is adjacent to x . ) (5) If TG is rooted at a BT node with children A,, . . . ,A k , where these children are in a pruning order of T (i.e. A k is the root of T) and xi is the cutnode between A iand the path in T leading to the root then perform the following algorithm. Note that each A iis a or B, node.

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D . G . Corned, L. K . Srewarf

Set D to 0 /*do a pruning order scan of A l , . . . , A k - ] * / for i : = l to k - 1: if D dominates A i\ { x i } then continue else if D U { x i } dominates A i then add xi to D and continue else if there exists z E A i \ { x i } such that D U {z} dominates A i\ { x i } then add z to D and continue else if there exists z E Ai\{x} such that D U { x i } U { z } dominates A i then add z and xi to D and continue else find y , z E A,\ { x i } such that D U {y} U {z} dominates A i , add y and z to D and continue end / * now handle the root A k * / i f D dominates G then halt else if there exists x € A ksuch that D U { x } dominates G then add x to D and halt else find x , y E A k such that D U { x } U { y } dominates G , add x and y to D and halt As an example of this algorithm, consider the graph in Fig. 1. Let the pruning order be Al = (1, 2, 3, 9, lo}, A 2 = { 5 , 6 , 7) and A 3 = {3,4, 5, S}, where x 1 = 3 and x 2 = 5. When i = 1, D = 0 initially, and x , = 3 does not dominate A l . However vertex 1 dominates A l \ (3). Thus D = (1). When i = 2, D = (1) initially, which does not dominate A Z . The addition of x 2 = ( 5 ) to D does dominate A 2 , so D now is { 1,5}. When i = 3, since { 1,5} does not dominate G , another node (for example 4) is added, yielding a minimum dominating set {1,4,5). We now sketch a proof that Algorithm 3.2 does in fact find a minimum dominating set for a 1-CUB in polynomial time.

Lemma 3.6. Algorithm 3.2 rum in polynomial time. Proof. Since the operations of complementation, connected components and search for cut points all may be performed in polynomial time, a 1-CUB tree TG can be constructed in polynomial time. Steps 1, 3 and 4 of Algorithm 3.2 clearly may be performed in polynomial time. In step 5 for each i at worst all pairs of vertices have to be checked to see if they dominate A i or Ai\{xi}. A simple induction argument establishes the polynomial time requirement of step 2. 0 Theorem 3.7. The D calculated by Algorithm 3.2 is a minimum cardinality dominating set of the 1-CUB G .

Proof. The theorem is trivially true for the cases where the root of TG is not a BT node. Thus we assume that the root is a BT node with children A l , . . . ,A k with

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this ordering being a ‘pruning’ order of T which is rooted at Ak. From Fact 3.4 we know that each A; (1 C i =sk ) is rooted at a or a B, node. We let GjC denote the subgraph of G induced by the subtree of T rooted at Ai.The bonding node joining Gf to the rest of G is xi (by convention X k = 0). Note that xi may equal xi for i # j . Finally we let Dj denote the value of D after A i has been processed. Thus Didlis the value of D on input to the processing of A; (by convention Do = 0 denotes the value of D on input to the processing of Al). DT is defined to be Din GjC. In order to prove Theorem 3.7 we use the following lemma:

Lemma 3.8. 0: is a minimum dominating set of GjC \ {xi} (1 s i s k ) where DjC contains xi if possible. Proof. (By induction). Assume Ai is a leaf of T (note Gf = A i ) . If i = 1 (i.e. Do = 0) then clearly D1=DT is a minimum dominating set of Al\{xl} which will contain x1 if possible. If i > 1 (and A; is a leaf of T) DiW1 n GjC is either 0 or {xi}. In both cases it is clear that 0’ is a minimum dominating set of GjC\ {xi}and will include xi if possible. For i > 1 we now assume that for all j G i - 1, 0 7 is a minimum dominating set of Gf\{xi} which contains xi if possible. Since 0 7 contains xi if possible, Di-l dominates as much of A ias possible without losing minimality. Note that the only nodes of A; which may be dominated by DiP1 are .xi’s (1 S j S i - 1) which are in A; or vertices adjacent to such an xi. Since A; is a or B, node, clearly the algorithm will add the minimum number of nodes to Di-l in order to complete the domination of Ai\ { x i } and will include xi if possible. 0 We now return to the proof of Theorem 3.7. 0: = Dk(the final value of D ) is a minimum dominating set of G i \ { x , } = G as required. 0 We note that Algorithm 3.2 can be modified to produce a minimum cardinality connected or total dominating set for 1-CUBs. The dominating cycle and dominating clique problems can also be solved with a similar algorithm. Some of the key observations are the following, where x is a B T node. Any connected dominating set for G, must contain all of the cutpoints. In calculating a total dominating set, we must be careful not to leave any isolated vertices in the set which cannot subsequently have a neighbour added to the set. Any dominating cycle must be entirely contained within a single 2-connected component, and any dominating clique must be contained within a block. We now turn to the problem of total domination on permutation graphs.

3.2.2. Total domination of permutation graphs In this section we develop a different type of dynamic programming algorithm which finds a minimum cardinality total dominating set in a permutation graph. The preceding domination algorithm relies on a decomposition of a 1-CUB,

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D . G. Corned, L. K . Stewart

whereas the following algorithm relies on a representation for the entire permutation graph and a theorem describing the structure of the particular sets that we are looking for. The algorithm of this section is similar to the other known domination algorithms for permutation graphs [lo, 12, 16, 18, 261. After this paper was written, it was brought to our attention that a different, independently discovered, algorithm for total domination of permutation graphs appears in [lo, 111. A permutation graph is a graph G ( V , E) for which there exists a labelling ( Y ) , v 2 , . . . , Y , } of V and a permutation n of (1,2, . . . , n} such that i appears before j in exactly one of (1, 2, . . . , n} and n if and only if (vi,v j ) E E [39]. A widely used representation for a permutation graph is the permutation diagram [23]. The permutation diagram for a graph G with n vertices is formed by writing in a column the integers (1, 2, . . . n} in order and, to the right, another column containing the integers {1,2, . . . , n} in the order in which they appear in n. n refers to a permutation which gives rise to this permutation graph, as described in the definition. We then add lines joining i in the left column with i in the right column for all 1 S i 6 n. We are left with a set of n line segments, each of which corresponds to a vertex of the graph, and two line segments cross if and only if the corresponding vertices are adjacent in the graph. There may be an exponential number of permutation diagrams for a particular permutation graph, but for our purpose, any one will suffice. Given a graph, the O(n2)algorithm of Spinrad [41] will determine whether o r not it is a permutation graph and, if so, will produce a permutation diagram for the graph. The following result tells us that any permutation graph which has a total dominating set must have a minimum cardinality total dominating set with a very specific structure.

Theorem 3.9. Let G be a permutation graph for which there exists a total dominating set. Then there exists a minimum cardinality total dominating set (mctds) of G which consists of the union of disjoint non-trivial paths (simple paths, each of size two or more).

Proof. Let T be any total dominating set for a permutation graph G. Let G ( T ) be the subgraph of G induced by the vertices of T and let H be any connected component of G ( T). H must have two or more vertices since T is a total dominating set. Claim 1. H contains no vertex of degree > 2. Proof of Claim 1. Suppose H had a vertex Y of degree b 3. Let us examine Y together with any three of its neighbours in H. All possible permutation diagrams (up to symmetry) for Y and these three adjacent vertices are shown in Fig. 3. In each case, we can identify at least one line which corresponds to a vertex which is redundant in T. All such lines are labelled r in the figure. The existence of these redundant lines contradicts the minimality of T. Thus Claim 1 is proved.

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Fig. 3.

From Claim 1, we conclude that H must consist of a simple cycle o r a simple path. Since G is a permutation graph, we know that any induced cycle has 3 or 4 vertices. By examination of the permutation diagram, it can be seen that H will not be a C, because this contains a vertex which would be redundant in T. Thus H is a simple path or H is isomorphic to C4. Claim 2. Any connected component of G ( T ) which is a C4 can be replaced in T by a P4, resulting in a total dominating set T' with the same cardinality as T. Proof of Claim 2. Let H be a connected component of G ( T ) which is isomorphic to C4. Let Y be any vertex of H and let w be a vertex in V - T which is adjacent to Y but to no other vertex of T. Such a vertex must exist since otherwise Y would be redundant in T, contradicting the minimality of T. The permutation diagram for H U {w} must be symmetric to that of Fig. 4. But from the diagram, we can see that H' = H - {x} U { w } dominates all vertices that H dominates, and that H' = P4. Furthermore, IT'J= ITI, where T' = T - {x} U {w}. Thus Claim 2 is proved. From the proof of Claim 2, we can also conclude that H' is guaranteed to be a connected component of T' = T - {x} U {w} since w is not adjacent to any vertex of T except Y. Let TI' be the set of lines which results from T by replacing each C4 of G ( T) by a P4, as described. Now T" is a mctds for G which consists of the union of disjoint non-trivial paths. 0

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D.G. Corned, L . K . Stewart

I

I

Fig. 4.

From the theorem we see that for a permutation graph G, a dominating set consisting of a collection of disjoint nontrivial paths, with minimum cardinality over all such sets, is a mctds for G. Thus, our algorithm need only construct such a minimum cardinality collection of paths. We do this by identifying a set of lines in the permutation diagram which corresponds to such a dominating set of disjoint paths in the graph. Some of the notation concerning the permutation diagram follows. A dominating set of lines in a permutation diagram is a set of lines L such that every line not in L crosses at least one line of L. There is a one-to-one correspondence between dominating sets of vertices in a permutation graph G and dominating sets of lines in a permutation diagram representing G. For a line x in a diagram D, left(x) refers to the position of x on the left side of D and right(x) is the position of x on the right side, where the top position on both sides is 1 and the bottom position is n, the total number of lines in D. The first line of a set of lines Y in D is the line x E Y such that left(x) G left(y) for all lines y E Y .

Lemma 3.10. For a permutation diagram D , let DS be a dominating set of lines in D . Let e be the first line of DS and let 1 be the line of D with right(1) = 1. Then left(e) s left(1). Proof. Any line x in D crosses I if and only if left(x) < left(1). This is because all lines have right endpoints greater than right(1). Since I must be dominated, there must be some line in DS with left endpoint sleft(1); hence, we must have left(e) s left(1). 0 In light of this lemma and the previous theorem, we might use the following approach to calculate a mctds in a given permutation diagram. We begin by finding all potential first lines for a mctds. These are the lines with left endpoints sleft(l), where I is the line with right(l) = 1. Then for each line e with

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left(e) s left(l), we calculate a mctds with e as the first line, and finally choose the minimum cardinality set obtained over all choices of e. For a particular choice of first line e, we proceed as follows. Let x be the line of minimum right endpoint such that left@) < left(e). If right@) < right(e) then x is the line highest on the right side of the diagram which is not dominated by e. Let f be the line of maximum left endpoint such that right(f)< min[right(x), right(e)]. If no such line exist or if e and f do not cross then e cannot be the first line of a total dominating set for D.Otherwise, the lines e andfcross.

Lemma 3.11. If there exists a minumum cardinality collection of disjoint nontrivial paths which dominates D and which has e as the first line, then there exis& a minimum cardinality collection of disjoint non-trivial paths which dominates D, has e as the first line, and contains f. Proof. Let T be a minimum cardinality collection of disjoint nontrivial paths which dominates D. Suppose T has e as the first line and suppose that f 4 T. Now T must contain some line y with right(y) < min[right(x), right(e)] and left(e) < left(y) < left(f ). The set { e, f } clearly dominates all lines which are dominated by {e, y} and possibly more. Thus, T' = T - {y} U {f} is a mctds in D. All components of T' are the same as those of T, except for the component C' of T' which contains f. Let C be the component of T which contains y. From Claim 1of Theorem 3.9, we know that each line of T (T') crosses at most two other lines of T (T'). If y crosses two lines in T then f crosses the same two lines in T', since f crosses at least as many lines as y, but cannot cross more than two lines of T'. Thus C' has the same configuration as C, that is, a path. If y crosses only e in T then either C = {e, y} or C is a path of length greater than 2, {y, e, . . .}. Suppose C = {e, y}, and let w be the first line of T - C. If w crosses two lines of T, then f cannot cross w , because then w would have three neighbours in T'. In this case, C' has the same configuration as C, namely, a path. If w crosses only one line of T, then f may or may not cross w. Iff crosses w then C' consists of the first two components of T combined into a path of length 2 4 . Iff does not cross w then C ' = {e, f } and T I - C ' = T - C. If C is a path of length 3, { y, e, Y } , then f cannot cross v as this would render v redundant. Thus C' is the same as C, a path. If C is a path of length 4, {y, e, v, z } , then iff does not cross z we have a path in C'. The line f cannot cross Y since this would give f three neighbours in T'. Iff does cross z, we have a C4. But then z must dominate some line w in D - T which is not dominated by any other line of T, for otherwise, z would be redundant in T. Now, T' - { v } U {w} is a minimum cardinality collection of nontrivial paths which dominates D, has e as its first line, and contains f.

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If C is a path of length >4, { y , e, Y,z , . . .}, then f cannot cross z since this would give z three neighbours in T'

For a particular first line e, we can calculate the line f, resulting in a path of length 2, {e, f }, which we know is part of a mctds for D if any mctds exists for D having e as its first line. We wish to extend this Pz to a total dominating set for D by adding the minimum number of lines. We now consider a general incremental or recursive formula for constructing a mctds in a permutation diagram by extending a partial total dominating set to a complete one by adding the minimum number of lines. Let TD(Zm,rm,Zs,r,,type) be the minimum number of lines which must be added to TD, to form a total dominating set for D,where TD, is any subset of the lines of D with the following properties. (i) TD, consists of a collection of disjoint nontrivial paths. (ii) TD, dominates exactly those lines of D with left 6 I, or right < r,,, or both. (iii) The maximum left endpoint of a line in TD, is I, the maximum right endpoint of a line in TD, is r,, the second maximum left endpoint in TD, is Z,, the second maximum right endpoint in TD, is r,. (iv) The structure of the path in the connected component of TD, which is lowest in D is given by type as follows: ' Qi

2 type =

1

r

ifTD,=0 , if the lowest component of TD, has size 2, if the lowest component has size >2 and the line with left = I, is an endpoint of the path, if the lowest component has size >2 and the line with right = r,,, is an endpoint of the path.

We now state a formula for TD(l,, r,,,,L,,r,, type) in terms of TD(lL, rL, Z:, r:, type') where 1; 3 I,, rk 2 r, and 1L + rk >I, + r,. For a particular set of endpoints I,, r,, I, and r,, we define the lines g, h, e i , h, lex and rex, with respect to these endpoints as follows. Let ND = { x 1 left(x) >I, and right@) > r,}. Now g is the line in ND with minimum left endpoint and h is the line in N D with minimum right endpoint. For any i such that left@) S i S left(h), we let eirefer to the line having left endpoint equal to i, and let J refer to the line with maximum left endpoint such that crosses all lines x with left(g) < left(x) 6 i and right(h)sright(x)Sright(e,). The line lex is the line of maximum right endpoint under the constant that 1, < left(1ex) < I, and right(lex) > r,. Similarly, rex is the line of maximum left endpoint such that r, < right(rex) < r, and left(rex) > I,.

Lemma 3.12. The following formula holds:

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TD(

7

rm

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type) if neither g nor h exists,

1 s > rs 7

jn MIN

1+ TD(l,, right(lex), left(lex), r,, r) ifg, h and lex exist and type E (2, I}, w otherwise,

1+ TD(left, (rex), r,, I,, right(rex), I) i f g , h and rex exist and type E (2, r}, 00 otherwise,

+

MIN[2 TD(left(h), right(e,), i, right(jJ, 2)], i = left(g), . . . , left(h), where right(e,) 2 right(h), ifg and h exist and g # h, 00 otherwise.

Proof. In any case, if neither g nor h exists then all lines are dominated by TD, and thus no additional lines are required. Otherwise, the two possibilities are to extend the path of TD, whose lines have maximum endpoints in D or to start a new path. Case (i): type = @. This corresponds to TD, = 0 . We must have I, = r, = Is = r, = 0 in this case, for otherwise, these endpoints would indicate an impossible configuration. If g and h exist and g = h then the top line in D corresponds to an isolated vertex in the graph and no total dominating set exists. We indicate this by assigning the artificial value m. If g and h exist and g # h then g must be the top left line of D and h is the top right line. By Lemma 3.10 and 3.11, the formula is correct. There is no possibility of extending an existing path, here, since no paths exist in TD,. Case(ii): type = 2. Here we have two choices: we can extend the lowest path of TD, or start a new component. We will choose the minimum result of these two approaches. Examination of the permutation diagram for P2 shows that there are two different ways to extend such a path. We can add a line x with 1, < left(x) < I, and right(x)>r,, thereby extending on the left, or we can add a line y with r,I,, which is called extending on the right. Of course, we can only extend on the left (right) if a line such as x ( y ) exists. If such a line does exist then choosing x (y) to be the line which reaches furthest down on the right (left) in the diagram guarantees that as many lines as possible are dominated, and that any total dominating set which is eventually found will have the minimum possible cardinality. The fact that the line which reaches furthest down in the diagram can be in a minimum cardinality collection of dominating non-trivial paths follows from arguments similar to those of the proof of Lemma

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3.11. Thus, the correct value for TD(l,, r,, I,, r,, 2) in'this case is one plus the minimum number of lines needed to complete TD, U {x} or TD, U { y}. Whether or not g = h has no bearing on path extension. However, suppose g = h and we wish to start a new path. This means that all lines with left < left(g) and all lines with right
Input: D , a permutation diagram representing graph G, Output: TD(O,O, 0, 0, a), the size of a mctds for G. for l m = n , . . . , 0 : f or r , = n, . . . , 0: for 1, = 1, - 1, . . . , 0 : for r, = r, - 1, . . . , 0 : for type = @, 2, I, r:

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ifl,, r,, I,, r,, and type are consistent and reflect a possible TD, then Calculate g, h, lex, rex. Calculate ei, fi for i = left(g), . . . , left(h). Calculate TD(l,, r,, l,, r,, type) using the formula of Lemma 3.12.

end end end end end The correctness of Algorithm 3.3 follows from Lemma 3.12.

Theorem 3.13. A mctak for a permutation graph can be calculated in polynomial time.

Proof. Notice that the I , and r, indices move from the bottom to the top of the permutation diagram. Thus, every time that a TD value Y is used in the calculation of another TD value, we know that Y has already been calculated and can be directly accessed in constant time. All steps within the five loops can certainly be performed in polynomial time, and these steps will be executed at most O(n') times. Thus the algorithm is polynomial. 0 We have made no attempt to produce an efficient algorithm; instead, simplicity was our concern. We note, however, that this algorithm can be improved by the following technique. Instead of generating all possible (l,, r,, l,, r,, type)-tuples, and testing each one for consistency, we can actually calculate all consistent and valid tuples from the indices I , and r,. We find that the valid tuples for I , and r, are the following, where i is the line with left(i) = I , and j is the line with right(j) = r,:

(l,, r,, O , O , @) (l,, r,, left(j), right(i), 2)

if 1, = r, = 0, if left(j) < I , and right(i) < r,,

(l,, r,, k, right(i), 1 ) if left(j) < I , and right(i) < r,, where k = left(j), . . . ,I,, (l,, r,, left(j), k, r) if left(j) < I , and right(i) < r,, where k = right(i), . . . , r,. The pairs of lines corresponding to l,, r, pairs with left(j) < I , and right(i) < r, are exactly the pairs of crossing lines. Each such pair corresponds to exactly one valid l,, r, pair and to exactly one edge in the graph. And, as can be seen from above, each such pair has O(n) tuples associated with it. Thus, this improvement leads to an O(max[ne, n']) algorithm, where e = (El.

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4. Glossary

Bipartite graph: A graph such that the vertices can be partitioned into two independent sets. Chordal graph: A graph in which every cycle of length four or more has a chord. Cograph: A graph which can be constructed from single vertices using only the operations of complement and union. Comparability graph: A graph for which there exists a transitive orientation. CUB: A graph which can be constructed from single vertices using only the operations of complement, union and bonding, where bonding two graphs G, and G2 is accomplished by identifying a clique of GI with a clique of G2, where the two cliques are the same size. Directed path graph The intersection graph of directed paths in a directed tree. Interval graph: The intersection graph of intervals on a line. k-CUB: A CUB in which all bonding operations are required to identify cliques of cardinality less than or equal to k. k-tree: A graph which can be constructed from a k-clique by repeatedly adding a vertex adjacent to some k-clique. Meyniel graph A graph in which every odd cycle of length five or more has at least two chords. Parity graph: A graph with the property that, for every pair of vertices u, v, all of the minimal paths joining u and v have the same parity. Perfect graph: A graph in which every induced subgraph has the property that the maximum clique size is equal to the chromatic number. Perfectly orderable graph: A graph for which there exists a linear ordering of the vertices such that, for every induced subgraph with the same relative vertex ordering, the Grundy number equals the chromatic number. A Grundy numbering is obtained by scanning the vertices in order and assigning to each vertex the smallest positive integer which is not already assigned to one of its neighbours. The Grundy number of the graph is the largest integer so assigned. Permutation graph: A graph for which there is a labelling {v,, v2, . . . , v,,} of the vertices and a permutation n of (1, 2, . . . , n } such that ( i - j)(n-'(i)n-'(J))< 0 if and only if (vL,v,) is an edge, where n-'(i) can be read as 'the position in n where i appears'. Split graph: A graph in which the vertices can be partitioned into a clique and an independent set. Strongly chordal graph: A graph G ( V , E) for which there exists an ordering { v I , v 2 , . . . , vn} of V satisfying the following two conditions for all i, J , k , I: if i <j < k and (v,,v,), (vl,Yk) E E, then (v,, v k ) E E ; if i <j < k < I and (vl,v k ) , (v,,vI),(v,, vk) E E, then (v,, vl)E E . Strongly perfect graph: A graph for which every induced subgraph H contains an independent set of vertices which intersects every maximal clique of H.

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Undirected path graph: The intersection graph of a set of undirected paths in a tree. Weakly chordal graph: A graph with the property that neither the graph nor its complement contains an induced chordless cycle with five or more vertices. Acknowledgement The authors gratefully acknowledge financial assistance from the Natural Sciences and Engineering Research Council of Canada.

References [ l ] S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math. 23 (1989) 11-24. [2] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin Luther Univ. Halle Wittenberg Math-Natur. Reihe (1961) 114. [3] C. Berge and P. Duchet, Strongly perfect graphs, Ann. Discrete Math. 21 (1984) 57-61. [4] A.A. Bertossi, Dominating sets for split and bipartite graphs, Inform. Process. Lett. 19 (1984) 37-40. [5] A.A. Bertossi, Total domination in interval graphs, Inform. Process. Lett. 23 (1986) 131-134. [6] A.A. Bertossi and M.A. Bonuccelli, Hamiltonian circuits in interval graph generalizations, Inform. Process. Lett. 23 (1986) 195-200. [7] T. Beyer, A. Proskurowski, S. Hedetniemi and S. Mitchell, Independent domination in trees, in: Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and computing (Utilitas Mathematica, Winnipeg, 1977) 321-328. [8] K.S. Booth, Dominating sets in chordal graphs, Research Report CS-80-34, University of Waterloo, 1980. [9] K.S. Booth and J.H. Johnson, Dominating sets in chordal graphs, SIAM J . Comput. 11 (1982) 191-199. [lo] A. Brandstadt and D. Kratsch, On the restriction of some NP-complete problems to permutation graphs, Technical Report N/84/80, Sektion Mathematik der Friedrich-Schiller-Universitat Jena, 1984. [ l l ] A. Brandstadt and D. Kratsch, On the restriction of some NP-complete problems to permutation graphs, in: Proceedings of FCT'85, Lecture Notes in Computer Science 199 (Springer, Berlin, 1985) (extended abstract of [lo]) 53-62. [12] A. Brandstadt and D. Kratsch, On domination problems for permutation and other graphs, Theoret. Comput. Sci. 54 (1987) 181-198. [13] M. Burlet and J.P. Uhry, Panty graphs, Ann. Discrete Math. 21 (1984) 253-277. [14] V. Chvatal, Perfectly ordered graphs, Ann. Discrete Math. 21 (1984) 63-65. [15] E.J. Cockayne, S. Goodman and S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975) 41-44. [16] C.J. Colbourn, J.M. Keil and L.K. Stewart, Finding minimum dominating cycles in permutation graphs, Oper. Res. Lett. 4 (1985) 13-17. [17] C.J. Colbourn and L.K. Stewart, Dominating cycIes in series-parallel graphs, Ars Combin. 19A (1985) 107-112. [18] C.J. Colbourn and L.K. Stewart, Permutation graphs; connected domination and Steiner trees, Research Report CCS-85-02, Faculty of Mathematics, University of Waterloo, February 1985, to appear in Ann. Discrete Math. [ 191 D.G. Corneil and D.G. Kirkpatrick, Families of recursively defined perfect graphs, Proceedings of the Fourteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (1983) 237-246.

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[20] D.G. Corneil and Y. Perl, Clustering and domination in perfect graphs, Discrete Appl. Math. 9 (1984) 27-39. [21] D.G. Corneil and J.M. Keil, A dynamic programming approach to the dominating set problem on k-trees, SIAM J. Algebraic Discrete Methods 8 (1987) 535-543. [22] A.K. Dewdney, Fast Turing reductions between problems in NP 4, Report 71, University of Western Ontario, 1981. [23] S. Even, A. Pnueli, and A. Lempel, Permutation graphs and transitive graphs, J . ACM 19 (1972) 400-4 10. [24] M. Farber, Independent domination in chordal graphs. Oper. Res. Lett. 1 (1982) 134-138. [25] M. Farber, Domination, independent domination, and duality in strongly chordal graphs, Discrete Appl. Math. 7 (1984) 115-130. [26] M. Farber and J.M. Keil, Domination in permutation graphs, J. Algorithms 6 (1985) 309-321. [27] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [28] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [29] R.B. Hayward, Weakly triangulated graphs, J. Combin. Theory Ser. B 39 (1985) 200-209. [30] D.S. Johnson, The NP-completeness column: an ongoing guide, J. Algorithms 6 (1985) 434-451. [31] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher, Eds, Complexity of Computer Computations (Plenum Press, New York, 1972) 85-103. [32] J.M. Keil, Total domination in interval graphs, Inform. Process. Lett. 22 (1986) 171-174. [33] J.M. Keil, private communication, 1986. [34] D. Kratsch, private communication, 1987. [35] R. Laskar and J. Pfaff, Domination and irredundance in split graphs, Technical Report 430, Clemson University, 1983. [36] R. Laskar, J. Pfaff, S.M. Hedetniemi and S.T. Hedetniemi, On the algorithm complexity of total domination, SIAM J. Algebraic and Discrete Methods 5 (1984) 420-425. [37] H. Meyniel, The graphs whose odd cycles have at least two chords, Ann. Discrete Math. 21 (1984) 115-119. [38] J. Pfaff, R. Laskar and S.T. Hedetniemi, NP-completeness of total and connected domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, 1983. [39] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canad. J . Math. 23 (1971) 160-175. [40] G. Ramalingam and C.P. Rangan, A unified approach to domination problems on iterval graphs, Inform. Process. Lett. 27 (1988) 271-274. [41] J. Spinrad, Transitive orientation in O(n’) time, in: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (1983) 457-466. (421 K. White, M. Farber and W.R. Pulleyblank, Steiner trees, connected domination, and strongly chordal graphs, Networks 15 (1985) 109-124.

Discrete Mathematics 86 (1990) 165-177 North-Holland

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UNIT DISK GRAPHS Brent N. CLARK and Charles J. COLBOURN Department of Computer Science, Universig of Waterloo, Waterloo, Ontario, N2L 3Gl Canada

David S. JOHNSON AT& T Bell Laboratories, Murray Hill, NJ, USA Received 2 December 1988 Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational geometry. We show that many standard graph theoretic problems remain NP-complete on unit disk graphs, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs. In contrast, we give a polynomial time algorithm for finding cliques when the geometric representation (circles in the plane) is provided.

1. Preliminaries Consider a set of n equal-sized circles in the plane. The intersection graph of these circles is an n-vertex graph; each vertex corresponds to a circle, and an edge appears between two vertices when the corresponding circles intersect (tangent circles are assumed to intersect). Such intersection graphs are called unit disk graphs, and the set of n circles is an intersection model. Intersection graphs have been widely studied (see, for example, [6]); for many classes, efficient algorithms for standard graph problems have been devised. Many of the intersection families previously studied form sublclasses of the class of perfect graphs, and many of the efficient algorithms arise because the problems are efficiently solvable for arbitrary perfect graphs. One of our primary motivations in studying unit disk graphs is that they need not be perfect; in particular, any odd cycle of length five or greater is a unit disk graph but is not perfect. Similarly, although unit disk graphs have a representation as points in the plane, they need not be planar; in particular, any complete graph is a unit disk graph. A second motivation for studying unit disk graphs is that they arise in a variety of settings. Another graph-theoretic definition is the following. For n equal-sized circles in the plane, form a graph with n vertices corresponding to the n circles, and an edge between two vertices if one of the corresponding circles contains the other’s center. This is a containment model of unit disk graphs. A purely geometric definition is also available. For n points in the plane, form a graph with n vertices corresponding to the n points, and an edge between two vertices if and 0012-365X/!30/$03.50

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only if the Euclidean distance between the two corresponding points is at most some specified bound d . This gives a proximity model of unit disk graphs. Transforming between intersection and containment models is simply a matter of doubling or halving the diameter. Transforming between the intersection and proximity models involves only an identification of the circle centers with the points in the plane and the circle diameter with d . Hence given any of the three models, we can produce the other two in linear time. The complexity of recognizing unit disk graphs is open, however, as is the complexity of building one of these models, although we strongly suspect that both problems are NP-hard. Most potential applications of unit disk graphs arise in broadcast networks, where the model is implicit. If we imagine that each point is a transmitter/receiver station, one can view the effective broadcast range of the transmitter as a circle. Further, if each station has the same power, the circles will be approximately equal in size. This model of broadcast networks is somewhat naive, because it assumes that no interference from weather, physical obstacles, and so on occurs. Nevertheless, the model is employed in the solution of important problems on broadcast networks [7,12,20]; the advent of cellular telephone systems has made analysis of problems via this model valuable. Two examples of note are frequency assignment [7] and emergency senders [181. In the frequency assignment problem, one is to assign different frequencies to transmitters whose ranges intersect. Using the intersection model of unit disk graphs, we see that frequency assignment (in its simplest form) is coloring. In the emergency senders problem, one is to find a minimum set of transmitters which can (in an emergency) transmit to all remaining stations. Using the containment model of unit disk graphs, this is domination. Finally, a clustering problem of interest is to find a maximum subset of points so that no two are at distance exceeding d ; using the proximity model, this is a maximum clique in the unit disk graph. With these applications in mind, we study the effect of the restriction to unit disk graphs on the complexity of the following problems, known to be NP-complete for general graphs: coloring, clique, independent set (vertex cover), domination, independent domination, and connected domination. We also consider the effect of the further restriction to ‘grid graphs’, where a grid graph is a unit disk graph in whose intersection model all the disks have centers with integer coordinates and radius 1/2. Table 1 summarizes what is now known about these restrictions, including for completeness two additional standard problems whose complexity has previously been resolved for unit disk and grid graphs. The new results of this paper are those marked by asterisks. For the case of clique, we present a polynomial time algorithm that will find a maximum-sized clique in a unit disk graph, given an intersection (containment, proximity) model for the graph. For the sake of completeness, we also briefly sketch proofs of the results attributed to [7,9,10] in Table 1, as these references merely reported the results, and to our knowledge no proofs have appeared. Our proof of the result

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Grid graphs

CHROMATIC NUMBER CLIQUE INDEPENDENT SET DOMINATING SET CONNECTED DOM. SET HAMILTONIAN CIRCUIT STEINER TREE

NP-complete [7,9] Polynomial [ * ] NP-complete [ * ] NP-complete [16] NP-complete [IS] NP-complete [S] NP-complete [4]

Polynomial Polynomial Polynomial NP-complete NP-complete NP-complete NP-complete

[ 10) [*] [8]

[4]

from [lo] also implies NP-completeness for the previously open independent dominating set problem for grid graphs. The remainder of the paper is divided into sections, one per problem. We shall use the prefix ‘UD’ to specify that the problem in question is restricted to unit disk graphs and the prefix ‘GRID’ for restrictions to grid graphs. When a problem name is given in all capital letters, as in ‘UD DOMINATING SET,’ we refer to the decision problem version of the problem (i.e., given G and an integer k , is there a dominating set of size k or less?), rather than the optimization version (i.e., given G, find a minimum size dominating set). We must consider decision problems since NP-completeness is formally defined only in terms of such problems. In proving NP-completeness we will omit the required proof of membership in NP, since this always follows from the membership of the unrestricted problem. (We assume the reader is familiar with these notions; if not, see [5].)

2. UD CHROMATIC NUMBER A graph G = (V, E) is k-colorable if there is a partition of the vertices into k sets V,,. . . , V, such that no edge joins two vertices in the same set. In UD CHROMATIC NUMBER we are given a graph G and an integer k and are asked whether G is k-colorable. This problem is of interest primarily because of its relevance to frequency assignment problems in broadcast networks [7]; with the proximity model, it is also equivalent to a geometric problem, DISTANCE-d PARTITION OF POINTS IN THE PLANE [9]. We show that UD CHROMATIC NUMBER is NP-complete, even if k is fixed at 3. Hale [7] references a proof of this due to Orlin, and Johnson [9] references a proof due to himself and Burr. Since neither proof to our knowledge has appeared in the literature, we include the latter proof for completeness. Note that CHROMATIC NUMBER is trivial for grid graphs, since all grid graphs are bipartite and hence two-colorable by definition.

Theorem 2.1. UD CHROMATIC NUMBER is NP-complete, even for fixed k = 3.

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Proof. We sketch a polynomial time transformation to the given problem from PLANAR GRAPH 3-COLORABILITY with maximum degree 3, which is shown to be NP-complete in [3]. We transform a planar graph G = (V, E) with maximum degree 3 into a unit disk graph G’ such that G is 3-colorable if and only if G’ is 3-colorable. We construct an intersection model for G’ by making use of the following result from [19]:

Lemma 2.1 (Valiant [19]). A planar graph G with maximum degree 4 can be embedded in the plane using O ( l V ( )area in such a way that its vertices are at integer coordinates and its edges are drawn so that they are made up of line segments of the form x = i or y = j , for integers i and j . Algorithms to produce such embeddings efficiently are given for example in [l, 81. Using one of them we construct such an embedding of G, adjusting the scale so that the horizontal and vertical straight line segments that make up edges are each of length at least 10. The vertices of G are modeled by circles of radius 1/2 centered at the locations of the vertices in this embedding. The edges of G are replaced by chains of radius-1/2 circles, specified as follows. If e, is the edge between vertices u and v , then the set of circles used to represent it is c [ e , ]= {d1,cY1,cll,clZ,c E zC,Z, , . . . , c,’kC,cL,, c,~,}, where k, depends on the length of the embedding of e,. These circles are positioned so that they yield an intersection pattern like that shown in Fig. 1 for an edge made of a single horizontal line segment. The reader may verify that this representation for a horizontal edge can be modified to bend around corners if the edge to be represented consists of both horizontal and vertical segments (given that each segment by construction is of length at least lo), and that the following two properties hold: (1) Any proper 3-coloring of C [ e , ]U {u, v} assigns u and v different colors. (2) For all possible pairs ( x , y ) of different colors from the set { 1,2,3}, there exists a proper 3-coloring of C[e,]U {u,v} which assigns u and 21 colors x and y respectively. It is an easy matter using these properties to see that G is 3-colorable if and only if G’ is 0 1

I!

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A clique in a graph G = (V, E) is a subset of the vertices, each pair of which is joined by an edge. Note that the problem of finding a maximum clique in a grid graph is trivial; a grid graph can have no cliques of size greater than 2. The problem is less trivial for other classes of intersection graphs, but most still have polynomial time algorithms. In many cases, this is a consequence of the efficient clique algorithm for perfect graphs [6]. In other cases, it is a consequence of the ‘Helly property’. This holds for many classes of intersection graphs, and requires that when n objects intersect pairwise, their n-fold intersection is a non-empty set. If an intersection graph obeys the Helly property and the number of potential ‘intersection regions’ is sufficiently small, we can thus find maximum cliques for it ‘quickly’. We simply compute for each region the set of objects containing it, and output the largest such set found. Unfortunately, sets of equal-sized disks in the plane do not necessarily satisfy the Helly property. In Fig. 2 there are two sets of three disks. In the first set three disks intersect at a common region. In the second, three disks intersect pairwise, but not at a common region. Nevertheless, despite the fact that unit disk graphs need neither obey the Helly property nor be perfect, the clique problem remains tractable for them. In the remainder of this section we show how to find a maximum clique in a unit disk graph G in polynomial time, given a proximity model for G. The algorithm we present may well not be the most efficient possible. However, we are here interested only in demonstrating polynomial time solvability, and will leave running time improvements to future researchers. Our approach is based on the following straightforward observations. Let A and B be a pair of points in the model for G whose distance D ( A , B ) < d , where d is the critical distance specified by the model. Let RA, denote the intersection of two closed disks of radius D ( A , B), one centered at A and one centered at B. See Fig. 3. Let HAB = R,, f l V .

Fig. 2

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Fig. 3

ObStm"ti0n 3.1. If A and B are maximally distant points in a set V ' , then V ' HAB. Corollary 3.1. If C is the vertex set for a maximum-sized clique in G , then C c HAB for some A , B E V , with D ( A , B ) =z d . Now partition RAB into RfiB and R$B as shown in Fig. 4 with the line segment from A to B assigned to region RfiB. Let H i , consist of the points in R i B n V and H:B consist of the points in R i s f l V . See Fig. 4.

Fig. 4

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ObSeNatiOn 3.2. If X and Y are points in H i B ( H i B ) , then D ( X , Y )S d . Corollary 3.2. The subgraph of G induced by graph.

HAB

is the complement of a bipartite

Observation 3.3. A maximum independent set in a bipartite graph on n vertices can be found in time Proof. This is perhaps not as straightforward as the previous observations, but is a well known application of bipartite matching. Recall that an independent set in a graph G = (V, E) is a subset V‘ E V such that no edge in E joins two members of V’, and a matching is a set of edges, no two of which share an endpoint. Suppose M is a maximum matching in a bipartite graph G. Given M, we can build a maximum independent Z set as follows. We start by including the set Z, of vertices not contained in the matching. Note that these vertices must be independent or else we could find a larger matching. Then for each edge in M, we take an endpoint that is not adjacent to any vertex chosen so far. (It can be shown that at least one endpoint must satisfy this property, as a failure would imply the existence of an augmenting path, and thus contradict the maximality of M.) In this way we construct an independent set of size n - [MI (in linear time, given M). This is the largest sue possible since at most one of the endpoints of each of the edges in M can be in an independent set. Since a maximum matching in a bipartite graph can be found in time O ( n 9 by the techniques of [2], the observation follows.

Corollary 3.3. Given the proximity model of a unit disk graph G = (V, E ) , one can find a maximum clique for G in time O(lV1“’). Proof. By Corollary 3.1, we need only consider the subgraphs induced by H A B for pairs of vertices A, B E V with D ( A , B) =sd. There are O(lV1’) such pairs. A maximum clique in such a subgraph is a maximum independent set in its complementary graph, which by Corollary 3.2 is bipartite. Thus to find a maximum clique in the subgraph induced by HAB, we need only construct the appropriate bipartite graph (in time that is certainly O(lVl’)) and then apply the O(n2.5) algorithm of Observation 3.3, for an overall bound of O((V/(2.5). Multiplying by the potential number of A, .B pairs gives the claimed running time.

4. UD VERTEX COVER and UD INDEPENDENT SET We have already defined ‘independent set’ in the proof of Observation 3.2 above. A vertex cover in a graph G = (V, E) is a subset V’ of the vertices such

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that for all edges e E E, at least one endpoint of e is in V'. It is an easy observation that if V' is a vertex cover for G, then V - V ' is an independent set. Thus for any class of graphs, the problem of finding a minimum vertex cover is polynomial equivalent to that of finding a maximum independent set. By Observation 3.2, both are solvable in polynomial time for grid graphs. Both are NP-hard for unit disk graphs as a consequence of the following theorem.

Theorem 4.1. UD VERTEX COVER is NP-complete.

Proof. The reduction is from PLANAR VERTEX COVER with maximum degree 3, which was shown NP-complete in [4]. As before, we transform the planar graph G with maximum degree 3 to a unit disk graph G ' such that G has a vertex cover S with IS1 S k if and only if G' has a vertex cover S' with IS'l S k'. We draw G in the plane using Lemma 2.1. We then replace each edge {u,v } by a path having an even number 2k,, of intermediate vertices, in such a way that an intersection model can be constructed. (This is clearly easy to do. Note, however, that a grid graph embedding will not be possible unless G is bipartite, which is why this construction does not work for grid graphs.) It is straightforward to verify that G has a vertex cover S such that IS1 G k if and only if G' has a vertex cover S' such that IS'[6 k + CuusE(G) kuu. 0

5. UD DOMINATING SET A dominating set in a graph G = (V, E) is a subset V' of vertices such that every vertex v E V is either in V' or adjacent to some member of V'. U D DOMINATING SET models a network problem of locating emergency senders [15], locating emergency services [16, 181, and a geometric problem sometimes called OPTIMAL BOMB TARGETING [9]. The NP-completeness of U D DOMINATING SET was proved in [16] and an unpublished proof of NPcompleteness for GRID DOMINATING SET was attributed to Leighton in [lo]. We here sketch our version of the latter proof, using a transformation that also yields NP-completeness for G R I D INDEPENDENT DOMINATING SET (An independent dominating set is a dominating set that is also an independent set).

Theorem 5.1. GRID DOMINATING SET is NP-complete. Proof. We sketch a transformation from PLANAR DOMINATING SET of maximum degree 3, which is known to be NP-complete [S]. Given a planar graph G with maximum degree 3, we construct a unit disk graph G ' such that G has a dominating set D with (DI s k if and only if G ' has a dominating set D' with [Dl4 k'.

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Using Lemma 2.1 we can embed G in the plane with line segrfients parallel to the x - or y-axis. It is an easy matter to ensure that no two parallel lines are closer than two units apart, that each line segment has integer length, and that the total line length for the line representing an edge {u, v} is of the form 3k, 1 for some integer k,. A grid graph G‘ is induced by this drawing, containing exactly those integer points lying on a line in the drawing. It is an easy exercise to verify that there exists a dominating set D in G with ( D ( s k if and only if there exists a dominating set D ’ in G’, with JD’J=s k + CuveE(G&. 0

+

A dominating set for the grid graphs constructed in the above proof is also an independent dominating set, as long as k , > 1 for each edge. Thus GRID INDEPENDENT DOMINATING SET is also NP-complete.

6. UD CONNECTED DOMINATING SET A dominating set is connected if the subgraph induced by it is connected. U D CONNECTED DOMINATING SET has been studied by Lichtenstein [15], who states that the problem has been posed by Spira in connection with packet radio network design. It is equivalent to the problem of locating a connected set of emergency senders in a network. Lichtenstein proved the problem to be NP-complete for unit disk graphs; we show that it remains NP-complete even when restricted to grid graphs.

Theorem 6.1. GRID CONNECTED DOMINATING SET is NP-complete. Proof. The reduction is from PLANAR CONNECTED VERTEX COVER with maximum degree 4, which was shown NP-complete in [4].We transform a planar graph G = (V, E) with maximum degree 4 into grid graph G’ such that G has a connected vertex cover S with I S I S k if and only if G’ has a connected dominating set D‘ with ID’I s k’.We assume without loss of generality that G is connected. Using Lemma 2.1, we first embed G in a 2-dimensional grid with edges drawn using line segments of length at least four, and with parallel lines at least 4 grid squares apart. The set V’ of vertices of our grid graph G’ will be made up of three sets: V,, the set of grid points corresponding to vertices in G, V,, the set of grid points that are internal to the paths corresponding to edges of G, and V,, a set consisting of one unique new neighbor for each member of V, that is not adjacent to a member of Vl. In what follows, we shall refer to the vertices of V, that are adjacent to vertices in V,, and hence to no vertices in V,, as connector vertices. For each vertex u E V, we shall denote the vertex corresponding to u by f ( u ) . For each edge {u, v} in G we shall denote the set of non-connector vertices

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from V2 in the path corresponding to edge { u , v } by f { u , v}, the connector vertex adjacent to f ( u ) by c(u, v), and the connector vertex adjacent to f ( v ) by c(v, u). Fig. 5 shows what a subgraph of G' corresponding to an edge { u , v } of G might look like. Note that the vertices of V, are chosen so that each is adjacent to precisely one vertex of V2(although it may be adjacent to as many as two other vertices of V,). Such choices are possible because by assumption all parallel lines are at least 4 grid cells apart in the embedding of G, and all line segments are of length at least 4. The construction of G' can clearly be accomplished in polynomial time. To complete our proof, we claim that there exists a connected vertex cover C in G with ICI S k if and only if there exists a connected dominating set D in G ' with ID1 S k IV2( - (El 1VI - 1. First suppose that the desired connected dominating set C exists. Since C is 'connected,' there is a set Ec of JC1- 1 edges which forms a spanning tree for the subgraph induced by C. Since C is a dominating set, this can be extended to a spanning tree E, for all of G in which all vertices except those in C have degree 1. Our connected dominating set D for G' then consists of the following four classes of vertices: (1)f ( u ) for each u E C , (2) f { u , v} for all edges { u , v} in G, (3) both c(u, v ) and c(v, u ) for all edges { u , v } in ET, and (4) a single C E {c(u, v), c(v, u ) } for each { u , v } in E - ET, that c chosen so that it is adjacent to a vertex of type (1). (Such a vertex exists because C was a dominating set for G.) Note that (DI = ( C (+ IV21- /El + 1V1- 1c k + IV,( - \El + (VI - 1 (since JCIS k, by assumption). We argue that D is a dominating set as follows: Each member of V, is dominated by its neighboring vertex from V,. Each vertex f ( u ) for u E V - C is dominated by the neighboring connector vertex c(u, v), where { u , v } is the edge of ET that contains u. Each omitted connector vertex c(u, v ) is dominated by its neighboring vertex in f{u, v}. All other vertices are in D. The subgraph of G' induced by D is connected because all vertices in the embedding of ET are present (except for verticesf(v) where v is not in C and hence has degree 1 in ET), and because the setsf{u, v } in D are all connected by connector vertices in

+

+

d Fig. 5

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D to vertices in { f ( u ) :u E C } Thus D is a connected dominating set for G’ of the appropriate size, as claimed. Suppose conversely that there is a connected dominating set D with JDIs k IVz(- IEl+ ( V (- 1. We shall show that G has a connected vertex cover of size k or less. We proceed by a series of observations that show that D can be assumed to be of a standard form.

+

Lemma 6.1. There ex& a minimum-sized connected dominating set for G‘ that contains no vertex from V,.

Proof. Let D be a minimum-sized connected dominating set containing the minimum possible number of vertices from V,, and assume this number exceeds 0. Let {u, v} be an edge of G whose embedding in G‘ contains a vertex from V, r l D. Consider the pairs (x, y ) where x E f {u, v} and y is its neighboring vertex from V,. If D contains at least one member from each pair, then the set D’ obtained from D by omitting all second components in pairs ( x , y ) and taking all first components will be a connected dominating set with 10’1s ID1 that contains at least one less element of V,, contradicting our choice of D. Thus D omits both vertices from some pair (xi,yi). Let z be the vertex in D that dominates yi, and note that z must itself be a member of V,. Note further that, since D is connected, it must contain all the vertices on a path from z to either f ( u ) or f (v), say f ( u ) . Let P consist of all the pairs (x, y ) through which this path passes, including the pair of which z is the second component. Note that D must contain at least one member of each pair in P. Moreover, since this path starts with a vertex in v, and must pass through at least one vertex of Vz (the neighbor of the connector vertex c(u, v)), it must contain both members of some pair (x’, y ’ ) in P. Thus if D’ is the set obtained from D by omitting all second components of pairs in P and taking instead xi together with all the first components, then D‘ will be a connected dominanting set with 10’1s ID1 that contains at least one less member of V,, again a contradiction. This exhausts the possibilities and proves the lemma. 0 Lemma 6.2. Let D be a connected dominating set satisfying Lemma 6.1. Then: ( 1 ) For each edge { u, v} in G , D contains all vertices in f { u, v} together with at least one connector vertex c(u, v) or c(v, u ) and the associated endpoint ( f ( u ) or

f (v)), (2) For each v E V such that f (v) is not in D, there must exist some edge { u , v } in G such that D contains both connectors c(u, v ) and c(v, u ) and such that f ( u )E D . ( 3 ) Let C = { u E V : f ( u ) E D } , and let Ec be the set of edges in G that have both their endpoints in C and both their connector vertices in D. Then Ec spans the subgraph of G induced by C and hence has at least ICI - 1 members.

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Proof. (1) follows immediately from the fact D is a connected dominating set and contains no members of V,. (2) holds for any connected dominating set D. For (3), let f ( C ) = { f ( u ) : u E C } = V, n D . Any path in D that connects two vertices in f ( C ) cannot pass through a member of V, - f ( C ) or through the embedding of an edge that does not have both its connector vertices in D. Thus, since D connects together all the vertices in f ( C ) , E , must connect together all the vertices of C. 0 To complete the proof of Theorem 6.1, we claim that C is the desired connected dominating set for G. C is a dominating set by part (1) of Lemma 6.2; it is connected because, by part (3), its induced subgraph is connected. Finally, by all three parts of the Lemma,

P I 3 ICI + N 2 l - (El+ (IVI - ICO + (ICI - 1). By assumption, however,

ID]sz k

+ IV2l-

[El + (IVl- 1).

Consequently, ICI s k . Thus the desired vertex cover exists if and only if the desired dominating set exists, and the proof is complete. 0

7. Concluding remarks

In this paper we have extended our knowledge about the relative complexity of problems under the restriction to unit disk graphs and to grid graphs. From these complexity results, it would seem that unit disk graphs are more closely related to planar graphs in terms of complexity than to grid graphs. For all of the problems mentioned here, the complexities for unit disk graphs and for planar graphs agree. Are there any problems for which the two classes yield different complexities? Graph isomorphism is a candidate, being solvable in polynomial time for planar graphs, but currently remaining open for unit disk graphs. A perhaps more significant open problem is determining the complexity of unit disk graph recognition. As remarked above, we suspect that the problem is NP-hard. It appears that the corresponding problem for grid graphs is NP-hard [14], and an extension of the proof to unit disk graphs is currently under study. Finally, we observe that the kinds of questions considered here can and have been studied for many other classes of graphs. Recent surveys of such results can be found in [lo, 111.

Acknowledgements We would like to thank Ehab El Mallah and Marlene Jones for helpful comments on this research. The second author would like to thank the University

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of Toronto for hospitality during the time this paper was written. The research of the second author is supported by NSERC Canada under grant A0579.

References [l] B.N. Clark, Unit disk graphs, M. Math. Thesis, University of Waterloo, 1985. [2] J. Edmonds and R.M. Karp, Theoretical improvements in algorithmic efficiency for network Row problems, J. ACM 19 (1972) 248-264. [3] M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoret. Comput. Sci. 1 (1976) 237-267. [4] M.R. Garey and D.S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32 (1977) 826-834. [5] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). (71 W.K. Hale, Frequency assignment: Theory and applications, Proc IEEE 68 (1980) 1497-1514. [8] A. Itai, C.H. Papadimitriou and J.L. Szwarcfiter, Hamilton paths in grid graphs, SIAM J. Comput. 11 (1982) 676-686. [9] D.S.- Johnson, The NP-completeness column: An ongoing guide, J. Algorithms 3 (1982) 182-195. (101 D.S. Johnson, The NP-completeness column: An ongoing guide, J. Algorithms 6 (1985) 434-451. [ll] D.S. Johnson, The NP-completeness column: An ongoing guide, J. Algorithms 8 (1987) 438-448. [12] K. Kammerlander, C 900 - An advanced mobile radio telephone system with optimum frequency utilization, IEEE Trails. Selected Areas in Communication 2 (1984) 589-597. [13] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher, eds, Complexity of Computer Computations (Plenum Press, New York, 1972) 85-104. [14] J. Kilian, personal communication. [15] D. Lichtenstein, Planar formulae and their uses, Siam J. Comput. 11 (1982) 329-343. [16] S. Masuyama, T. Ibaraki and T. Hasegawa, The computational complexity of the M-center problems in the plane, Trans. IECE Japan E64 (1981) 57-64. [17] R.H. Mohring, Algorithmic aspects of comparability graphs and interval graphs, in: I. Rival, ed., Graphs and Order (Reidel, Dordrecht, 1985) 41-101. [IS] C. Toregas, R. Swain, C. Revelle and L. Bergeman, The location of emergency service facilities, Oper. Res. 19 (1971) 1363-1373. [19] L.G. Valiant, Universality considerations in VLSI circuits, IEEE Trans. Computers 30 (1981) 135-140. [20] Y. Yeh, J. Wilson and S.C. Schwartz, Outage probability in mobile telephony with directive antennas and macrodiversity, IEEE Trans. Selected Areas in Communication 2 (1984) 507-51 1.

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PERMUTATION GRAPHS: CONNECTED DOMINATION AND STEINER TREES Charles J. COLBOURN Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ont., Canada N2L 3Gl

Lorna K. STEWART Department of Computing Science, University of Alberta, Edmonton, Altn., Canada

Received 2 December 1988 Efficient algorithms are developed for finding a minimum cardinality connected dominating set and a minimum cardinality Steiner tree in permutation graphs. This contrasts with the known NP-completeness of both problems on comparability graphs in general.

1. Introduction A dominating set of vertices in a graph G = (V, E) is a set S for which every vertex of V - S has a neighbour in S. A dominating set is independent if the subgraph induced on S has no edges, total if the subgraph induced on S has no isolated vertices, and connected if the subgraph induced on S is connected. The size of a minimum cardinality dominating set in a graph is called the domination number; analogously, one defines the graph’s independent domination number, total domination number, and connected domination number. Finding minimum cardinality dominating sets of these various types has attracted much attention. In the following table, we summarize the current status of these four domination problems. Note that many of the results are implied by the inclusions: (1) {bipartite graphs} 5 {comparability graphs}, (2) {split graphs} c_ {chordal graphs}, (3) {interval graphs} c {directed path graphs} E {undirected path graphs} c_ {chordal graphs}, (4) {interval graphs} c {strongly chordal graphs} c chordal graphs}, ( 5 ) {cographs} E {permutation graphs} {comparability graphs}. Connected domination involves finding a smallest connected subgraph which dominates the remainder of the vertices. This bears some similarity to the Steiner tree problem, a central problem in network analysis and design. Given a graph G = (V, E), we identify a set T c V of target vertices. Then a Steiner tree for T in G is a set S c V - T of vertices, such that S U T induces a connected subgraph. A minimum cardinality Steiner tree is a set S of smallest cardinality; we call the

=

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Table 1. Complexity of domination problems Graphs

Domination

Bipartite Comparability Split Chord a1 Strongly chordal Interval Directed path Undirected path Series-parallel Cographs Permutation

Independent

Connected

Total

NP-c [17] NP-c [17] NP-c [15] NP-c [15] p ~ 5 1 p ~ 5 1

NP-c [ 171 NP-c [17] NP-c [15] NP-c [16]

p [251 p [51 this paper

problem of finding such a set CARDST to distinguish it from the usual edge-weighted Steiner tree problem widely studied in the networks and graph theory literature. White, Farber, and Pulleyblank [25] observe that whenever the complexities of CARDST and connected domination are currently known, they are the same. In fact, CARDST is NP-complete for bipartite and comparability graphs [12], split and chordal graphs [25], but can be solved efficiently for strongly chordal graphs [25] and series-parallel graphs [7,20,24]. In this paper, we examine the connected domination and CARDST problems on permutation graphs, and develop efficient algorithms for each. The solutions are remarkably similar; however, we develop different methods in the two cases, exploiting the structure of minimum cardinality connected dominating sets. It is perhaps important to note that the solution for connected domination given here differs considerably from the Farber-Keil approach used in other domination problems on permutation graphs. The definition of the various families referred to are standard, and can be found in many of the references; we repeat the necessary ones here. A permutation graph is a graph for which there is a labeling {v,,. . . , v,} of the vertices and a permutation x of { 1, . . . , n } for which (i - j ) ( x ( i )- n(j))< 0 if and only if (vi,vj) is an edge. All permutation graphs have a transitive orientation, and hence are comparability graphs; in fact, a graph is a permutation graph if and only if both the graph and its complement are comparability graphs [19]. A subclass of permutation graphs which has been studied extensively is the class of cographs. A graph is a cograph if and only if it has no induced subgraph which is a 4-vertex path. Many equivalent characterizations are known 14,221.

2. Connected domination In this section, we develop a simple algorithm for finding a minimum cardinality connected dominating set (MCCDS) in a connected permutation

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graph. Connected domination is NP-complete for comparability graphs. However, it is not hard to see that connected domination has a trivial solution for cographs [ 5 ] . In fact, a cograph which does not have a single dominating vertex must have a pair of adjacent vertices which forms a dominating set; otherwise, a four vertex path would be induced. This is of interest here, since {cographs} E {permutation graphs} c {comparability graphs}. The algorithm for finding a MCCDS in a permutation graph employs a geometric representation. Consider two columns, each consisting of the integers (1,. . . ,n} in order, and a permutation JC. A line connects i in the left column with n(i) in the right. A permutation graph is obtained by taking the n lines as vertices; edges exist between crossing lines. Each line e has endpoints in the left and right columns; left(e) and right(e) denote the indices of these endpoints. Then the left-span of a set L = {el, . . . , ek} of lines is a set LEFT(L) = {i 1 i 3 min(left(e) I e E L) and i max(left(e) I e E L)}. The right-span of L, RIGHT(L) is defined analogously. The pair (LEFT(L), RIGHT(L)) is the span of L. Two spans (L, R ) and (L‘, R‘) are said to intersect if one of the following holds: (1) L n L’ #to, (2) R n R ’ f 0 , or (3) max(L’) > max(L) and min(R’) < min(R). Two sets of lines L, and L2 are said to intersect if their spans intersect. A set of lines is connected if there is no nontrivial partition into two non-intersecting sets of lines. Given any permutation graph, Spinrad’s algorithm will produce this geometric representation in O(n’) time [21], an improvement on the earlier O(n3) algorithm [19]. Hence, it suffices to determine dominating sets in this geometric setting.

Lemma 2.1. A connected dominating set is a connected set of lines L such that every line not in L intersects some line in L. Proof. A connected dominating set induces a connected subgraph, and therefore the set of lines in the dominating set must be connected. Moreover, any line not in the dominating set which does not intersect a line in the dominating set is not dominated in the corresponding permutation graph. 0 The structure of MCCDS is somewhat special; we explore this in a sequence of preliminary lemmas.

Lemma 2.2. Let M be an MCCDS. Then M contains no three lines all intersecting. Proof. Let e l , e2, and e3 be the three lines, with left(e,) < left(e,) < left(e,). Since all three cross, right(el) > right(e2) > right(e3). But then any line crossing e2 must cross either e, or e3 as well. Then M - e2 is a connected dominating set of smaller cardinality than M. 0

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Lemma 2.3. Let M be an MCCDS. Then there is an MCCDS of the same cardinality in which there are no four lines which induce a four-vertex cycle.

Proof. Suppose there are four lines which induce a four-vertex cycle. Geometrically, they appear as two nonintersecting pairs el and e2, fl and fi. with each ei crossing each 6. This is illustrated here:

Suppose that M contains more than just these four lines. Then it must contain a fifth line h that intersects one of these four; without loss of generality, we may assume this is e l . Then h cannot cross f i or f2 by Lemma 2.2. Thus h, f l , and f2 form three ‘parallel’ lines all intersecting el, and the middle of the three is redundant, contradicting the minimality of M. Thus M just contains the four lines shown. In order for el to be required, some line h not in M intersects el but none of the other three. But then replacingf, by h yields a connected dominating set of the same cardinality which does not contain a 4-cycle. 0 Since induced subgraphs of permutation graphs are themselves permutation graphs, and since permutation graphs have no induced cycles of lengths five and greater [13], Lemmas 2.2 and 2.3 establish that a MCCDS induces a tree. In fact, we can establish an even stronger result:

Theorem 2.4. An MCCDS in a permutation graph induces a path. Proof. In view of Lemmas 2.2 and 2.3 we need only exclude stars on three edges. Such a star, geometrically, is a triple of three pairwise nonintersecting lines e l , e2, and e3 and a fourth line f crossing all three. Supposing that left(e,) < left(e2)< left(e3), it is immediate that e2 can be removed from the set. 0 The algorithmic importance of Theorem 2.4 is that, in order to find a MCCDS in a permutation graph, we need only find the minimum cardinality dominating induced path. An algorithm to do this for a connected permutation graph is quite straightforward, and is outlined here. A line e for which there is no line f having both left(f) < left(e) and right(f) < right(e) is termed initial. Notice that the

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MCCDS need not contain any of the initial lines, but if it does not, it must then contain, for each initial line, some line which crosses it. {locate set I of initial lines} let 1 be the line with left(1) = 1 let r be the line with right(r) = 1 I = { I , r} minr = right(1) for i = 1 to left(r) do let q be the line with left(q) = i if right(q) < minr then Z=IU {q} minr = right(q)

{Inow contains all initial lines} minsize = n {all n lines form a connected dominating set} for each line e in turn do {try e as the first ‘left’ line of the path, i.e. the line with lowest left( ) in the Path} ifeEZ then Lo = (1, . . . , left(e)} R1 = (1, . . . ,right(e)} L1= Ro = 0 i=l else let f be the initial line not crossing e for which right(f) is minimum let g be the line crossing both e and f for which left(g) is maximum; if no such line exists or le€t(g)< left(e), abandon e as a possible first line L1 = L2 = {left(e) 1, . . . , left(g)} R1 = R2= {right@) 1, . . . , right(e)} i=2 {now the first two lines are e and g , in that order} endif

+ +

{expand two possible paths in parallel; the two paths arise from a ‘left-rightleft- -’and a ‘right-left-right- . - -’alternation}

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done = false while not done if either every line 1 has left(/) 6 max(L,-l) or right(/) G max(R,) or every line 1 has left(1) 6 max(L,) or right(/) S max(R,-l) then done = true minsize = min(i, minsize) else i=i+l maxleft = max{j I some 1 has left(/) = j and right(1) E (undefined when R j - l = +) L j = {max(L,-,) 1, . . . , maxleft} (empty when LiP2= 0 or maxleft undefined) maxright = max{j I some 1 has right(/) = j and left(/) E Lip,} (undefined when L j P 1= 0) R, = {max(R,-,) + 1, . . . , maxright} endif endwhile endfor

+

{result is minsize) Minsize gives the connected domination number; this' follows directly from Theorem 2.4 and the observation that minsize is the length of a shortest dominating path. It is a simple matter to retain the lines themselves and produce the MCCDS. This algorithm demonstrates that

Theorem 2.5. A MCCDS in an n-vertex permutation graph can be found in O(n2)time. Proof. The geometric representation can be produced in O(n2) time [21]. Once done, initial lines can be classified in O(n) time. Each of the n lines is selected as the first line; we must show that O ( n ) time is spent per selection. When the first line is initial, the sets are constructed in 0(1)time by retaining only the smallest and largest members of each set. The update operation requires time linear in the size of the set; O ( n ) operations are required in total, since each line is considered in at most four sets (two left, two right). When the first line is not initial, the only important note is that there is a unique second line which can be located in O(n) time. 0

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3. Steiner trees We again employ the geometric representation of a permutation graph in solving the CARDST problem. Initially, we have a classification of lines into two types: target lines and nonturger lines. The CARDST problem can be formulated as requiring the selection of the minimum number of nontarget lines, which when included with all of the target lines induces a connected set. We can recast this problem as follows. We are to determine the minimum number of nontarget lines required to connect all target lines intersecting the span ((1, . . . ,n}, (1, . . . , n}). To do this, we determine, for all i, j E (1, 2, . . . , n}, the minimum number of nontarget lines required to connect all target lines intersecting the span ((1, . . . , i}{l, . . . ,j}), the i, j-span. Whenever there is a target line not intersecting the i,j-span (but some target line intersects the i,j-span), we further insist that the nontarget lines chosen, together with the target lines intersecting the i,j-span, intersect all lines that have exactly one end in the i,j-span. This minimum number is then denoted @ ( z , j ) . We observe first that @ ( l , 1) = 0; in fact, if there is an initial target line from i to j , @(i,j ) = 0. We develop some simple constraints on @ ( i , j ) . To do this, we require some auxiliary definitions. Let GT(i, j ) be the minimum of @(p, q ) over all selections of k, I , p , q so that there is a target line from k to 1, and either k d p = i and j = I > q or i = k > p and j = q 3 I . Let &(i, j) be one plus the minimum of @(p, q ) over all selections of k, 1, p, q so that there is a nontarget line from k to I, and either k s p = i and j = l > q or i = k > p and j = q ” l . Let & ( i , j ) be the minimum of @(k, l ) , taken over all k, I-spans properly containing the i, j-span. Finally, let GF(i, j ) be ~0 if for every p,q-span properly contained in the i,j-span, there is a target line not intersecting the p,q-span; otherwise, @ F ( i ,j ) is the minimum of #(p, q), taken over all p,q-spans properly contained in the i,j-span, which in addition intersect every target line. The following lemma is straightforward (but necessary): Lemma 3.1. For every i,j-span,

Proof. If no target lines intersect the i,j-span, the statement is vacuous. So consider an i,j-span which at least one target line intersects. Every set of lines intersecting a span properly containing the i,j-span certainly intersects all target lines intersecting the i,j-span and all lines with one exactly one endpoint in the i,j-span; hence @(i,j ) d &(i, j). Furthermore, once all target lines intersect the p,q-span, the same holds for all spans properly containing it; hence @ ( i , j ) s #~(i,j). Now suppose that we have chosen a set of lines which intersect some p,q-span properly contained in the i,j-span. Suppose without loss of generality that p = i

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and q < i (the other case is symmetric). In order to intersect all lines in the +span, it suffices to add a line from k to 1 to our set for which k s i and 1 a j ; this covers precisely the i,j-span (and no larger span) when 1 =j. If the (k, 1 ) line is a target line, we have that @(i, j ) s &(i, j ) s @(p,q ) ; if it is a nontarget line, we have that @(i, j) S @ N ( i j, ) + 1S @(p,q ) + 1. 0 Hence evaluating @ provides an upper bound on the number of non-target lines needed. We next verify that this number of lines is, in fact, also necessary.

Proof. Let qj denote the set of target lines intersecting the i,j-span. First we consider the case of i,j-spans which have no proper sub-span which every target line intersects (in this case, &(i, j ) = m). Let mij= min(@T(i,j), GN(i, j ) , GR(i,j ) ) . Let us suppose to the contrary that for some smallest i,j-span, there is a set L of fewer than mij nontarget lines for which (1) Tj U L induces a connected subgraph, every line with one endpoint in the i,j-span intersects a line of U L, and (2) there are lines ( i , q ) and (p, j) in qj U L with p s i and q 6 j. If (i, q ) is a target line, consider the p,j-span. We have @ ( i , j ) S @ T ( i , j ) S @ ( p ,j ) s (LI, a contradiction. If (i, q ) is a non-target line, L\{(i, q)} is sufficient (together with T,.) to cover the p,j-span, and hence @(i, j ) s @ N ( i j, ) s @ ( p ,j ) 1s ILI, a contradiction. Next, if this does not occur but (to the contrary) there is some smallest i,j-span and set L with fewer than mjj lines satisfying (1) but not (2) above, then Tj U L covers a span, say the k,l-span, properly containing the i,j-span. But then @(i, j ) c &(i, j ) c @ ( k ,I ) , and again we have a contradiction. Finally, if all target lines intersect a span, say the p,q-span, properly contained in the i,j-span, then @(i, j ) d &(i, j ) s @(p,q ) , as required. 0

xj

+

It is now a simple matter to compute @ ( n ,n ) using dynamic programming techniques; we present a somewhat different technique here, by using a fairly standard conversion of dynamic programming to finding shortest paths in a graph. Of course, we need not write down all of the constraints given by Lemma 4.1, since many imply others by transitivity. Construct a directed graph with n2 vertices {(i, j ) I 1s i s n , 1~j s n } . Directed edges from (i 1,j ) to ( i , j) and from (i, j + 1) to ( i , j ) appear with cost 0. A directed edge from (1,l)to (i, j) of cost 0 appears whenever there is an initial target line from i to j. A directed edge from (i, j ) to ( n , n ) of cost 0 appears whenever every target line intersects the i , j-span. Whenever there is a target line from i to j , an edge of cost 0 from (i, k) to ( I , j ) is added for each k s j and 1 a i; symmetrically, an edge of cost 0 from

+

Permutation graphs: Connected domination and Steiner trees

187

(k, j) to (i, I) is added for each k 6 i and I aj. Finally, for any non-target line from i to j , the same edges are added, but each with cost 1. The Steiner tree is now easy to find; one simply finds a minimum cost path from ( 1 , l ) to (n, n) in this digraph. The cost of the path is the number of non-target lines chosen. Moreover, from the edges of cost 1 chosen, one can produce an actual selection of non-target lines.

Theorem 3.3. A minimum cardinality Steiner tree in an n-vertex permutation graph can be found in O(n3) time.

Proof. The required digraph can easily be constructed in O(n3) time. By Lemma 3.2, a Steiner tree with @(n,n) target lines is the smallest; it is easy to see that the digraph constructed ensures that @(i,j ) satisfies the constraints in Lemma 3.1 Hence only timing is at issue. Minimum cost paths can be found using breadth-first search, for example, in time proportional to the number of edges in the digraph. 0

4. Weighted Connected domination

In many practical applications, there is a cost, or weight, associated with the inclusion of a particular vertex in the dominating set. Thus much consideration has been given to the solution of weighted domination problems. Here each vertex has a weight, and the objective is to find a dominating set with minimum weight. The ideas of Sections 2 and 3 combine nicely to yield an O(n3) algorithm for weighted connected domination in permutation graphs. We sketch such an algorithm in this section. In (cardinality) connected domination, one could assume that the next line selected caused the largest increase in the span covered so far; in the weighted case, this need not be true. This consideration of all lines, rather than just those which maximize increase in covered span, is easily handled using ideas from the CARDST algorithm. j ) to be the weight of a minimum weight connected dominating We define set covering all lines intersecting the i,j-span. Then we have the following inequalities for

v(i, +.

Lemma 4.1. (a) v ( 0 , O ) = 0; (b) if there is an initial line from i to j of weight k, W ( i , j ) S k ; (c) if there is a pair of lines, one from i to j of cost k, and another from i’ to j ‘ of cost k‘ which together intersect all initial lines, q(i, j ‘ ) s k + k ‘ ; (d) if there is a line from i to j of cost k which intersects the i’,j’-span, q(max(i, i’), max(j, j ’ ) ) ) G v(i’, j’)+ k ; (e) if there is no line not intersecting the i,j-span, v(n, n) = v(i, j).

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C .J . Colbourn, L. K . Stewart

Proof. All follow directly from the definition.

0

The details from this point on parallel the method for CARDST very closely and are omitted here. The same ‘shortest paths’ approach leads to an O(n3) algorithm for weighted connected domination.

5. Conclusions The methods in this paper extend the research of Farber and Keil [ll] to include a further domination problem which has been widely studied, and they also support the contention of White, Farber and Pulleyblank [25] that cardinality Steiner tree and connected domination are algorithmically closely related problems. The topic which we believe is of most significance for future research is to account for the remarkably similar behaviour of CARDST and connected domination algorithmically.

Acknowledgements Research of the first author is supported by NSERC Canada under grant number A0579. Thanks are due to both referees for suggesting improvements in the presentation.

References [l] K.S. Booth, Dominating sets in chordal graphs, Research Report CS-&0-34, University of Waterloo, 1980. [2] K.S. Booth and J.H. Johnson, Dominating sets in chordal graphs, SIAM J. Comput. 11 (1982) 191-199. [3] D.G. Corneil and J.M. Keil, A dynamic programming approach to the dominating set problem on k-trees, to appear. [4] D.G. Corneil, H. Lerchs and L. Stewart Burlingham, Complement reducible graphs, Discrete Appl. Math. 3 (1981) 163-174. [5] D.G. Corneil and Y. Perl, Clustering and domination in perfect graphs, Discrete Appl. Math. 9 (1984) 27-40. [6] D.G. Corneil and L.K. Stewart, Dominating sets in perfect graphs, preprint, 1986. [7] G. Cornuejols, J. Fonlupt and D. Naddef, The graphical travelling salesman problem and some related integer polyhedra, Research Report 378, Laboratorie d’hformatique et de Mathematiques appliquees de Grenoble, 1983. [8] A.K. Dewdney, Fast Turning reductions between problems in NP 4, Report 71, University of Western Ontario, 1981. [9] M. Farber, Independent domination in chordal graphs, Oper. Res. Lett. 1 (1982) 134-138. [lo] M. Farber, Domination, independent domination and duality in strongly chordal graphs, Discrete Appl. Math. 7 (1984) 115-130. [ l l ] M. Farber and J.M. Keil, Domination in permutation graphs, J . Algorithms 6 (1985) 309-321.

Permutation graphs :Connected domination and Steiner trees

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[12] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [13] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [14] T. Kikuno, N. Yoshida and Y. Kokuda, A linear algorithm for the domination number of a series-parallel graph, Discrete Appl. Math. 5 (1983) 299-311. [15] R. Laskar and J. Pfaff, Domination and irredundance in split graphs, Technical Report 430, Clemson University, 1983. [16] R. Laskar, J. Pfaff, S.M. Hedetniemi and S.T. Hedetniemi, On the algorithmic complexity of total domination, SIAM J. Algebraic Discrete Methods 5 (1984) 420-425. [17] J. Pfaff, R. Laskar and S.T. Hedetniemi, NP-completeness of connected and total domination and irredundance for bipartite graphs, Technical Report 428, Clemson University, 1983. [18] J. Pfaff, R. Laskar and S.T. Hedetniemi, Linear algorithms for independent domination and total domination in series-parallel graphs, Technical Report 441, Clemson University, 1984. [19] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canad. J. Math. 23 (1971) 160-175. [20] R.L. Rardin, R.G. Parker and M.B. Richey, A polynomial algorithm for a class of Steiner tree problems on graphs, ISE Report 5-82-5, Georgia Institute of Technology, 1982. [21] J. Spinrad, Transitive orientation in O(n2)time, in: Proceedings Fifteenth ACM Symposium on the Theory of Computing (1983) 457-466. [22] L. Stewart, Cographs: a class of tree representable graphs, M.Sc. Thesis, University of Toronto; also technical report 126/78, University of Toronto, 1978. [23] L. Stewart, Ph.D. Thesis, Department of Computer Science, University of Toronto, 1985. [24] J.A. Wald and C.J. Colbourn, Steiner trees, partial 2-trees, and minimum IF1 networks, Networks 13 (1983) 159-167. [25] K. White, M. Farber and W.R. Pulleyblank, Steiner trees, connected domination, and strongly chordal graphs, Networks 15 (1985) 109-124.

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Discrete Mathematics 86 (1990) 191-198 North-Holland

191

THE DISCIPLINE NUMBER OF A GRAPH

v. CHVATAL Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, USA

W. COOK Graduate Schoof of Business, Columbia University, New York, N Y lOm7,USA

Received 2 December 1988

1. Introduction The domination number y ( G ) of a graph G is the size of a smallest set D of vertices such that every vertex outside D has at least one neighbour in D ; Fink, Jacobson, Kinch, and Roberts [4] defined the bondage number b ( G ) of a graph G as the least number of edges whose deletion makes y ( G ) increase. As we are about to point out, computing b ( G ) amounts to solving an integer linear program. Define a whip in a graph G as any spanning subgraph F of G such that each component of F is a star and F has precisely y ( G ) components; let E ( G ) denote the set of edges of G and let W ( C )denote the set of all whips in G. Obviously, b(G) is the optimal value of the problem

C {x,: e E E ( G ) ) subject to C {x,: e E E ( F ) } 2 1

minimize

for all F in W ( G ) ,

x, 5 0

for all e in E ( G ) .

x, = integer

for all e in E ( G ) .

By the fractional bondage number b*(G) we shall mean the optimal value of the ‘linear programming relaxation’ of (l),

subject to

{xe: e E E ( F ) } 2 1 for all F in W ( G ) ,

x, 2 0

(2)

for all e in E ( G ) .

By the duality theorem of linear programming, b*(G) equals the optimal value of 0012-365X/90/$03.50 @ 1990-Elsevier Science Publishers B .V.(North-Holland)

192

V . Chvatal. W. Cook

the dual of (2), maximize

C { y F :F E w(G))

subject to

2 { y,:

e E E ( F ) ) s 1 for all e in E ( G ) ,

YF20

for all F in W(G).

(3)

Since (3) can be seen as the linear programming relaxation of

C { y F :F E w ( G ) ) subject to C { y F :e E E ( F ) } 6 1

maximize

YF

a.

yF = integer

for all e in E ( G ) ,

(4)

for all F in W ( G ) , for all F in W(G),

problems (1) and (4) are in a sense dual. Therefore we refer to the optimal value of (4) as the discipline number dis(G) of G. We have 1s dis(G) S b * ( G )S b ( C )

(5)

for all graphs G. Apart from establishing upper bounds on b ( G ) , Fink et al. computed the bondage number of cycles, paths, and complete multipartite graphs and studied the bondage number of trees (several of these results can also be found in Bauer, Harary, Nieminen, and Suffel 111). The purpose of this paper is to provide ties with analogous results for the fractional bondage number and for the discipline number.

2. The fractional bondage number The principle restraining device of this section goes as follows.

Theorem 1. Zf G has n vertices and m edges then b * ( G ) m / ( n - y ( G ) ) . Proof. Observe that the constraints of (2) are satisfied by x, = l / ( n - y ( G ) ) for all e. 0 As usual, let A(G) denote the largest degree of a vertex in G. Fink et al. conjectured that b ( G )G A ( G ) + 1.

Theorem 2. b*(G)S A(G).

The discipline number of a graph

193

Proof. Consider any maximal set S of pairwise nonadjacent vertices: trivially, every vertex outside S has at least one neighbour in S and the number of edges in G is at most the sum of the degrees of all the vertices outside S. Hence the desired conclusion follows from Theorem 1 : we have y ( G ) = s ( S ( and m s - PI). 0 Let C,, denote the cycle with n vertices. Fink et al. proved that b(C,,)= 3 if n = l mod3, and b ( C , , ) = 2 otherwise. Now we shall prove a theorem that includes a formula for b*(C,,) as a special case. Recall that a graph G is called edge-transitive if for every choice of its edges e l , e2 some automorphism of G sends e , onto e2.

Theorem 3. If G is edge-transitive with n vertices and m edges then b * ( G )= m / ( n - Y(G)).

Proof. Since G is edge-transitive, ( 2 ) has an optimal solution with all x, equal to each other. Hence b * ( G ) is the optimal value of the problem minimize mx

subject to ( n - y ( G ) ) x

3 1, x 2 0.

0

Since C,, is edge-transitive and y(C,,) = [ n / 3 ] , Theorem 3 yields b*(C,,)= n/ 12n/3J. Let P,, denote the path with n vertices. Fink et al. proved that b(P,) = 2 if n = 1 mod 3 , and b(P,) = 1 otherwise.

Theorem 4. b*(P,) = 3 i f n = 1 mod3, and b*(P,,)= 1 otherwise.

Proof. We may assume that n = 3 k + 1 , for otherwise the desired conclusion = k + 1, Theorem 1 guarantees that b*(P3k+l) s follows from ( 5 ) . Since to prove the reversed inequality, we only need exhibit a feasible solution of (3) in which precisely three variables have value 4. To put it differently, we only need to find three whips in P3k+l so that each edge belongs to precisely two of the three whips. For this purpose, label the edges of P3k+l as e l , e2, . . . , em in such a way that ei and ei+l share an endpoint whenever 1zs i =s m - 1. Now the jth whip arises by deleting all the edges ei with i =j mod 3 . 0

a;

In dealing with complete multipartite graphs, we shall distinguish between those having a positive number k of classes of size one and. those in which all classes have size at least two. For the first kind, Fink et al. proved that the bondage number equals r k / 2 ] .

Theorem 5. Let G have n vertices and let precisely k vertices of G have degree n - 1. If k = 1 then b * ( G )= 1 ; if k 2 2 then b*(G)= k / 2 .

V . Chucital, W. Cook

194

Proof. We may assume that k a 2 , for otherwise the desired conclusion follows from ( 5 ) . Setting x, = l/(k - 1) if both endpoints of e have degree n - 1, and x, = 0 otherwise, we obtain a feasible solution of ( 2 ) ; hence b * ( G )S k / 2 . On the other hand, there are precisely k whips; setting y , = 3 for all of them, we obtain a feasible solution of (3); hence b * ( G )2 k / 2 . 0 For complete multipartite graphs G with class sizes n , , n2, . . . , n, such that 2 s nl S n2 S * * * 6 n,, Fink et al. proved that b ( G )= n - n, unless n , = n2 = . . . = n, = 2, in which case b ( G ) = n - 1.

Theorem 6. Let G be a complete multipartite graph with n vertices and m edges. If all classes of C have size at least two then b * ( G ) = m / ( n - 2). Proof. Theorem 1 guarantees that b * ( G )=sm / ( n - 2 ) ; to prove the reversed inequality, we shall exhibit an appropriate feasible solution of (3). For this purpose, let S,, S,, . . . , S, denote the classes of G; write nk = ISk\. By a center of a star, we shall mean a vertex in the star adjacent to all the other vertices in the star (unless the star has precisely two vertices, its center is uniquely determined); by a pointed whip, we shall mean a whip with a center distinguished in each of the two components; the pointed whip is of type (i, j ) if its two centers belong to S, and S,. Clearly, there are precisely 2"-(",+n,) I 1

pointed whips of type ( i , j ) ; each edge with one endpoint in S, and the other endpoint in S, belongs to precisely

(n, + n - 2)2"-(",+n,) I

pointed whips of type (i, j ) , to precisely nk2n- ( n , + n t ) - 1 pointed whips of type ( i , k) with k # i , j , and to precisely nk2n - ( n , + m ) -

1

pointed whips of type ( k , j ) with k # i, j . Hence the desired feasible solution of (3) can be obtained by setting first

for every pointed whip H of type (i, j ) , and then YF

=

c

zH

with the summation running through all pointed whips H such that E ( H ) = E(F). Fink et al. proved that b ( T )S 2 for every tree T.

The discipline number of a graph

195

Theorem 7. b*(T ) S (n - 1)/ [n/21 for every tree T with n vertices.

Proof. As S.T.Hedetniemi pointed out to us, Theorem 13.1.3 in Ore’s book [5] implies that y(G)S [ n / 2 ] for every graph without isolated vertices; the rest follows from Theorem 1. 0

To show that the bound of Theorem 7 cannot be improved (at least not for even values of n ) , consider the tree with vertices ui, vi (1 s i s k ) and edges uiui+l(1G i s k - l), uivi (1 =Z i =Z k ) . We shall refer to any such tree as a Justine (71. (One of the referees pointed out that the same trees have been called combs by Fink et al. [3]. However, combs is also the name of graphs used by Padberg and Rinaldi [6] in solving a traveling salesman problem. To avoid confusion, we prefer the descriptive and unambiguous term Justine.) Theorem 8. 6 * ( T )= 2(n - l)/n for the Justine T with n vertices. Proof. By virtue of Theorem 7, we only need prove that b * ( T )3 2(n - l)/n; to do this we only need exhibit a feasible solution of (3) in which precisely n - 1 variables have value 2/n. To put it differently, we only need find whips F,, 4,. . . , FZk-l in a Justine with 2k vertices so that each edge belongs to precisely k of these whips. We propose to do so by induction on k. The case of k = 1 is trivial; now assume that appropriate whips 4, F,, . . . ,F2k--3have been found in the Justine with 2k - 2 vertices. Without loss of generality, assume that F,, 4,. . . , Fk-2 do not include the edge u k - 2 u k - l . Next, observe that each of these k - 2 whips must include the edge u k - l v k - 1 . Extend each 6 with 1S i s k - 2 by adding the edge u k - $ k and extend each 4 with k - 1s i S 2k - 3 by adding the edge u k v k . Finally, let F2k-2 consist of all uivi with i odd, all uiui+l with i even and less than k, and u k - 1 U k . Let Fzk-l consist of all uivi with i even, all uiui+lwith i odd and less than k , and u k - l u k . 0

3. The discipline number

Theorem 3 combined with ( 5 ) implies that dis(C,) = 1 whenever n 3 5; Theorem 7 combined with ( 5 ) implies that dis(T)= 1 for every tree T; in addition, it is easy to see that dis(G) = 1 whenever y(G) = 1. However, we are about to show that dis(G) can be arbitrarily large even when y(G) = 2.

Theorem 9. Let G be a complete multipartite graph with no classes of size one, a classes of size two, and 6 classes of size at least three. If a Lb/2J 3 3, then

+

dis(G) = a + [ b / 2 ] .

If (a, b ) = (0, 4), (0, 5 ) or (1, 3), then dis(G) = 3 . Zf (a, b ) = (0, 3), (1, l), (1, 2), (2, 0) or (2, l), then dis(G) = 2. If (a, b ) = (0, 2 ) , then dis(G) = 1.

V . Chvdtal, W. Cook

196

Proof. Enumerate the classes of G as S,, S,, . . . , Sa+b so that ISj[ = 2 whenever 1 s i < a and ISil3 3 whenever a + 1Si < a + b. Claim 1. dis(G) 2 a + [ b / 2 ] . Proof of Claim 1. Write S, = { u t , v,} for i = 1, 2, . . . , a and choose vertices u,+], v,+] with 1 6j =G Lb/2] so that u,,, E Sa+21-1, v,+] E S,+zl. For every choice of i and j such that 1S i <j S a + l b / 2 ] ,set U,U,

€6, V1VIE E ,

U,V]E

F;,

U,”,

Eq.

For all the remaining vertices w, set wu,E E if w and and wv,E F; otherwise. 0

v,belong to the same Sk,

Claim 2. If a + b 3 4, then dis(G) z=3. Proof of Claim 2. Choose vertices u l , u2, u3 so that ui E Sj and choose a vertex x in S,. Set ~

1 E FI, ~

2~

2

E 4, ~

3~

3

E &, ~

For all the remaining vertices w, set wx

Claim 3. Zf a 3 1, then dis(G)

E

1 U1x

E 4,

U ~ E X

&,

U ~ EX Fl.

6 if w E Si and wuj E fi otherwise. 0

2 2.

Proof of Claim 3. Write S, = {xl,x , } . For every vertex w outside S1, set wxl E Fl and W X ~ E F , . 0 These three claims guarantee that the values stated in Theorem 9 provide correct lower bounds on dis(G); now we shall establish the upper bounds. For this purpose, consider arbitrary pairwise edge-disjoint whips 4,4, . . . , Fk in G. For each i = 1, 2, . . . , k, choose vertices ui, vi that are centers of the two components of F;. Write

Q = {UI, Claim 4. Zf

lQl

2/1,

u21 2 / 2 2 .

= 2k, then

..

k
9

uk, uk}.

+ Lb/2].

Proof of Claim 4. Consider the graph H whose set of vertices is Q, two vertices being adjacent in H if and only if they are adjacent in some F;. Since no ui is adjacent to v i in H, all the remaining pairs of vertices must be adjacent in H: we have ( ik)- k = k(2k - 2 ) and each F; contributes 2k - 2 edges to H . Now call an Si special if it includes at least two vertices from Q. As we have just observed, each special Si includes some uiand vi and it includes no other vertices from Q ; since each vertex outside Q is adjacent to at least one of ui and vi, we must have ISi( = 2. It follows that lQl s 2a + b. 0

The discipline number of a graph

Claim 5. Zf k

2 3,

197

then l Q l 3 4.

Proof of Claim 5. Assume the contrary: k 3 3 but lQl s 3. Since G has at least four vertices, some vertex w lies outside Q; since F,, 4,. . . , Fk are edge-disjoint, w is adjacent to at least k distinct vertices in Q. Hence lQl = k = 3. Now no Sj can include a vertex from Q and a vertex w outside Q ( w has to be adjacent to at least three distinct vertices in Q); since lSjl 3 2 for all j , it follows that Q = Sj for some j . Finally, this Sj includes some vertex w distinct from u1 and vl, a contradiction: w must be adjacent to at least one of u1 and v l . 0

Claim6. Z f k 2 4 , t h e n k S a + L b / 2 ] . Proof of Claim 6. By virtue of Claim 4 , we only need show that lQl = 2k. For this purpose, assume the contrary: without loss of generality u1= u2. Write Qo=

{ui, 211,

u2, 212, u3, 213, u4,

21,)

and consider the graph Ha whose set of vertices is Q,, two vertices being adjacent in Ha if and only if they are adjacent in some E with 1s i S 4. Since each & with 1s i s 4 contributes lQol - 2 edges to H,, we have

observing that lQol ~7 (since u , = u2) and lQol 2 4 (by Claim 5 ) , we conclude that lQol = 7. Now Ha has twenty edges, which is a contradiction: = 21 and no ui with 2 s i s 4 is adjacent to v i in Ho. 0

(z)

Claim7. I f k = 3 , then k < a + l b / 2 ] o r a + b 3 4 . Proof of Claim 7. Claim 4 allows us to assume that IQI s 5; Claim 5 guarantees that lQl z 4. Defining H as in the proof of Claim 4, observe that H has 3( lQl - 2) edges. It follows that H (and hence also G) contains four pairwise adjacent vertices. Claim8. I f a = O a n d b = 2 , t h e n k Z 2 . Proof of Claim 8. Assume the contrary: k = 2 but a = 0 and b = 2. Claim 4 implies that lQl S 3 and so, without loss of generality, u l = u 2 e S 1 . Since S1 includes a vertex distinct from both u l , vl but adjacent to at least one of them, we must have ~ , E S , ; a symmetric argument shows that v ~ E S , .But then S, includes a vertex outside Q and adjacent to only one vertex in Q , a contradiction. 0 This ties down the proof of Theorem 9.

0

198

V. Chvdtal, W. Cook

The reader interested in additional results in a similar vein is directed to [2, Chapter 51.

Acknowledgement We thank the two referees for their thoughtful comments which helped to improve the presentation of our results.

References [l] D. Bauer, F. Harary, J. Nieminen and C.L. Suffel, Domination alteration sets in graphs, Discrete Math. 47 (1983) 153-161. [2] K.J. Devlin, Constructibility (Springer, Berlin, 1984). [3] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination numbers half their order, Per. Math. Hungar. 16 (1985) 63-69. [4] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, The bondage number of a graph, preprint. [ 5 ] 0. Ore, Theory of Graphs, Amer. Math. SOC. Colloq. Publ. 38 (Amer. Mathematical SOC., Providence, RI, 1962). (61 M. Padberg and G. Rinaldi, Optimization of a 532-city symmetric traveling salesman problem by branch and cut, Oper. Res. Lett. 6 (1987) 1-8. [7] D.A.F. de Sade, Oeuvres Completes, 2-3 (J.-J, Pauvert, Paris, 1955).

Discrete Mathematics 86 (1990) 199-214 North-Holland

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BEST LOCATION OF SERVICE CENTERS IN A TREELIKE NETWORK UNDER BUDGET CONSTRAINTS* James McHUGH and Yehoshua PERL C.I.S. Department, N.J.I. T., Newark, NJ 07102, USA Received 2 December 1988 We consider the problem of locating service centers in a treelike network in order to maximize the serviced population under budget constraints. We show that the problem is NP-hard. In the case where the costs of establishing the service centers are equal for all n cities we obtain the maximum weight k-domination problem. An O(nk2) dynamic programming procedure is given. Then an O(nB2) pseudo-polynomial dynamic programming procedure is presented for the original problem, where B is the budget constraint. Finally a variation of the new left-right dynamic programming technique is applied to obtain a more efficient pseudo-polynomial procedure.

1. Introduction Let G(V, E) be a graph, where the vertices represent cities, and there is an edge between a pair of cities if the corresponding cities are mutually easily accessible. We assume there is a weight associated with each city which represents its population. We consider the problem of locating service centers at the cities, under the assumption that each service center can serve only the home city itself and its adjacent cities. We wish to optimize the location of the service centers, so that the centers service the maximum possible population. We remark that if a city is serviced by more than one service center, its population is still only counted once. The best possible outcome would be that every city was serviced by at least one service center, either in the city itself or in an adjacent city. However, it often occurs that a budget precludes attaining this maximum possibility. Thus, suppose there is a cost c(v) to set up a service center in city v , for every v in V. Suppose also there is a total budget B which can be spent establishing service centers. Then, we may look for a set S of cities with COST(S) = Cuesc(v) s B and such that the population which will be served by service centers in S will be maximized. The relevant mathematical concept is that of Domination. A set of vertices S is said to dominate all the vertices which are adjacent to S. For our application, a vertex is considered as dominating not only its. adjacent vertices, but also the Supported in part by the State of New Jersey SBR Grants

0012-365X/90/$03.50 0 1990-Elsevier Science Publishers B.V. (North-Holland)

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vertex itself, corresponding to the fact that a service center serves not only its adjacent cities, but its home city as well. Thus, in our case we consider weighted domination where each dominating vertex is responsible for the weight of itself and its adjacent vertices. Thus, we are looking for a maximum weighted dominating set S under the budget constraints Csc ( u ) s B. This is the constrained maximum weight domination problem. The original domination problem was to find a minimum set of vertices which together dominate all the other vertices in a graph. This was shown to be NP-hard by Karp 17). Recently, much attention has been given to variations of the Domination Problem. See, for example, the extensive bibliography by Hedetniemi and Laskar [ 5 ] . It is clear that our extension of this problem is difficult. Actually, we shall show the problem is NP-hard even for trees, for which case there is a linear algorithm for the original domination problem [l]. The minimum cost domination problem for weighted trees is solved in linear time by Natarajan and White [lo]. In this problem the cost of the domination is the sum of the costs of the dominating vertices and the cost of the least edges from all other vertices to a dominating vertex. We refer to [4] for a bibliography on domination on trees. In this work, we limit ourselves to trees. First, we show a reduction from the knapsack problem [7] to our problem for a tree, thus showing our problem is NP-hard. Then we consider the case where the cost of an installation of a service center at every vertex is the same. Thus, we have equal costs c = c(u). In this case, the budget constraint is translated into building k = LB/c] service centers. In this case, the problem is to find a set of k cities maximizing the sum of the populations of those k cities and their adjacent cities. Let D ( S ) denote the set of vertices in V-S which are adjacent to vertices in S. Then, our problem is to find a set S of k vertices such that WEIGHT(S) = CVSUD(S)w(v) is maximized. Denote this problem as the maximum weight k-domination problem. We present an O(nk2) dynamic programming algorithm for the problem. The special case where w(v) = 1 for every vertex u in V was solved by Hsu [3]. Note the difference between our model and some other models of optimum location of the service centers in a network. For example in [8] and [2] the centers provide services to all vertices and the objective is to minimize the average service cost. In our model only part of the vertices, namely those assigned service centers and their adjacent vertices, are receiving service and the objective is to maximize the population served by the k centers. Finally an O(nB2) pseudo polynomial dynamic programming procedure is presented for the constrained maximum weight domination problem. We apply a variation of the new left-right dynamic programming technique [6] to obtain a more efficient pseudo-polynomial procedure. To apply the left-right technique, we have to overcome some difficulties inherent in the nature of the domination problem.

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2. NP-completeness result In this section we show that the maximum weight domination problem, under the cost constraint Csc(v) s B, where S is the dominating set, is NP-hard even when the graph is a tree. Let us first formulate the corresponding decision problem. The constrained weight domination problem: Given a tree T with n vertices having non-negative integral weight w,and cost c, associated with each vertex i, and two integers M and B, is there a dominating set S in T satisfying COST(S) s B and WEIGHT(S) 3 M? This problem is clearly in NP. The reduction is from the NP-complete knapsack problem (see [7]): Given a set U of elements ul, u2, . . . , u, with integer weight w ( u , ) and cost c(u,) associated with each u,, and two integers B and M, is there a subset U ’ of U such that: w(u,) 3 M U’

c(u,) s B?

and U’

The reduction is as follows: Construct a rooted tree T of 2n + 1 vertices where the root r , with w ( r ) = 0 and c ( r ) = B + 1, has n sons vl, . . . , v, with weights w ( v , ) = O and cost c,. Each son v, has a single son v,’ with w(v,’)= w, and c(v,‘) = B + 1. Refer to Fig. 1 for an illustration. Clearly a dominating set of cost at most B uses only a subset U‘ of the vertices v,, since all the other vertices violate the cost constraint. The weight of such a dominating set is just C I E Uw, r and its cost is just Clcurc,. Thus, if we can solve the constrained weighted domination problem in polynomial time, then we could solve the knapsack problem in polynomial time. This establishes that the constrained weight domination problem is NP-complete even for trees.

3. Dynamic programming procedure for the maximum weight k-domination problem In the previous section we showed that the constrained maximum weight domination problem is NP-hard. In this section we consider the special case of

v‘2 (B+l.W2)

. . . .

Fig. 1. Knapsack reduction.

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equal costs which as we showed earlier is actually the maximum weight k-domination problem. Note that transforming the tree into a rooted tree does not change the problem. Thus we may assume that a tree rooted at a vertex r is given. We shall describe a dynamic programming procedure which is polynomial in n and k. The algorithm applies a bottom-up scanning technique to the (rooted) tree, calculating for each vertex u and integer k', 0 =G k' G k, the weight WD(u, k') of a maximum weight k'-dominating set in the subtree rooted at u. In case k ' is greater than the size of the subtree at u, we define WD(u, k) = --cD. The last computed value WD(r, k ) is the weight of the maximum weight k-dominating set in the original tree. Let the subtree T ( u ) be the complete subtree of T rooted at the vertex u, and let x l , . . . , x , be the m sons of u. Let T ( x , ) , . . . , T(x,) be the complete subtrees rooted respectively at xl, . . . , x,. Refer to Fig. 2 for an illustration. The development of the algorithm is complicated by two features. First, the analysis depends critically on whether the root u of T ( u ) does or does not lie in a maximum weight dominating set, as well as on whether the root is or is not dominated by a maximum weight dominating set. Secondly, the straightforward dynamic programming procedure is computationally intractable. For example, in the case that the root of the subtree T ( u ) is neither in, nor dominated by, the dominating set, the principle of optimality which gives the basis for the dynamic programming procedure can be applied in a straightforward manner. In this case, a maximum weight k-dominating set for T ( u ) is the disjoint union of maximum weight k,-dominating sets for the subtrees T ( x , ) , . . . , T(x,), where ,C ; k, = k. But, even in this simplified case, if we follow a straightforward dynamic programming approach, then for each sequence k l , . . . , k, such that CE1ki = k, the corresponding maximum weight ki-dominating sets must be found. Since there are O(km-') such sequences to be considered, this is intractable for large values of rn. We can avoid the computational difficulties of this direct approach by using the following propagation technique. We let T ] ( u ) , i s m, denote the u-rooted subtree of T spanned by T ( x l ) ,. . . , T ( x i ) . We then iteratively construct a maximum weight k'-dominating set, for each k' 4 k, for each of the subgraphs Tl!(u), l ~ i a m .For i = m , we obtain the desired maximum weight k'dominating set for the tree T ( u ) . This technique leads as we shall see, to a

Fig. 2. Tree rooted at u.

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polynomial time algorithm. For another example of such a propagation technique see [ll]. We address the complications caused by the possible states of u (in the dominating set or not, dominated by the dominating set or not) by defining subproblems appropriate to the status of u. We then solve these interrelated subproblems by a bottom-up, propagating dynamic programming scheme that iterates over all the subtrees Tf(u),for each vertex u in T. We propose the following definitions: WDO(u, k', i): the weight of the maximum weight k'-dominating set in the tree T f ( u ) ,where u does not lie in the dominating set. WDl(u, k', i): the weight of the maximum weight k'-dominating set in the tree T!(u), where u is required to lie in the dominating set. WD2(u, k', i ) : the weight of the maximum weight k'-dominating set in T:(u), where u does not lie in the dominating set, but is dominated by the dominating set. WD3(u, k', i): the weight of the maximum weight k'-dominating set in T!(u), where u does not lie in the dominating set, and is not dominated by the dominating set. WDO(u, k') : the weight of the maximum weight k'-dominating set in T ( u ) , where u is not in the dominating set. the weight of the maximum weight k'-dominating set in T ( u ) , WDl(u, k'): where u is in the dominating set. WD(u, k'): the weight of the maximum weight k'-dominating set in T ( u ) . WD(r, k) is the quantity actually sought. The initializations for i = 0 are as follows: 0 for k'=O, WDO(u, k', 0) = --co for k ' > 0 . W ( u ) for k' = 1, WDl(u, k', 0) = -m for k ' < > l .

{ {

WD2(u, k', 0) = --OO 0 WD3(u, k', 0) =

{

--oo

for all k'. for k'=O, for k'>O.

(1.3)

These assignments also suffice to define the initial values for the endpoints of the tree. The general dynamic programming relations between the variables and the proofs of these relations follow. The first relation we establish is: WDl(u, k', i

+ 1) =

max {WDl(u, k", i ) + WDl(xi+l, k' - k"),

0rk"Gk'

WDl(u, k", i ) + WD2(xi+1, k' - k"), , WDl(u, k", i ) + WD3(xi+l, k' - k") + ~ ( ~ i + j ) } (2) where xi+l refers here to the i + 1 son of u.

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The principle of optimality applies here as follows WDl(u, k', i + 1) gives the weight of the maximum weight k ' - dominating set in T,'+,(u)that includes u. We can partition such a set into that part of it that lies in T,!(u)and that part of it that lies in the complete tree rooted at x , + ~ viz., , T ( X , + ~The ) . part of the dominating set in Tz!(u)includes u by definition, and may be chosen to be the maximum weight such set. Its size can be any integer k" from 0 to k'. The remaining part of the dominating set lies in T(x,+,) and has size k' - k". This part has to be handled differently depending on the status of x , + ~ viz., , according as: (i) x , + ~is in the dominating set, or (ii) x , + ~is not in the dominating set but is dominated by it, or (iii) x , + ~is neither in the dominating set, nor is dominated by it. One must exercise care in calculating the weight of the combined parts of the dominating set. Case (i) is straight forward. The weight of the maximum part in T ( X , + must ~ ) be WDl(x,+,, k' - k''). In this case, x , + ~is in the dominating set, so its weight is already accounted for, and hence the weight dominated by the combined dominating set is given by the first expression in the maximization in (2). Case (ii) is similar. However, in case (iii), the weight of x i + l is not dominated by that part of the dominating set lying in T(xZ+*).Since u is in the part of the dominating set in T:(u), then x,+, is dominated by the combined parts. Thus, its weight must be added on explicitly as in the third expression in (2). The following relation obviously holds: WDO(u, k', i

+ 1)= max {WDj(u, k', i + l)}.

(3)

j=2,3

This is not an application of the principle of optimality. It merely reduces the i + 1-order WDO to a pair of i + 1-order WD2 and WD3 problems. The following relations solve these problems.

+ WD1(xi+l, k ' - k"), WD2(u, k", i) + WDO(x,+,, k ' - k"), WD3(u, k", i) + WDl(x,+l, k ' - k") + w ( u ) } .

WD2(u, k', i + 1) = max {WD2(u, k", i) 0sk"sk'

(4)

+

Considerations similar to those for calculating WDl(u, k', i 1) apply here. The vertex u is dominated by, but does not lie in, the dominating set. There are three cases according as u is dominated: (i) both by the part of the dominating set in T,'(u) and the part in T ( X ~ +or ~), ), (ii) by the part in T:(u), but not the part in T ( X , + ~or (iii) by the part in T ( X ~ + but ~ ) ,not the part in T,'(u). Case (i) corresponds to the first expression in (4). For case (ii), observe that requiring that u is not dominated by x , + ~implies x , + ~is not in the part of the ). the second expression in (4) uses the dominating set lying in T ( X , + ~Therefore, term WDO(X,+~, k' - k"), which excludes x , + ~from the part of the dominating set

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in T ( X ~ +Finally, ~). for case (iii), u is excluded and xitlis included in the dominating set, whence the first two terms in the third expression in (4). The combined parts dominate u, giving the third term in this expression. The WD3 relation is: WD3(u, k‘, i

+ 1)=

max WD3{(u, k“, i)

0Sk“Sk’

+ WDO(xi+l, k’ - k”)}.

(5)

This relation merely states that u must be both excluded from and not dominated by the dominating set in Tz!(u),whence the term WD3(u, k”, i). Also, xi+l cannot lie in the part of the dominating set in T ( X ~ +otherwise ~), u would be dominated by it, whence the term WDO(X~ +k’ ~ ,- k”), which excludes x i + l . The final relations to observe are: WDi(u, k’)= WDi(u, k’, deg(u))

(i = 0, l),

WD(u, k’) = max {WDi(u, k’)}. i=O,1

(6)

(7)

The calculation of WD(r, k) depends on a sequence of previously calculated WD-terms of the form: WDj(u, k’)and WDj(u, k’, i), where u varies over the n vertices of T, k’ varies from 0 to k, i varies from 1 to deg(u), and j runs from 0 to 3. The overall complexity of the calculation is O(nk2).To prove this, we argue as follows. Referring to (2), observe that calculating WDl(u, k’, i + 1) entails O(k‘) operations on the previously calculated WD-terms. WDl(u, k’) is just WDl(u, k‘, deg(u)), so its calculation requires O(k’ deg(u)) operations on previously found WD-terms. Since this must be done for all 0 s k’ s k, the work at each vertex is O(k2deg(u)). Summing over all the vertices gives CallaO(k2 deg(u)) = O(nk2)operations in all to calculate WD(r, k). It is straightforward to maintain, for every variable the appropriate list of dominating vertices and still remain within the complexity bound. Thus the maximum weight k-dominating set is obtained together with WD(r, k).

4. Pseudo polynomial algorithm for the constrained maximum weight domination problem The algorithm analysis for the maximum k-domination problem shows the dynamic programming procedure for WD(r, k ) actually defines a pseudopolynomial algorithm for the constrained maximum weight domination problem. The notation for the solution for the general problem is similar to that for the maximum k-domination solution. The term WDl(u, B‘, i) now refers to the maximum weight dominating set in Tz!(u),where u is included in the dominating set and the cost of the set is at most B‘. The other terms are defined similarly. For the defining relations, we merely replace occurences of k in the given relations with occurences of a general integer cost B, and make some minor modifications at the initialization phase. WD(r, B) is the weight of the constrained maximum weight dominating set. The resulting algorithm has

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206

performance O(nBz).The defining relations are: WDO(u, B ' , i

+ 1) = max{WDj(u, B', i + 1)). j=2,3

WDl(u, B', i

+ 1) =

+ WD1(xi+l, B' - B"), WDl(u, B", i) + WD2(x;+l, B' - B"), WDl(u, B", i) + WD3(Xi+l, B' - B") + W ( X ; + , ) } .

max {WDl(u, B", i)

0sB"zsB'

(9) WD2(u, B ' , i + 1) = max {WD2(u, B", i)

+ WDl(xi+l, B' - B"),

WD2(u, B", i)

+ WDO(xi+L, B' - B"),

WD3(u, B", i )

+ WD1(xi+l, B' - B") + w ( u ) } .

0GB"sB'

(10) WD3(u, B', i

+ 1) =

WDi(u, B')

max {WD3(u, B", i)

OG B"
= WDi(u,

B', deg(u)) (i = 0, 1).

WD(u, B ' ) = max WDi(u, B'). i=O, 1

+ WDO(xi+,, B' - B " ) } .

(11) (12) (13)

The validity of these relations is established analogously to relations (2)-(7).

5. Left-right dynamic programming procedure for the constrained maximum weight problem Johnson and Niemi [6] introduced a nonstandard approach to dynamic programming called left-right dynamic programming. In left-right dynamic programming, the vertices of the tree are ordered according to depth first search order, and the dynamic programming algorithm processes the vertices in a manner which is a combination of depth first order and bottom-up order. Johnson and Niemi apply this approach to a knapsack problem on a precedence tree and to a minimum capacity tree partitioning problem. In their Tree Knapsack problem, vertices have weights and values. The problem is to determine a maximum value subtree, which contains the root of the tree, and with weight bounded by some given knapsack capacity. Using the left-right approach, they solve this problem in time O(nP), where P is the value of an optimal solution. In contrast, the bottom-up method has complexity O(nP2). In their Tree Partitioning Problem, vertices have weights and edges have capacities. The problem is to partition the tree into parts, where the total weight of each part is constrained by some bound, and the sum of the capacities of the edges between the parts is to be

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minimized. Let Co denote this minimum capacity. Then whereas the standard bottom-up dynamic programming procedure of Lukes [9] for this problem has complexity O(nCg), the solution of Johnson and Niemi has a complexity of O(Con2).Per1 and Snir [12] subsequently applied this technique to another tree partitioning problem to obtain an O(Con4) procedure rather than the O(Ch3) complexity of the previous bottom-up procedure, where again co represents a bound on the sum of the edge capacities of each component. This is an improvement in the usual case that Co> a. A straightforward application of left-right dynamic programming is not possible for the constrained maximum weight domination problem, because of the way in which subproblems must be combined. For example, we have already seen that for the bottom-up propagation method a solution for a tree rooted at v, containing its first i children and all their descendants can be combined with a solution for the tree rooted at a child xi+l of v, provided component solutions are available for all states of v and x i + l ; viz., dominating, not dominating but dominated, and neither dominating nor dominated. Call such vertices as v and x i + l , where multiple-state solutions for the corresponding subtrees have to be maintained, contact vertices. Contact vertices serve as contact points between separate subproblems. Because of the way the propagation approach works, there are only two contact vertices at any stage. The state dependent solutions for the parent contact vertex v have to be maintained from one propagation stage to the next, because v will again act as a contact vertex, at the next stage, for its next child. On the other hand, the state dependent solutions for the tree rooted at the child vertex x i + l , do not need to be maintained at the next stage, because x i + l will no longer be a contact vertex at that stage. Thus, under the propagation method, we need to maintain only the solutions for four subproblems (cf. to (8)-(11), for each of two subtrees, and for each B ’ < B at each stage of the procedure. The underlying feature of the propagation method that allows this, is that the method appends a complete subtree at a time, and, to a fixed vertex. This obviates the need to maintain multi-state solutions, or indeed any solutions at all, for the appended subtree, after that stage. Furthermore, the solutions that must be maintained from one stage to the next depend only on the state of the fixed contact vertex v. While the propagation method adds a complete subtree at a time, the left-right approach adds only a single vertex at a time, together with the connecting edge of that vertex. As we shall see, this improves the performance of the algorithm as a function of B, in exchange for poorer performance as a function of n. Following [6], we define the sequence of subtrees considered in the left-right approach as follows. Let T be the tree to be scanned, and let v be a vertex in the tree T. We define Tt!(v)to be the tree consisting of T ( v ) together with all the vertices in T that precede v in depth-first order. Fig. 3 and Fig. 4 show a tree and its left-right subtrees, respectively. Following [6] we have ordered the subtrees so that: T,!(v)

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J . McHugh, Y. Per1

Fig. 3. Example tree.

precedes TI!+l(v), for every v , and i satisfying 0 i < deg(v); as well as so that if w is the ith child of v , then TI!-,(v)precedes Th(w) and T&,,(w) precedes Tl!(v).Certain subtrees (such as Tf(v)and T&+,,)(w), where w is the ith child of v) are identical as trees, although we shall consider them for convenience as separate problem states. A left-right dynamic programming approach based directly on [6] would work as follows. Starting with the initial trivial tree in Fig. 4a, we would add an edge at a time, extending a subtree along a depth first search path determined by the original tree T (refer to the trees in Figs. 4a, 4b, 4c for example). At each stage, we would obtain the optimal B' solutions, for every B', for the corresponding subtree. The subproblems to be combined in order to obtain that optimal solution would be handled just like in the propagation method, except that in the left-right case, the appended tree would always consist of only a single vertex, and the newly appended vertex u would have to be considered as a contact vertex for the next extension of the search path. There would always be exactly two partitions of B' that would have to be considered: {B' - c ( u ) , c ( u ) } and { B ' , 0}, corresponding to the cases where the appended vertex was considered to be dominating or not dominating. Instead of B' partitions, as in the propagation method, we would have only a fixed number (two) of partitions to consider. This is essentially the advantage of the left-right approach. It reduces the number of B partitions that have to be considered, in the same way that the propagation method reduces the number of partitions that have to be considered over the direct approach. However, a problem arises with the procedure we have described as soon as one retracts the depth first search path being used to generate the subtrees, and attempts to advance the path in a new direction from a previously passed vertex w. Recall that we viewed the endpoint of the search path as the contact vertex. But, in order to advance the search path from w along a new edge, we need to have available all the solutions for the subtree generated up to this stage as a function of the possible states of the contact vertex w. Furthermore, and this is critical, this is true, for the same reasons, for any outstanding contact vertex on the current depth first search path. One might try to circumvent this apparent need for solutions as functions of all the contact vertices by reparametrizing the problem. For example, we might maintain solutions only for every combination of every state of the endpoint of the search path and (say) every state of its first preceding contact vertex, on the

209

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'1

'6

'6

'6

Fig. 4. Left-right subtrees for Fig. 3

'5

'6

'6

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current search path. Then, one could try to bootstrap oneself back up the tree whenever the search path needs to retract to a previous contact vertex. But, this approach fails too. The same problem just reoccurs at an earlier contact vertex. The only resolution is to explicitly maintain optimal solutions, for every value of B',for the current left-right subtree, and for every combination of states of the current contact vertices on the search path. The resulting procedure will have complexity O(nB3"), where h is the height of the tree. For example, if h =log,(n), such as for a balanced d-ary tree, then the complexity is O(Bn[1+'o*(3)1.) For a balanced binary tree this is approximately O(Bn2.6). This contrasts with the O ( n B 2 )complexity of the previous bottom-up procedure. Let P ( v ) be the path from the root of T to v. Let { P ( v ) }be the ordered set of vertices on P ( v ) . Let Status ( P ( v ) ) be the vector of length equal to the cardinality of { P ( v ) } and whose j-th entry defines a state requirement on the j-th vertex of { P ( v ) } .Each vertex can be in any of three states: State 1. The vertex lies in the dominating set. State 2. The vertex is not in the dominating set, but is dominated. State 3. The vertex is not in the dominating set, and is not dominated. Let W(T,!(v), Status(P(v)), B') denote the weight of a maximum weight dominating set in T l ! ( v ) of , cost B', wherein the vertices in { P ( v ) } are in the states given by the status vector Status(P(v)). If a particular combination of state requirements is unrealizable, then the corresponding problem is undefined. For example, referring to Fig. 3, if v1 is in State 1 (dominating), then v 2 cannot be in State 3 (not dominated). We next turn to the dynamic programming relations for W. Let the tree be T, ! ( v )with tree path P ( v ) . Let (u, v ) be the edge to be added, which then forms the tree TA(u). The new path vertex set is { P ( u ) } = { P ( v ) } U { u } . We must determine the constrained maximum weight solution for every state of every (contact) vertex in this extended path set. For convenience, we introduce the following notation. Let x be a vertex, then: x(D) indicates x is dominating, x ( - D , D ) indicates x is not dominating, but dominated, x ( - D , -D) indicates x is neither dominating nor dominated. If S is an instance of a state vector, then S + x ( - .) denotes the state vector obtained by appending the indicated state requirement x(- . -) to S. In the following, S v(. . .) denotes an instance of the state vector for P ( v ) , and we wish to extend the corresponding solutions to the case where u is appended, i.e. to the state vector S + v(- . + u ( - . .). The appended vertex u can be either dominating or not dominating, and the recurrences vary accordingly. The recurrences for the extended solution W(T,!,(u),S v(. -) u(. .), B') follow. If u and u are both dominating, the recurrence is straightforward:

-

+

0

)

+

W(Th(u),S + u ( D ) + u p ) , B') = W ( T t ! ( v )S,

+ v(D),B' - c ( u ) ) + w ( u )

+

-

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If u is dominating, but v is not, then v must be dominated in the extended solution. There are two ways this can happen, as indicated by the recurrence: W(T&(u), s + v(-D, D)+ u(D),B ’ ) = max{W(Tl(v), S + v ( - D , D),B‘ - c ( u ) ) + w ( u ) , W(Tl!(v),S + v(-D, -D),B’ - c(u)) + W ( V ) - w ( u ) } . (15) If u is not dominating, but v is, then u must be dominated in the extended solution: W(T&(u), s + v ( D )+ u ( - D , D),B ’ ) = W(TI(V),s

+ v(D),B ’ ) + w ( u ) .

(16)

If both u and u are not dominating, but v is dominated, then: W(T&(u),S + v(-D, D)+ u ( - D , -D),B ‘ ) = W(Tl!(V),s

+ v(-D,

D ) , B’).

If neither u nor v are dominating or dominated, then: W(T&(u), s + v(-D, -D) = W(TI(V),s

+v(-D,

+ u(-D,

-D),B ’ )

-D),B ’ ) .

This shows how to advance the status vector thru the tree. The retraction process is simpler and merely involves coalescing states. That is, whenever the path P retracts from a vertex x , then x is removed from the status set. This is done by optimizing over all the states of x , for each status vector. A brief illustration of these calculations follows. A weighted tree is shown in Fig. 5a. Each vertex is labelled with its cost c and weight w ,as (c, w). The budget is B = 3. We show how to generate the solution for T&(v4)from those for T ;(v2 ). The contact vertices are v1 and v 2 , as shown in Fig. 5b. For convenience, we introduce the notation (s1, s2,

.

* *

;b, w )

to denote a solution of cost b and weight w ,where vertex vi is in state si. States are only specified for vertices on the current search path. The other undefined entries are indicated by a dash. T ‘ ( v 2 )Solutions. The solutions for T i ( v 2 )follow. Some state vector instances do not arise because they correspond to infeasible combinations of states or violate cost constraints. For example, for no instance is it possible to have v,(D) and v2(-D, -D).Again, no instance with vl(D), v 2 ( D )can arise, because it would exceed the budget. The T ; ( v 2 )solutions are: (1) (v1(-D, -D),v2(-D, -D),-; O , O ) , (2) (Vl(D), v2(-D, D),-; 3,2), (3) ( v , ( - D , D ) , v2(D), -; 1, 31, (4) (v1(-D, D),v2(D), -; 2, 31, ( 5 ) (Vd-D, -D)9 v2(-D, D),-; 192).

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J . McHugh, Y.Per1

(a) Tree with COSTS and WEIGHTS shown.

V3

/

\

\

”4’:

(b) T ; ( u , ) with contact vertices u , and u,.

Fig. 5. Left-right example.

These solutions correspond to the dominating sets: 0, {v,}, {vz}, {v2, u g } , {v3}, respectively.

T1(v4) Solutions. If we append the vertex v4 at the contact vertex vz, we obtain the solutions for T;)(v4). The extensions depend on whether v4 is dominating or not. v4 Dominating. The solutions obtained by appending v4(D) to solutions 2 and 4 above are beyond the budget. That is, their costs do not correspond to partitions of the budget, B = 3, so they are excluded. (1)’ Appending v4(D) to solution 1 gives:

(vd-D, - D ) , v z ( - D , D ) , -> u4(D); This follows from recurrence (15), specifically, the second term in the max{. . . , . . .} expression. The first term in that expression does not arise, because there is no solution for T1(v2)which has uz(-D, D ) and cost zero. (2)’ Appending v4(D) to solution 3 gives: (v,(-D, D ) , uz(D),

-7

v4(D); 3,4).

This uses recurrence (14). (3)’ Appending v4(D) to solution 5 gives:

(Vd-D,

-01, uz(-D,

D ) , -, v4(D); 3, 3).

This also uses (15), but for a different value of b.

Best location of service centers

213

v4 Not dominating. (4)’ Appending v4(-D) to solution 1 gives: (v1(-D, -D),V , ( - D , -D),-, v4(-D, -D);0,O).

This uses recurrence (18). (5)’ Appending v4(-D) to solution 2 gives: (Vl(D), V2(-D,

D), v4(-D, -0); 3,2). -7

This uses recurrence (17). (6)‘ Appending v4(-D) to solution 3 gives:

(%(-D,D),v2(D),

-9

V,(-D, D);1, 4).

This uses recurrence (16). (7)’ Appending v4(-D) to solution 4 gives: ( V 4 - 0 , D),v2(D), ->

Vd-0,

D);2, 4).

This also uses recurrence (16). (8)’ Appending ~ ~ ( to - solution 0 ) 5 gives: (Vd-0,

-D),V2(-D, D), v4(-D, -D);1,2). -9

This uses recurrence (17). These solutions correspond to the dominating sets: 0, {v2,v,}, {v3,v,}, {v,}, {v,}, {vz,v,}, { v d , respectively.

6. Concluding remarks In this paper, we have considered the cost constrained maximum weight tree domination problem. We have shown this problem to be NP-hard. For the special case of equal costs, we have described an O(nk2) dynamic programming procedure for the problem of finding a maximum weight k-dominating set. A pseudo polynomial dynamic programming procedure is then presented for the general constrained maximum weight domination problem for trees. The new left-right dynamic programming technique is then applied to obtain a more efficient pseudo-polynomial procedure. In applying the left-right technique we have to overcome some difficulties caused by the nature of the dominating set problem.

References [l] E.J. Cockayne, S.E. Goodman and S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975) 41-44. [2] M.J. Fischer, L.J. Guibas, N.D. Griffeth and N.A. Lynch, Optimal placement of identical resources in a distributed network, in: Proceedings 2nd International Conference on Distributed Computing Systems, Paris, April 1981, 324-336.

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[3] W.-L. Hsu, On the domination numbers of trees, North-western University, Dept. Ind. Eng. and Management Sci., Tech. Rept., July 1982. [4] S.T. Hedetniemi, S. Hedeteniemi and R. Laskar, Domination in trees: Models and algorithms, in: Y. Alavi et al., eds., Graph Theory and its Applications to Algorithms and Computer Science, (Wiley, New York, 1985) 423-441. [5] S. Hedetniemi and R. Laskar, A Bibliography on Domination, Clemson Univ., November 1985. 16) D.S. Johnson and K.A. Niemi, On knapsacks, partitions, and a new dynamic programming technique for trees, Math. Oper. Res. 8 (1983) 1-14. [7] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher, eds., Complexity of Computer Computations (Plenum, New York 1972) 85-104. [8) 0. Kariv and S.L. Hakimi, An algorithmic approach to network location problems part 1: The P-centers, SIAM J . Appl. Math. 37 (1979) 513-538. [9] J.A. Lukes, Efficient algorithm for the partitioning of trees, IBM. J . Res. Develop. 18 (1974) 217-224. [lo] K.S. Natarajan and L.J. White, Optimum domination in weighted trees, Inform. Process. Lett. 7 (1978) 261-265. [Ill Y. Per1 and Y. Shiloach, Efficient optimization of monotonic functions on trees, SIAM J . Algebraic Discrete Methods 4 (1983) 512-516. [I21 Y. Per1 and M. Snir, Circuit partitioning with size and connection constraints, Networks 13 (1983) 365-375.

Discrete Mathematics 86 (1990) 215-224 North-Holland

215

DOMINATING CYCLES IN HALIN GRAPHS* Mirosiawa SKOWRONSKA Institute of Mathematics, Copernicus University, Chopina 12/18, 87-100 Torun, Poland

Maciej M. SYStO institute of Computer Science, University of Wrocbw, Przesmyckiego 20, 51-151 Wroctaw, Poland Received 2 December 1988 A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Halin graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin graph H are characterized in terms of the cycle graph, the intersection graph of the faces of H.

1. Preliminaries The various domination problems have been extensively studied. Among them

is the problem whether a graph has a dominating cycle. All graphs in this paper have no loops and multiple edges. A dominating cycle in a graph G = ( V ( G ) ,E ( G ) ) is a subgraph C of G which is a cycle and every vertex of V(G)\V(C) is adjacent to a vertex of C. There are graphs which have no dominating cycles, and moreover, determining whether a graph has a dominating cycle on at most f vertices is NP-complete even in the class of planar graphs [7], chordal, bipartite and split graphs [3]. If a graph contains a dominating cycle then the problem is to find one of minimum length. Efficient algorithms exist for 2-trees [ 6 ] , 2-connected outerplanar graphs [7], permutation graphs [2] and series-parallel graphs [3]. A related domination problem results when each edge of a graph is required to have at least one endvertex on the cycle; the cycle is called a D-cycle. It was noticed in [lo] that the D-cycle problem is also NP-complete for planar graphs and then proved to be NP-complete also for chordal and bipartite graphs [3]. Efficient algorithms exist for series-parallel graphs and split graphs [3]. In this paper we study both dominating cycle problems in the class of Halin graphs, a subclass of Hamiltonian graphs. A Hafin graph is a plane graph obtained from a plane tree with no vertex of degree 2 by drawing a cycle through all endvertices [l,51. With every Halin graph we can associate a skirted graph and *This research was partially supported by the grant RP. 1.09 from the Institute of Computer Science, University of Warsaw. 0012-365X/90/$03.50 0 1990 -Elsevier Science Publishers B .V.(North-Holland)

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M . Skowroriska, M.M. Sysfo

every skirted graph can be obtained from a triangle by a simple procedure [S]. These two observations are crucial for determining a minimum dominating cycle in Halin graphs. In the next section, we present recursive formulae for finding a minimum dominating cycle in a Halin graph. The formulae can be easily modified to solve the D-cycle problem, too. Then in Section 3 we provide a complete characterization of dominating cycles and D-cycles in Halin graphs in terms of the associated outerplane graph. The dominating cycle problem for Halin graphs was independently solved in [12] by applying a general method designed for partial k-trees (a Halin graph is a partial 3-tree), whereas our approach uses a more natural recursive definition of Halin and skirted graphs.

2. Recursive formulae

Let T, be a plane tree with at least three vertices, rooted at x and with no vertex, except possibly the root x, of degree 2. The choice of the root in T, induces the natural father-son relation between adjacent vertices. Moreover, for every non-endvertex u of T,, the counterclockwise ordering of all sons of u as viewed from u determines their linear ordering with sons numbered in their increasing ordering, and also, the linear ordering of all endvertices of T,. Let y and z denote, respectively, the first and the last endvertex of T, in this ordering. We define a skirted graph G(x, y, z ) as a plane graph obtained from T, by drawing the path from y to z through all endvertices of T,. If deg,x = 2 then the left and right subtrees of T, are the trees T,, and T,,, where x, and x, are the first and last sons of x, respectively. If degzx 3 3 then the left subtree of T, is the tree T,,, where x, is the first son of x and the right subtree of T, is the tree T, = T,\T,, with n, = x as the root. The subgraphs G ( x , , y,, z/), G(x,, y,, z,) induced in G(x, y, z ) by the vertices of T,, and T,, with y,, z,, y,, z, as the first and last endvertices in T,,, T,,, respectively, we call the left and right subgraphs of G(x, y, z). They are also skirted graphs. Observe that yl = y and z, = 2. Let H be a Halin graph defined by a tree T and let x E V ( T ) be a nonendvertex of T adjacent in T to exactly one vertex of degree at least 3. Denote by y and z two endvertices of T adjacent to x in T such that yz is an edge of H. Then G(x, y, z ) = H\yz is a skirted graph and we call it a skirted graph associated with Halin graph H. In this section, by a path in a graph G we mean a subgraph P = ({xl,. . . ,x k } , {xixi+,, i = 1, . . . , k - 1)) of G. A subgraph F of G dominates a vertex x of G if x E V ( F ) or there exists y E V ( F ) adjacent to x in G. Let G I , . . . , G,,, n 2 1 , be subgraphs of a graph G. We define MINL{G,, . . . , G n } to be a graph Gi,1 6 i 6 n, with the minimum number of

Dominating cycles in Halin graphs

217

vertices among G1, . . . , G,,. If S is a property of graphs then by a minimum graph satisfying S we mean a graph with the minimum number of vertices which has property S. If G , , G2 are two graphs, then

G1 u G2 = (V(G1) u V(G*), E(G1) u E(G2)). L e t a , b , c , d , e , f E { O , l}. Definition 2.1. For three vertices x , y, z of a graph G, let CG(xab,yCd,z e f ) denote a minimum cycle C in G satisfying the following conditions: (i) x E V(C) iff a = 1; (ii) y E V(C) iff c = 1; (iii) z E V(C) iff e = 1; (iv) C dominates x iff b = 1; (v) C dominates y iff d = 1; (vi) C dominates z iff f = 1; (vii) C dominates every vertex u E V(G)\ { x , y , z } . Note that in Definition 2.1, if a = 1 ( c = 1 or e = 1) then b = 1 (d = 1 or f = 1, resp.). Therefore, if a = 1 ( c = 1 or e = 1) we can write x 1 (y' or zl, resp.).

Definition 2.2. For three vertices x , y , z of a graph G, let P$(zef) denote a minimum path P in C with endvertices x , y satisfying: (i) z E V(P) iff e = 1; (ii) P dominates z iff f = 1; (iii) P dominates every vertex u E V(G)\{z}.

Proposition 2.3. Let H be a Halin graph and let G = G(x, y , z ) denote a skirted graph associated with H . Then the cycle

cDoM = M I N L{ C~ (X~ y l, , z ~ ) cG(xl, , y l , zol), cG(xl, yol, ~ l ) cG(xl, , yol, zol), CC(xO',Y O 1 ,

ZO1),

PyCr(xO1) u ( { Y ,

z>,{ Y Z > ) >

is a minimum dominating cycle in Halin graph H .

Proof. Let H be a Halin graph and G(x, y , z ) = G be a skirted graph associated with H. If C is a minimum dominating cycle in H then C satisfies one of the following conditions: (1) x, Y , z E V(C); (2) x , y E V(C) and z $ V(C); (3) x , z E V(C) and y 4 V(C); (4) y , z E V(C) and x 4 V(C); ( 5 ) x E V(C) and y, z 4 V(C); (6) x, Y , 4 U C ) .

M. Skowrohka, M . M . Syslo

218

It is easy to see, by the definition of G(x, y, z ) , that in case (1) we have C = C G ( x ' , y 1 ,z ' ) , in case (2): C = C G ( x 1 , y ' ,zol), in case (3): C = CG(x',yo', z'), in case (4): c = P,G,(xO') u ({y, z } . {yz}), in case (5): c = CG(x',yo', zol) and in case (6): C = CG(x0',yo', zol). Hence

c = M I N L { C ~ ( ~yl,' , z]),cG(xl,yl, zol), cG(xl, yo', z'), CG(X"',Y O ' , ZO'), PycZ(X"')

cG(xl, yo', zol),

u ({Y, z}, {YZ})}.

0

Proposition 2.3 implies that the minimum dominating cycle problem for a Halin graph H can be transformed to finding some cycles and paths satisfying the conditions of Definitions 2.1 and 2.2, respectively, in a skirted graph associated with H. To this end, we now give recursive formulae which determine CG(xub, yCd,z e f )and P:(kgh) for i, j , k E { x , y, z } and a, 6, c, d , e, f,g, h E (0 , l} in a skirted graph G(x, y, z), as combinations of cycles and paths in the left C(xl,yl, z / ) and the right G(x,, y,, z,) subgraphs of G(x, y, z ) which are also skirted graphs. Since the smallest skirted graph is a triangle and the subgraphs G(x,,yl, zl) and G(x,, y,, z,) of G(x, y, z ) are defined with respect to the value of degG(x), we distinguish the following three cases: (1) degG(x) = 2 and ( V ( G ) (= 3, i.e., G is a triangle; (2) deg&) = 2 and IV(G)l> 3; (3) degG(X) 3 3. We present the formulae only for cases (1) and (2), since case (3) is very similar to case (2). Let C', P', C', P' denote cycles and paths in G(x,, y I , q), G(x,, y,, zr), respectively. Case (1). We have IV(G(x,,y l , zJ)l = IV(G(x,, y,, z,))l = 1, so xI = y / = zl = y and x , = y , = z, = z , and therefwe: CG(x',Y ' , 2') = ( { x , Y , z>,{ X Y , y z , P Z ( Z ' ) = ( { x , y , 21, { x z , z y > ) ;P$(Z0') = ( { x , Y } , { x y } ) ; P,",(Y') = ({x, Y , 21, { X Y , y z > ) ;m y " ' ) = ({x, 21, { x z > ) ; PycZ(x') = ( { x , y , 21, { Y X , x z } ) ; P$(X0') = ( { Y , 21, { Y Z ) ) .

The other cycles and paths are not defined in this case.

Yz

=

Y

z = z

z = z r

Fig. 1 . deg,n

Yz = 2,

=

Y

deg,x

3 3.

r

Dominating cycles in Halin graphs

219

Case (2). Let us assume that IV(G(x,, yl, z,))1, IV(C(x,, y,, zr))l > 1. (If IV(G(x,,Y / , z/))l= 1 (or IV(G(x,, y,, z,))l= 1) then xI = Y~= z/ = y (or x, = Y , = z, = z) the corresponding formulae are simple modifications of those given below.)

CG(x',y',

2')

= P ~ , ( y ' ) U Pi&')

U Q;

CG(x', yOd, z') = Pk,(yod) U Pi,&')

CG(x',Y

', 2")

U Q,

= p ~ , ( y ' ) UK , , ( z o f U ) Q,

d E (0,1);

f

CG(xl, yOd, zof) = pk,zl(~Qd) U p:,y,(zof) U Q,

E

(0,1);

d , f E {0,1);

where Q = ( { x , x l , x,, Y,, z,}, {-w,xx,, Z ~ Y , ) ) . Other cycles CG(xub, ycd, z e f )are not defined in this case.

pZ(zef)= MINL{P:,,(x:), ( ( 4x,,

p : , < x 3 , p:z,(xY)>u p:,,.(zef)u Zr,

Yr), {xx,,

Y,Z/>),

e, f E

(0,1);

pxCz(ycd)= ~ k , , ( y =u~MINL{P;,(~,~), ) p;,(xY), p;+(xol)}u ((4

q,Y , } ,

{w,Y r Z J ) ,

c, d E (0, 1);

u pi,(Y:)9 f$,(Z:) u p:,(yP'), p:&;) u P:+(YY), p:x,(zY)u p:&(Y;),p:x,(z:') u p:,(Yp'),

pZ<x'>= MINLv:,,(z:)

u pi,(Y;)} u ( { x , X I , xrl9 { X X / , xx,>); pg(xol)= MINL{P:,(X:) u p;,(x:), P:,(x:) u p;,(x;'), P:,(x:~) u p;Ad)> u (14, Y , } , { a Y r > ) ; p,",<x">= p:z,(x:') u p;,(xS') u ( { Z / , Y r > , { Z t Y r ) ) . p:X,(zY)

Using Proposition 2.3 and the above formulae one can design an algorithm for finding a minimum dominating cycle in a Halin graph H which works in O( IV(H)I) time. One can easily modify consideration above to hold for D-cycles in Halin graphs. First, in Definition 2.l(vii) and in Definition 2.2(iii), C and P should dominate all edges of G , resp. In a counterpart of Proposition 2.3 for D-cycles we remove the last but one term since no D-cycle in G ( x , y , z ) omits all three vertices x , y and z . Then, recursive formulae for shortest cycles and paths have to be modifieGwe leave this task to the reader.

3. Chacacte&ations The algorithm presented in the preceding section, although asymptotically very efficient, does not give any insight into the structure of (minimum) dominating cycles in Halin graphs. The purpose of this section is to characterize such cycles in terms of (fundamental) basic cycles generated by the interior (spanning) trees.

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M. Skowrotiska, M.M.Systo

The empty set, the set of all cycles and unions of edge-disjoint cycles of G, is a vector space called the cycle space of G, over GF(2), the field of integers modulo 2 with the vector addition of elements defined as the ring sum of sets of edges. In this section we shall sometimes use the same symbol to denote a cycle and the set of its edges. Let H = (V, E) be a plane Halin graph and % = { C i } ,denote the set of interior faces of H. Note that % is a cycle basis of H generated by the interior tree of H. We denote by G = B ( H , %) the cycle graph of H with respect to %. The graph G has the vertex set corresponding to % and two vertices are adjacent in G if the corresponding cycles share an edge. It is clear that G is isomorphic to H*-v,,~, where H* is the dual of H and ueXtdenotes the vertex of H* lying in the exterior of H. Therefore, we may assume that G is naturally embedded in the plane. Hence, there exists a one-to-one correspondence between the interior edges of H and the edges of G. It is easy to see that G is a 2-connected outerplane graph. Every cycle of H and in particular every Hamiltonian and dominating cycle in H is a linear combination of a subset of % over GF(2). For a cycle c of H, we denote by I ( c ) the subset of I such that c = Bier(=) Ci and say that Z(c) [or cycles in Z(c)] generates the cycle c. We first identify those subsets J of I which generate cycles in H. In what follows we assume that H is a Halin graph, % = {Ci}l is the set of interior faces of H, G = B ( H , %) is the cycle graph of H with respect to %, and J G I . We denote by G I J the subgraph of G induced by the vertices corresponding to cycles Ci for i E J.

Lemma 3.1. The edge set c=@liS,Ci forms a cycle in H subgraph G 1 J is connected.

if and only

if the

Proof. Note first that if G I J is disconnected then c cannot generate a connected subgraph of H, hence c is a union of cycles. To prove sufficiency we proceed by induction on IJI. If IJI = 1, for instance J = { j} then Cj is a cycle. Let us now assume that the lemma is true for every subset of I with less than k elements and let us consider J such that IJI = k and F = G J is connected. F contains at least one vertex j for which the graph F‘ = G I J’, where J’ = J - { j } is also connected. Therefore, by the inductive hypothesis, J’ generates a cycle, say c’. Let M denote the set of all neighbours of j in G I J. Since every two cycles in % have at most one edge in common, Cmn C, is an edge, say em, for every m E M. Note that em ( m E M ) does not belong to any other cycle of %, since every interior edge of H appears in exactly two cycles of %. We claim now that the set of edges K = { e m :m E M } constitutes a path in C,. Let us assume a contrario that there exists an interior edge f in C,, f $ F which lies between two connected components of F on C,. Hence, there exists exactly one cycle C, in % such that f E C, and 1 $ J. Since 1 and j are adjacent in G (which is a 2-connected outerplane graph), the edge lj disconnects G and also the graph

Dominating cycles in Halin graphs

G I J’ is not connected-a c’ G3 Cj is a cycle. 0

221

contradiction. Thus, F is a path on Cj and hence

One can easily demonstrate that Lemma 3.1 may not be true if H is an arbitrary plane graph, see [9] for details. If c = Ci is a cycle in H then its length satisfies

eie, IcI

= IJI

+ I{J,

I - J}I,

(1)

where {J,I - J} denotes the set of all edges in G whose one end belongs to J and the other is in I - J. We first recall from [ll] a characterization of Hamiltonian cycles (which trivially dominate also all vertices of the graphs) in Halin graphs in the terms of cycle graphs. If G = (V, E) is a plane graph then a subset W c V is called an independent face cover of G if every interior face of G contains a vertex of W but no two vertices of W belong to the same interior face of G.

Theorem 3.2. The edge set c = Bie,Ciforms a Hamiltonian cycle in H if I - J is an independent face cover of G .

if and only

In the characterization of dominating cycles to be developed, face independence of vertices in I - J will be dropped out and replaced by a forbidden configuration of vertices in J. Additionally, one exception will occur. For the sake of clarity we first prove some properties of dominating cycles in Halin graphs.

Lemma 3.3. If c = Bie, Ci forms a dominating cycle in H then G 1 J has no face each edge of which belongs to other faces of G I J .

Proof. Assume that G I J contains an interior face D of G. Hence, c does not pass the vertex w which corresponds to D in H. If D is surrounded in G 1 J by other faces of G then also no neighbour of w belongs to c. Therefore, w is not dominated by c-a contradiction. 0 Lemma 3.4. If the graph G 1 J is connected, no face of G I J is surrounded by other faces of G I J and every face of G contains an element of J then c = Bit,Ci forms a dominating cycle in H . Proof. Let J c I satisfy the conditions of the lemma. By Lemma 3.1, c is a cycle in H. We now show that c dominates every vertex v of H. If u is an interior vertex in H then G has a face D corresponding to u.By the assumption, D contains an element of J. If D has a vertex which does not belong to J then v belongs to c. Otherwise, since D is not surrounded by other cycles in G 1 J, v is adjacent to a vertex in c , so dominates v.

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M. Skowrokka, M . M . Systo

Fig. 2. Halin graph H (solid lines) and its cycle graph G (dotted lines). A dominating cycle in Hand the corresponding subgraph of G are in heavy lines.

Let u be an exterior vertex of H, and C, and C, denote the two basic cycles containing u (v is of degree 3). If k E J or 1 E J then u belongs to c. If k, 1 4 J then v is dominated by c since the face of G which contains the edge kl is covered by an element of J. 0 Fig. 2 shows a counterexample to the converse of Lemma 3.4; namely the subgraph G I J does not cover the three triangular faces of G. We now show that there is no other type of exception.

Theorem 3.5. The edge set c =

eieJ

C, forms a dominating cycle in H if and only if: (1) G I J is connected, ( 2 ) no face of G I J is surrounded by other faces of G I J , and (3) for every face D of G , either D contains an element of J or D is a pendant triangular face ( u , u , w), where deg(u) = 2, and J contains both the neighbours u’ and w ’of u and w , respectively, such that u’ # u, w ’# u, and u ‘ and w ’ belong to the exterior cycle of G. Proof. To show sufficiency, we have to demonstrate only that Lemma 3.4 holds also when its third assumption is relaxed to condition (3) of the theorem. Let D = ( u , u, w ) be a triangular face in G with deg(u) = 2 and v ’ and w ’ denote the neighbours of u and w , resp., on the exterior cycle of G. If D is not covered by the elements of J then J contains u’ and w’,what guarantees that the vertices of the triangle in H corresponding to u are dominated by the vertices which belong to the cycles which correspond to u’ and w ’ . Necessity. Conditions (1) and (2) follow by Lemmas 3.1 and 3.3. Let D be a face of G and u denote the vertex in H which corresponds to D.If u belongs to c , then at least one but not all of the vertices of D belongs to J. If u does not belong to c, then either all vertices of D belong to J or none. In the former case the

Dominating cycles in Halin graphs

223

proof is completed and in the latter case one can easily show that G - D has exactly one connected component since otherwise G I J would not be connected. Moreover D has to be a triangle since if D is of length at least 4 then an internal edge of the fan F in H corresponding to D would have a vertex which is not dominated by c. Hence, if c dominates all vertices of F then J must contain the neighbours of D on the exterior cycle of G. 0 We now characterize D-cycles in Halin graphs. A counterpart of the Lemma 3.3 for D-cycles has the following form.

Lemma 3.6. Z f c = BieJ Ci forms a D-cycle in H then no interior edge of G is also an interior edge in G I J .

Proof. If c = BiaJ Ci is a D-cycle and G I J contains an interior face D of G then the vertex u of H which corresponds to D does not belong to c. Let e = kl be an interior edge of G and D,, D, be the two faces of G which contain e. Let u, v denote the vertices of H which correspond to D,, 0,. If G I J contains D,, D, then u, v $ c, therefore e* = uv is not incident with c-a contradiction.’ 0 We are now ready to characterize D-cycles in Halin graphs.

eiSJ

Theorem 3.7. The edge set c = Ci forms a D-cycle in H if and only if: (1) G I J is connected, ( 2) every face of G contains an element of J , ( 3 ) no interior edge e of G is an interior edge in G 1 J , and (4)for every three consecutive vertices on the exterior cycle of G , at least one-of them belongs to J .

Proof. To prove necessity, it remains to show that conditions (2) and (4) are satisfied. Let us assume that c = Ci is a D-cycle in H. If there exist three consecutive vertices i l , i2, i3 on the exterior of G which do not belong to J, then the exterior edge in H which belongs to Ci,is not incident with c. Therefore, condition (4) must hold. To show that also condition (2) is fullfiled, let us assume that D is a face of G with no vertex of D belonging to J. Hence, G - D has only one component which shares an edge with D. Thus, G has three consecutive vertices (those of D ) which are not in J - a contradiction with (4). Sufficiency. Condition (1) guarantees that c is a cycle. By condition (4), c D-dominates every exterior edge of H. Let e be an interior edge of H. If e is not incident with an exterior vertex of H, then the corresponding edge of G is interior in G and, by condition (3), e is incident with a vertex of c. Let e be an interior edge which is incident with an exterior vertex of H and e* denote the corresponding edge in G. If any vertex of e* belongs to J then e is D-dominated by c, so let us assume that none of the endvertices of e* is in J . By condition (4),

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the vertices k and I incident with e* on the exterior cycle of G belong to J . If C, or C, is incident with e then c D-dominates e. Otherwise, let v denote the endvertex of e which is interior in G and D denote the face of G corresponding to v. By condition (2), J contains a vertex of D and hence, v belongs to c. Therefore also in this case e is D-dominated by c. 0

Acknowledgement The authors are indebted to the referees for their helpful suggestions which improved the presentation.

References [l] J.A. Bondy, Pancyclic graphs: recent results, in: A . Hajnal, R. Rado and V.T. Sos, eds., Infinite and Finite Sets, Coll. Math. SOC. JBnas Bolyai 10 (Kiado, Budapest and North-Holland, Amsterdam, 1975) 181-187. [2] Ch.J. Colbourn, J.M. Keil and L.K. Stewart, Finding minimum dominating cycles in permutation graphs, Oper. Res. Lett. 4 (1985) 13-17. [3] C h J . Colbourn and L.K. Stewart, Dominating cycles in series-parallel graphs, Ars Combin. 19 (1985) A, 107-112. [4] G. CornuCjols, D. Naddef and W. Pulleyblank, Halin graphs and the travelling salesman problem, Math. Programming 26 (1983) 287-294. [5] R. Halin, Studies in minimally connected graphs, in: D.J.A. Welsh, ed., Combinatorial Mathematics and Its Applications (Academic Press, New York, 1971) 129-136. [6] A. Proskurowski, Minimum dominating cycles in 2-trees, Internat. J . Comput. Inform. Sci. 8 (1979) 405-417. [7] A. Proskurowski and M.M. Syslo, Minimum dominating cycles in outerplanar graphs, Internat. J . Comput. and Inform. Sci. 10 (1981) 127-139. [8] M. Skowronska, The pancyclicity of Halin graphs and their exterior contractions, Ann. Discrete Math. 27 (1985) 179-194. [9] M.M. Sysio, An efficient cycle vector space algorithm for listing all cycles of a planar graph, SIAM J. Comput. 10 (1981) 797-808. [lo] M.M. Sysio, NP-complete problems on some tree-structured graphs: a review, in: M. Nag1 and J . Perl, eds., Proceedings of the WG’83, (Trauner, Linz, 1984) 342-353. [ l l ] M.M. Systo, On two problems related to the travelling salesman problem on Halin graphs, in: G. Hammer and D. Pallaschke, eds., Selected Topics in Operations Research and Mathematical Economics (Springer, Berlin, 1984) 325-335. [ 121 T.V. Wimer, Linear algorithms for the dominating cycle problems in series-parallel graphs, partial k-trees and Halin graphs, Congr. Numer. 56 (1987) 289-298.

Discrete Mathematics 86 (1990) 225-238 North-Holland

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FINDING DOMINATING CLIQUES EFFICIENTLY, IN STRONGLY CHORDAL GRAPHS AND UNDIRECTED PATH GRAPHS Dieter KRATSCH Sektion Mathematik der Friedrich-Schiller-Universitiit, Uniuersitatshochhaus, Jena, 6900, GDR

Received 2 December 1988 We study a new version of the domination problem in which the dominating set is required to be a clique. The minimum dominating clique problem is NP-complete for split graphs and, hence, for chordal graphs. We show that for two other important subclasses of chordal graphs the problem is solvable efficiently. We present an O(m . log n) algorithm for strongly chordal graphs and an O(n“)algorithm for undirected path graphs.

1. Introduction

An important task in the investigation of algorithmic graph problems is to find special classes of perfect graphs which admit polynomial time solutions of NP-complete graph problems and are as large as possible, or, on the other hand, to find special classes, for which the problem remains NP-complete, which are small. A lot of work was done for some special problems, as e.g. HAMILTONIAN CIRCUIT, MINIMUM CUT LINEAR ARRANGEMENT, SEARCH NUMBER (cf. [ll,161) and many others, but the “embroidery champion of the world” is the domination problem [17]. The problem itself and many variants in which the dominating set is required to be independent or a cycle, induce a subgraph which is connected or without isolated vertices are studied very extensively in the last years. The complexity status, i.e. whether the problem remains NP-complete or is solvable in polynomial time for the restricted class of graphs, is known for a lot of special classes. We refer the reader to [7,17,18]. The aim of this paper is to present results on the variant of the domination problem in which the dominating set is required to be a clique. This paper is organized as follows: Section 2 contains necessary notions and definitions. In Section 3 we summarize some results on the problem of determining a dominating clique. In Sections 4 and 5 we present polynomial time algorithms for locating a dominating clique of minimum cardinality for strongly chordal graphs and undirected path graphs, respectively. The main part of the paper shows how to use special structural properties of graph classes to design efficient algorithms. We will present an O(m . log n ) algorithm for strongly chordal graphs and an 0012-365X/90/$03.50 01990-Elsevier Science Publishers B.V. (North-Holland)

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O ( n 4 ) algorithm for undirected path graphs (where n denotes the number of vertices). We finish the paper with some remarks related to the notion of “separation in complexity” of graph classes, introduced by Johnson in [181.

2. Basic notions and properties

Throughout this paper we study only finite, undirected graphs G = (V, E) with no loops and no parallel edges, where V denotes the set of vertices of G; E, the set of its edges; n is the number of vertices; and m , the number of edges of G. A set V’ G V is said to be a clique if for all x , y E V ’ {x, y } E E holds; V’ is said to be a maximal clique if every proper superset of V’ is not a clique. A set V’ E V is said to be a dominating set if for all u E V - V’ there is a ‘u E V’ such that {u,v } E E. Let F be a finite family of non-empty sets. An undirected graph G is an intersection graph for F if there is a one-to-one correspondence between the vertices of G and the sets of F such that two sets have a non-empty intersection exactly when the corresponding vertices are connected by an edge in G. In this situation F is said to be an intersection model for G . A special issue of Discrete Mathematics [25] on interval graphs and related topics contains a lot of interesting results and attention is given to intersection graphs of subgraphs of a tree, e.g. subtrees and paths. The difference between vertex or edge intersection is studied there and also in [23]. We only consider the case of vertex intersection, i.e. the intersection of two subtrees is non-empty if they share at least a vertex. Now we mention some classes of perfect graphs. All definitions not given here are standard and can be found, e.g., in [2,15]. A graph G = (V,E) is perfect if for every subset V’ V, x(Gv.)= o ( G v . )holds, i.e. for each induced subgraph G,,., of G the chromatic number is equal to the size of a maximum clique. A graph G = (V, E) is chordal if every cycle of length exceeding three has a chord i.e. an edge joining two nonconsecutive vertices in the cycle. Chordal graphs are the intersection graphs of subtrees of a tree, i.e. the vertices of such an intersection graph correspond to subtrees and an edge connects two vertices if the corresponding subtrees share at least one vertex (cf. [12,151). In this manner three subclasses of chordal graphs can be defined: Undirected path graphs are the vertex intersection graphs of undirected path in a tree. These graphs are also called VPT-graphs [25] and UV-graphs [23]; in both cases to label the great difference to edge intersection graphs. Directed path graphs are the vertex intersection graphs of directed paths in a (rooted) directed tree, such that to each vertex corresponds a subpath from the root to a leaf of the tree [13]. In [23] they are called RDV-graphs.

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Directed Path

O(n+rn) Ip

(Intervah O(n+rn)

Fig. 1. Containment relations.

Znterval graphs are directed path graphs where the tree is itself a path. Usually they are defined as the intersection graphs of a family of intervals of the real line. Split graphs G = (V, E) are those graphs whose vertex set can be partitioned into a clique C and an independent set 1. It is known that G is a split graph iff G and G = (V, 8 ) are chordal graphs. In [lo] another important subclass of chordal graphs was introduced - the class of strongly chordal graphs: Let N(v)sf {y: {y, v} E E} and N[v]efN(v) U {v}. A vertex v is simplicia1 in G if N ( v ) forms a clique in G. A vertex v E V is simple if { N [ u ] :u E N[v]} can be linearly ordered by inclusion. It is known that a graph G is chordal iff every (nonempty) induced subgraph of G has a simplicia1 vertex (cf. [15]). Farber has also shown in [lo] that G is strongly chordal iff every induced subgraph of G has a simple vertex. (We can take this characterization of [lo] as the definition of the class.) It is also mentioned in [lo] that each simple vertex is also a simplicial one. For the definition of other classes of perfect graphs we refer to the well-known book of Golumbic [U]. Many results on the restriction of NP-complete graph problems to special classes of graphs can be found in [18], which is also a source for open problems in this field.

3. Dominating clique problems Shortly we consider the (existence) problem: find out, whether the graph has a dominating set which forms a clique, called dominating clique. DOMINATING CLIQUE (abbr. DC) ‘!Zf { G = (V, E): there is a V’ G V such that (1) for all u E V - V’ there is a v E V’ such that { u , v} E E, (2) for all vl, v 2 E V ’ {vl, vz} E E } . was introduced in [4,8,19]. The problem DC is NP-complete for perfect graphs (and even for weakly triangulated graphs) 141. This problem is of interest for

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graphs with diameter at most three, i.e. the maximum distance between two arbitrary vertices in a connected graph is at most three; since obviously all graphs with a dominating clique have diameter at most three. Recently Damaschke has shown that a chordal graph has a dominating clique iff it has diameter at most three [21]. DC is solvable in polynomial time for a lot of special classes of perfect graphs, e.g. comparability graphs and chordal graphs. The same is true for the problem of locating a maximum dominating clique (which is obviously NP-complete for the class of perfect graphs) [4]. The most interesting variant is the minimum dominating clique problem: locate a minimum dominating clique o r find out that there is no dominating clique. Its decision version MINIMUM DOMINATING CLIQUE (abbr. M D C ) s f

{(G, k ) : G = (V, E) graph, k that /V’I s k}

E Z,

G has a dominating clique V’ such

remains NP-complete for split graphs, which is a consequence of the basic polynomial transformation from the NP-complete problem VERTEX COVER to DOMINATING SET (cf. [6,7]). (Note that consequently MDC is NP-complete for chordal graphs!) For interval graphs and for comparability graphs the following statement holds: If a graph has a dominating clique, then it has a dominating edge or a dominating vertex, i.e. a dominating clique of size one or two. Consequently, there is a linear time algorithm for interval graphs, and an O(n . rn) algorithm for comparability graphs, for solving the minimum dominating clique problem. We should mention that, unlike a lot of other graph parameters, the following holds: If a graph has dominating cliques of size kl and k3 it does not necessarily have a dominating clique of size k,, for every k,, with k , < k , < k 3 , kiE Z.(see Fig. 2.) The main purpose of this paper is to settle the complexity status of the MDC problem for two important subclasses of chordal graphs, which have been open up to now: strongly chordal graphs and undirected path graphs. We present 9

‘ 0

4

5

I

2

Fig. 2. A graph with dominating cliques of sizes 3 ({l,2 , 3 ) ) and 5 ( { 4 , 5 , 6 , 7 , 8 ) ) , but none of size 4.

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polynomial time algorithms for both problems: O(m log n) for strongly chordal graphs and O(n4) for undirected path graphs. Thus the borderline between P and NP is fully determined for the classes in Fig. 1; the MDC problem is NP-complete for chordal and split graphs and is in P, i.e. solvable in polynomial time, for undirected path graphs, strongly chordal graphs, directed path graphs and interval graphs.

4. MDC for strongly chordal graphs For our algorithm we need a characterization of strongly chordal graphs which is similar to perfect elimination orderings for chordal graphs. Farber [lo] showed that G is strongly chordal iff it is possible to order the vertices vl, v2, . . . , v, in such a way that, for each i E (1, 2, . . . , n}, v, is a ,,.}. Such an ordering is called a simple simple vertex of G,gfG{,z, elimination ordering. Suppose that v l , v2, . . . , v, is a simple elimination ordering of V . Then it is a strong elimination ordering (str.e.0.) if it satisfies: For each i < j < k, if vl and v k belong to N,[v,] in GI then N,[vl]rN,[v,], where N,[v,] %fN[v,]n {vt, v,+*,. . . , v,} for 1 3 i - the neighbourhood in G,. In other words, for each vertex v, we require that the ordering of the vertices in N,[v,] is consistent with the ordering by inclusion of this nested family. Hence the rightmost element v, of N,[v,] in this ordering satisfies N,[v,]2 N,[u] for all u E N,[v,]. Farber [lo] shows that G is strongly chordal iff it admits a strong elimination ordering and describes an O(n5) algorithm for obtaining such an ordering. A simple elimination ordering can be obtained in time O(n’) [ l ] . Recently, the best known time bound of a recognition algorithm for strongly chordal graphs is O(m . log n). This algorithm of Paige and Tarjan additionally determines a strong elimination ordering [24]. For describing our MDC algorithm we need some additional concepts of graph theory: For V ’ E V we define N [ V ‘ ]= Uvcvr N [ v ] and N ( V ’ )= N [ V ’ ]- V ‘ . Thus V ‘ is a dominating set iff N [ V ’ ]= V holds. We shall say Vl E V dominates V, E V if N[V,] 2 V,, in the case of singletons we omit the braces, i.e., e.g. u dominates V, if N [ u ]3 V, holds. If there is a path in G containing u and v then the distance from u to v in G, denoted dc(u, v ) (or d(u, v ) when there is no ambiguity), is the smallest n such that there is a path of length n in G containing u and v. Otherwise, dc(u, v ) = m. G is connected if dc(u, v ) < 00 for every u, v E V . If G is connected then the diameter of G , denoted diam G, is max{dc(u, v): u, v E V } . We assume that the input for our MDC algorithm is a strongly chordal graph G = ( V , E ) and a strong elimination ordering u , , u2, . . . , v, of the vertices of G. The main idea of the algorithm is the following: We pass the str.e.0. from left to right and start with ‘processing’ the vertex v,. In each iteration, i.e. all steps of the algorithm when a fixed vertex v is under consideration, called ‘processing’, one vertex of the dominating clique is V,+,,

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determined, the algorithm stops since G has no dominating clique, or there are no changings. The candidates for the minimum dominating clique are stored in a set D. U = U(D)gf { u E V :d,(u, v ) a 2 for every v E D } is the set of vertices which have neither a neighbour in D nor belong themselves to D (the set of vertices which are ‘undominated’ with respect to D).Consequently, if u E U holds we have to incorporate a vertex of N [ u ] in D in order to make D a dominating set of G. On the other hand, D has to remain a clique in the whole process. Therefore we use the set S = S(D) N ( u ) as the set of all candidates s for extending the clique D in such a way that D U {s} remains a clique. Now we describe the algorithm.

Algorithm. MDC. Input: A strongly chordal graph G = (V, E) with strong elimination ordering U l , U 2 , . . . , v,. Output: “NO DOMINATING CLIQUE”, if G does not have a dominating clique. A minimum dominating clique, if G has a dominating clique. (0) Initially, S := V , U := V , D := 0 and i := 1. (1) Process vi and distinguish the following three cases: (1.1) u; E u. If N [ q ]f Sl = 0 holds then the algorithm stops after the output of “NO DOMINATING CLIQUE”. Otherwise perform: BEGIN j := max{s: v, E N [ v j ]n S}; D := D U { u , } ; u := u - “v,]; s := s n N ( u ~ ) . END If then U # 0 holds, increase i by one and go to (1). If U = 0 holds then the algorithm stops after the output of D as a minimum dominating clique. (1.2) u; E D. If S = 0 holds then the algorithm stops after the output “NO DOMINATING CLIQUE”. Otherwise perform: BEGIN k := max{s: u , E ~ S}; D := D U { v k } ;

u := u - N [ U k ] .

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23 1

END If U = 0 then output D. Otherwise output “NO DOMINATING CLIQUE” In both cases the algorithm stops. (1.3) vi$ D U U. Increase i by one and go to (1). We have to mention that this algorithm uses the basic idea of Farber’s algorithm for DOMINATING SET [9], but this idea has to be refined here. For this reason the correctness proof of Farber’s algorithm, which is easy to understand, cannot be transfered to our algorithm. We only give a sketch of the proof here, without any of the technical details. The interested reader is refered to [20] which also contains a PASCAL-like description of the algorithm.

Theorem. The algorithm MDC halts after O(n + m ) operations and is valid. The time bound is obvious. The correctness proof needs some additional notions and some technical lemmata. The main lemmata study the set { u l , u2, . . . , u r } . Thereby this set contains all those vertices uk, k E (1, 2, . . . , r } , with the property that during processing vi a new vertex w, is taken into D, except that processing vi is the last iteration for the considered performance of the algorithm.

Distance Lemma. For every ui, uj, i, j E { 1, 2, . . . , r } , i # j , we have: dG(ui,uj) = 3. Thus { u l , u 2 , . . . , u,} is a set of vertices of G with pairwise disjoint neighbourhoods (cf. [9]) and the size of a minimum dominating set of G, denoted y(G), is at least r.

Neighbourhood Lemma. Every clique D;, j E (1, 2, . . . , r } , which dominates { u l , u2, . . . , u,} and is of cardinality at most r, satisfies “Dl] G “ ( ~ 1 7

. , wj}].

~ 2 *,

*

Thus the clique {wl, w2,. . . , wj} is optimal in this sense of domination among all cliques dominating { u l , u2, . . . , uj} which have j vertices. A careful analysis of all cases of termination of the algorithm verifies then the correctness. It would be interesting to find a shorter correctness proof for this algorithm since “it seems to be valid from definition”. We should point out here that the strong elimination ordering can be constructed for directed path graphs in linear time [9]. Thus we get a linear time algorithm for MDC on directed path graphs, interval graphs and trees. For

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strongly chordal graphs we use the algorithm of Paige and Tarjan [24] to construct a str.e.0. in time O ( m log n ) . Thus MDC is also solvable in time O ( m . log n ) for strongly chordal graphs. Finally we mention the most interesting consequence of the Distance Lemma: Whenever the strongly chordal graph G has a dominating clique we denote by yclique(G)the size of a minimum dominating clique. Since we construct a minimum dominating clique of size r 1 for each clique dominated strongly chordal graph we have, because of the Distance Lemma, y ( G ) s yclique(G) s y(G) 1. This suggests to conjecture equality and surprisingly this is really true.

-

+

+

Theorem. For every clique dominated strongly chordal graph G h o l h :

The proof is not hard if one uses all structure lemmata of the proof of correctness [20]. A different proof is given in [21]. Similar results may not be expected for chordal graphs.

Theorem. There is a chordal graph G = (V, E ) with y(G) = 2, yclique(G)= k and I V J = 2 k + 1 ,forevery k 2 3 . Proof. We give a clique tree for G, i.e. for every vertex 21 E (1, 2, . . . , 2k + l} the corresponding subtree is the induced subtree of those vertices which contains v. (See Fig. 3.) It is easy to verify that the graph G has all the properties stated in the theorem. (1, 2k + 1} is a minimum dominating set and {1,2,3, . . . ,k} is a minimum dominating clique. 0 Even if we restrict the diameter of the chordal graph to be at most 2 we cannot force the equality of y ( G ) and yclique(G).

Theorem. There is a chordal graph C = (V, E ) with diam(G) = 2, y(G) = 2, yclique(G) = k and IVI = (k + 1)’+ 1, for every k 2 3. 2.3,_.., k - l .k,2k+l

Fig. 3.

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a.a

233

v

k ' k l '\2 '..'"kk

Fig. 4.

Proof. We give the graph G = (V, E) for k 3 3: def

V = {q j i: , j N, ~ 1S i , j s k } U { a ; 6 ) U {a,, 6,: i E N, 1~ i s k } ,

Esf{{vij,~ ~ ~ } : ( i , j ) # ( k , Z ) } U { { a{,6a,~6 }i ,} : 1 ~ i s k } U { { a , v i j } ,{ b , vi,}: l s i , j < k } U { { a i , v,,}, {vij,6,): lsz,j s k } . We show that G is chordal by giving a clique tree of G (See Fig. 4). {a, 6) is a minimum dominating set of G and {vii: 1< i < k } is a dominating clique. It remains to show that there is no dominating clique C of G with )C1< k. C cannot contain a and 6 and without loss of generality we assume 6 ff C. But then the independent set {61, 62, . . . , bk} has to be dominated by {vij: i, j E N, 1< i < k } U {61, 62, . . . , bk} which obviously requires k vertices. Finally note that for every i, j E { 1, 2, . . . , k } , i # j, there is a path a, - vij- bj, thus G has diameter 2. 0

5. MDC for undirected path graphs

We want to show in this section that it is possible to locate a minimum dominating clique in a given undirected path graph, i.e. a vertex intersection graph of paths in a tree, in polynomial time. It seems to be a nice example for showing the power of carefully studying the structural properties of graphs and extensively looking for the help of special cases. For each maximal clique C of G we transform the original problem into the problem of finding a minimum hitting set of a given collection of subsets of a set DZC. The problem HITTING SET is known to be NP-complete [ll], but because only paths are allowed in the intersection model we are again in a special case: This special case of HI'ITING SET can be solved by transforming it into the edge cover problem for a special graph constructed from the collection of subsets in

mc. The problem EDGE COVER is solvable in polynomial time by determining a maximum matching of the graph. So we get an algorithm which solves the minimum dominating clique problem in time O(n2-5 n - rn) if a characteristic

+

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tree, i.e. a special intersection model (definition below) is part of the input. Otherwise such a model can be computed in time O ( n 4 ) [14]. First we introduce some notions. The set of all maximal cliques of G will be denoted by (3 and the set of the maximal cliques containing a vertex v will be denoted by (8,. Clearly, (qU is the set of maximal cliques of G N U , the , subgraph of G induced by N [ v ]:= { w E V : {v, w } E E } U {v}.

Theorem [14,23]. A graph G is an undirected path graph if and only if there exists a tree T whose set of vertices is G ,so that for every vertex v E V , TcSu,the subgraph of T induced by the vertex subset @, is a path in T . Such a tree T will be called characteristic tree of G . [14] presents an algorithm which checks in time O ( n 4 )whether a given graph is an undirected path graph by constructing a characteristic tree T of G, if one exist. To simplify the notation we will use the same name for the vertices of T as for the corresponding elements of (si. Now to our question: How to locate a dominating clique of G in T?

Lemma. Let T be a characteristic tree of the undirected path graph G . Then a clique D V is a dominating clique of G if and only if every leaf C E (3 of T contains at least one vertex of D .

Proof. Let D E V be a dominating clique. Each leaf C E (3 contains at least one vertex v (a simplicia1 vertex of G) which is only a vertex of this maximal clique, since otherwise C is a subset of its father in the tree which is not allowed in a characteristic tree. (Note that C is a maximal clique!) Since D is a dominating set, u belongs to D or a neighbour u of v belongs to D . In both cases one vertex of C, v or u, belongs to D. Hence, D contains one vertex of each leaf C E (3 of T. Now let every leaf C E (3 of T contain a vertex of D. For a vertex w E D the vertex subset CYW E @ of T induces a path in T. Now in a tree there is exactly one path between two given vertices and the paths from a fixed vertex C (which we can call a root) to all leaves of the tree meet all vertices of the tree [2]. Thus D has one vertex in common with each element of @, i.e. with each maximal clique of G. Therefore, D is a dominating set of G. 0 The Lemma shows how to locate a minimum dominating clique: For each element C E G which is a dominating set we determine a minimum subset S c C with IS n CI 3 1 for all A E 2,where 2 is the set of all leaves of T. This problem can be reformulated as a problem for a collection of subsots: For C E (3 let rXn, be the set {A r l C: A E 2}.We ask for a minimum hitting set S E C of a, i.e. S satisfies IS n MI 3 1 for each M E rXnc.

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The problem HITTING SET (problem [SP 81 of [ll])is NP-complete, but now we have again a special case of an NP-complete problem. Since G is an undirected path graph each vertex v of a maximal clique C belongs to at most two leaves of T; otherwise (3” does not form a path in T. This means for the collection of subsets: each element of the set C belongs to at most two subsets of the collection 2Jlc. Fortunately, this transforms our subproblem, which we have to solve for each dominating set C E @ , into an edge cover problem, in a very natural way: H,sf(2Jlc, F,) is the intersection graph of the family 2Jlc, a collection of subsets of C. For each edge { B l , B,} E Fc we choose an arbitrary vertex of B1n B2 # B as label of this edge. Each isolated vertex of Hc corresponds to a set B E 2Jtc with B f l (U(%Rc { B}))= 0. Therefore each minimum dominating clique S c C contains exactly one vertex of B (an arbitrary one). Hence, for finding a minimum hitting set of (C, 2XC) we can omit all isolated vertices from H,, the resulting graph is called fi, = F,). Then the following lemma is an obvious consequence of the construction of H,, as the intersection graph of 2Jtc.

(ac,

Lemma. If F’ E Fc is a minimum edge cover of H , then the set of labels of the edges in F’ is a minimum hitting set of (C, and F’ U F is a minimum hitting set of (C, 2Jlc), where F contains exactly one vertex of each set B E 2Jlc-

a,-)

a,.

As is well known, the minimum edge cover problem has a strong connection to the maximum matching problem.

Theorem [2]. Let G be a graph without isolated vertices. Given a maximum matching E’ of G, a minimum covering is obtained by adding an edge incident to v for each vertex v, which is not incident with an edge of E ’ . Now we are able to describe an algorithm which locates a dominating clique of minimum cardinality: Algorithm. MDC.

Input: An undirected path graph G = (V, E) and a characteristic tree T of G. Output: “NO DOMINATING CLIQUE”, if G has no dominating clique. A minimum dominating clique U,otherwise. (0) If there is no dominating set C E C9 STOP with answer “NO DOMINATING CLIQUE” (1) u:=v

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do: (2) For each dominating set C E (2.1) Construct Hc and Gc; (2.2) Determine the label set S' of a minimum edge cover EL; (2.3) Form a new set S by adding to S' for each isolated vertex B of Hc an arbitrary vertex of G belonging to B ; (2.4) If IS1 < IUI then U:= S (3) Write U , STOP. l'he correctness of the algorithm is a consequence of the considerations above. What about its time complexity? Obviously step (0) can be done in time O(n * m ) [4]. Undirected path graphs, as special chordal graphs, have at most n maximal cliques (cf. [15]). Thus the loop in step (2) is carried out at most n times. How much time does the algorithm need for one execution of the loop, for a fixed C E The edges of Hc are in unique correspondence to the vertices of G, i.e. Hc has at most n edges. The vertices are in unique correspondence to the leaves of the characteristic tree T, i.e. Hc has at most n vertices. The sum of the cardinalities of all sets in 2Jlc is at most 2n, since each vertex of G can occur in at most two leaves. So the algorithm passes through the list of elements of the leaves and forms the list of vertices and the adjacency list of H,, together with the labels for the edges. We do not want to go into the details, but this can be implemented to run in time O(n). Micali and Vazirani [22] have shown that a maximum matching of a given graph G = ( V , E) can be determined in time O(lVl4 . IEI). Hc has at most n vertices and at most n edges, since we get it by removing all isolated vertices of H,. (Clearly, this removing need not to be done really by the algorithm; it only does not consider isolated vertices.) Therefore, a maximum matching of Hc can be determined in time O ( ~ Z ' . ~ ) , which is already presented by the corresponding label set S'. All the other operations of the loop can be done in time O(n), thus one execution of the loop (step (2)) takes time O(n'.')). Hence, the algorithm runs in time O(n2.5+ n * m ) .

a?

Theorem. There is an algorithm which solves the minimum dominating clique problem in time O(n2.' + n m ) for a given undirected path graph together with a characteristic tree. If the input is only the graph, the problem is solvable in time

o(~~). 6. Concluding remarks The MDC problem is the first problem which separates in complexity the undirected path graphs from the class of chordal graphs, i.e. MDC is NP-

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complete for chordal graphs but solvable in polynomial time for this subclass. The notion of “separation in complexity” of graph classes was introduced by Johnson [18], together with some questions to open separations among them the one of undirected path graphs from the chordal graphs; now an example is MDC. The most interesting question in this field seems to be the following: Find a problem which is NP-complete for chordal graphs, but solvable in polynomial time for all the three important subclasses, namely, split graphs, strongly chordal graphs and undirected path graphs. Unfortunately, MDC is NP-complete for split graphs. This is not surprising, since we know only one separation in complexity of split graphs from the chordal graphs, the clustering problem, introduced and extensively studied in [6]. Perhaps, this problem, is a good candidate since the reduction of [6] shows that it remains NP-complete for the class of such chordal graphs, which do not belong to one of the subclasses: split, strongly chordal and undirected path graphs.

References [l] R.P. Anstee and M. Farber, Characterizations of totally balanced matrices, J. Algorithms 5 (1984) 215-230. [2] C. Berge, Graphs (North-Holland, Amsterdam, 1985). [3] A. Brandstadt, On the domination problem for bipartite graph, in: R. Bodendieck and R. Henn, eds., Topics in Combinatorics and Graph Theory (Physica, Heidelberg, 1990) 145-152. [4] A. Brandstadt and D. Kratsch, Domination problems on permutation and other graphs, Theoret. Comput. Sci. 54 (1987) 181-198. [5] A. Brandstadt and D. Kratsch, On the restriction of some NP-complete graph problems to permutation graphs, in: L. Budach, ed., Proceedings of the FCT ’85 Conference, Lecture Notes in Computer Science 199 (Springer, Berlin, 1985) 53-62. [6] D.G. Corneil and Y. Perl, Clustering and domination in perfect graphs, Discrete Appl. Math. 9 (1984) 27-39. [7] D.G. Corneil and L.K. Stewart, Dominating sets in perfect graphs, this volume. [S] M.B. Cozens and L.L. Kelleher, Dominating cliques in graphs, this volume. [9] M. Farber, Applications of L.p. duality to problems involving independence and domination, Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 1982. 1101 M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (1983) 173-189. [ll] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [12] F. Gavril, The intersection graphs of subtrees in a tree are exactly the chordal graphs, J. Combin. Theory Ser. B 16 (1974) 47-56. 1131 F. Gavril, A recognition algorithm for the intersection graphs of directed paths in directed trees, Discrete Math. 13 (1975) 237-249. [14] F. Gavril, A recognition algorithm for the intersection graphs of paths in a tree, Discrete Math. 23 (1978) 211-227. (151 M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [16] D.S. Johnson, The NP-completeness column: An ongoing guide, J. Algorithms, series of papers, started in Vol. 2 (4) (1981). 1171 D.S. Johnson, The NP-completeness column: An ongoing guide, 10th edition, J. Algorithms 5 (1984) 147-160. [18] D.S. Johnson, The NP-completeness column: An ongoing guide, 16th edition, J. Algorithms 6 (1985) 434-451.

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[19] L.L. Kelleher, Domination in graphs and its applications to social network theory, Ph.D. Thesis, Northeastern University, Boston, MA, 1985. [ZO] D. Kratsch, A linear time algorithm for the minimum dominating clique problem on strongly chordal graphs, Forschungsergebnisse der FSU, Jena, N/87/29, 1987. [21] D. Kratsch, P. Damaschke and A. Lubiw, Dominating cliques in chordal graphs, submitted to Discrete Math. [22] S. Micali and V.V. Vazirani, An O ( V * E ) algorithm for finding maximum matching in general graphs, in: 21st Annual Symposium on Foundations of Computer Science (Syracuse, 1980), (IEEE Computer Society Press, New York, 1980) 17-27. [23] C.L. Monma and V.K. Wei, Intersection graphs of path in a tree, J. Combin. Theory Ser. B 41 (1986) 141-148. [24] R. Paige and R.E. Tarjan, Three partition refinement algorithms, SIAM J. Comput. 16 (1987) 973-989. [25] Special issue on interval graphs and related topics, Discrete Math. 55 (2) (1985) 113-244.

Discrete Mathematics 86 (1990) 239-254 North-Holland

239

ON MINIMUM DOMINATING SETS WITH MINIMUM INTERSECTION Dana L. GRINSTEAD* and Peter J. SLATER* * * Department of Mathematics and Statktics, The University of Alabama in Huntsville, Huntsville, A L 35899, USA

Received 2 December 1988 In the developing theory of polynomial/linear algorithms for various problems on certain classes of graphs, most problems considered have involved either finding a single vertex set with a specified property (such as being a minimum dominating set) or finding a partition of the vertex set into such sets (for example, a partition into the maximum possible number of dominating sets). Alternatively, one might be interested in the cardinality of the set or the partition. In this paper we introduce an intermediate type of problem. Specifically, we ask for two minimum dominating sets with minimum intersection. We present a linear algorithm for finding two minimum dominating sets with minimum possible intersection in a tree T, and we show that simply determining whether or not there exist two disjoint minimum dominating sets is NP-hard for arbitrary bipartite graphs.

1. Introduction Given a graph G = (V, E), a vertex subset S c_ V is independent if no two vertices in S are adjacent; @ ( G )will here denote the maximum number of vertices in an independent set; G is k-colorable if V = V, U V, U * . - U V, where each is independent; and the chromatic number x(C) is the minimum k such that C is k-colorable. Vertex subset D E V is a dominating set if each v E V - D is adjacent to at least one vertex in D; y(G) here denotes the minimum number of vertices in a dominating set; G is k-domatic if V can be partitioned into k sets V,, V,, . . . , V, such that each & is a dominating set for G; and the domatic number of G is the maximum k such that G is k-domatic. Determining if @ ( G ) > K is an NP-complete problem even for cubic planar graphs (Garey, Johnson and Stockmeyer [17]); deciding if G is K-colorable is NP-complete even for planar graphs of maximum degree four (Karp [23]); deciding if y ( G ) s K is NP-complete for planar graphs of maximum degree three (Garey and Johnson [19]) and for bipartite graphs (Dewdney [16]); and for the domatic number problem introduced by Cockayne and Hedetniemi [13] determining if the domatic number of G is at least K is NP-complete (Garey, Johnson and Tarjan [18]). Much of the extensive amount of research in graph algorithms has been concerned with developing polynomial time algorithms for NP-complete problems restricted to appropriate classes of graphs. Indeed, many linear time algorithms

x

* Research supported in part by the U S . Office of Naval Research Grant N00014-86-K-0745. ** Dr. Slater thanks those at Clemson University for their hospitality during the Fall of 1987 and gratefully acknowledges support under the U .S. Office of Naval Research Grant N00014-86-K-0693. 0012-365X/90/$03.50 0 1990 -Elsevier Science Publishers B.V. (North-Holland)

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have been developed. As examples, we have linear algorithms for minimum domination in trees (Cockayne, Goodman and Hedetniemi [12]), R-domination in trees (Slater [32]) and block graphs (Chang and Nemhauser [14]), independent domination in trees (Beyer, Proskurowski, Hedetniemi and Mitchell [S]), independent domination and total domination in seriesparallel graphs (Hedetniemi, Laskar and Pfaff [28]), domination in seriesparallel graphs (Kikuno, Yoshida and Kakuda [24]), locating-dominating sets in seriesparallel graphs (Colbourn, Slater and Stewart [ 15]), and dominating subforests of a tree (Lawler and Slater [26]). Many other domination related algorithmic papers have appeared, as have many related to finding independent sets in graphs. In general, much work has been done to develop polynomial/linear algorithms for finding a (minimum/maximum) vertex or edge set S with a specified property. Further, problems involving partitions of vertex set V have been investigated. For example, as reported in Johnson [22], Bodlaender [9] has developed a k-chromatic number algorithm for partial h-trees that is polynomial for fixed k and h. In fact, a general theory of linear algorithms is being developed. Especially notable is the thesis of Wimer [35],with other notable papers including Takamizawa, Nishizeki and Saito [33], Bern, Lawler and Wong [6], Arnborg and Proskurowski [l], and the work of Robertson and Seymour, including [29]. In this paper we introduce an intermediate type of problem. The general type of problem is defined by asking for more than one vertex set with required properties, but not necessarily for a partition of V. A general treatment of such problems is contained in Grinstead [20]. Some previous work on finding a pair of disjoint dominating sets having some property P appears in Bange, Barkauser and Slater [2-51. Here we relax the requirement of disjointness and ask for two minimum dominating sets with minimum possible intersection. We let M , ( G ) denote the minimum cardinality of the intersection of two minimum dominating sets in G. Note that if G has a unique minimum dominating set D , then M , ( G ) = y ( G ) = )DI. In the next section, we show that simply determining if there exist two disjoint minimum dominating sets is NP-hard for arbitrary bipartite graphs. Section 3 contains a linear algorithm for computing M , ( T ) for a tree T. The algorithm works by a single pass over the endpoint list of T (described in Section 3). Then in Section 4, we note that two such sets can actually be obtained by an additional backward pass through the endpoint list, and briefly discuss how the procedure can be extended to cover series-parallel graphs. 2. Determining M,,(G) is NP-hard

Having defined M J G ) to be the minimum cardinality of the intersection of two minimum dominating sets in G , we can pose the following decision problem. Given a graph G and a nonnegative integer K, is M , ( G ) s K? In this section we show that simply determining whether or not M , ( G ) = 0 is NP-hard for bipartite graphs G. As was pointed out to us by a referee, our DISJOINT MINIMUM DOMINATING SETS problem is in NPNP, the class of languages recognizable

On minimum dominating sets with minimum intersections

24 1

nondeterministically in polynomial time with the aid of an oracle from NP. Given an oracle to test if y ( G ) = k , such a nondeterministic algorithm is as follows: Guess at two sets D1 and 0 2 and verify that they are disjoint dominating sets. Using the oracle, verify that each has cardinality y ( G ) . We next describe a polynomial time reduction from NOT-ALL-EQUAL 3SAT (see Schaefer [30]) to the problem of determining if M J G ) = 0 for bipartite G , which implies that this DISJOINT MINIMUM DOMINATING SETS problem is NP-hard. The NOT-ALL-EQUAL 3SAT problem appears in Garey and Johnson [19, p. 2591, and [19] contains a complete discussion of the theory of NPcompleteness. NOT-ALL-EQUAL 3SAT

Instance: Set U of variables, collection C of clauses over U such that each clause c E C has IcI = 3. Question: Is there a truth assignment for U such that each clause in C has at least one true literal and at least one false literal? DISJOINT MINIMUM DOMINATING SETS

Instance: Graph G . Question: Does G have two disjoint minimum dominating sets? Let U = {ul, u2, . . . , u,}. Given C = c1 A c2 A * * . A c, where ci = (silv si2 v si3) and each sij is uh or iih for some 1G h G n, we show here how to construct a graph G (in time polynomial in m) such that U has a NOT-ALL-EQUAL 3SAT truth assignment for C if and only if G has DISJOINT MINIMUM DOMINATING SETS. Hence a polynomial time algorithm for the latter decision problem would imply a polynomial time algorithm for the former known NP-complete problem. The graph G will contain 3m copies of the graph H in Fig. 1. We need to note that H is bipartite with vertices labelled uii and ii, in the same set of the bipartition, and the only vehices in H adjacent to other vertices of G will be ujj and iiij (so that the degrees in G satisfy deg,(v) = deg,(w) = 7 and deg,(x,) = 2 for 1d i G 14). Each copy of H in G will be called an H-subgraph with designated vertices uij and iijj.Letting D be any minimum dominating set for G, the following observations are easy to verify. Set D must contain at least one of ujj and ziji (consider xl, x2, x3); ID n V(H)I 3 3; and if ID n V(H)I = 3 then D n V ( H ) is {v, w ,uij), {v, w,ii,}, {uij, w,x14) or { i i j j , v,~ 1 3 ) . In particular, if G has two disjoint minimum dominating sets, then one contains {uij,w , ~ 1 4 )and the other contains {iijj,v,x I 3 } . For each clause ci= (sil v si2v si3)we construct a graph Gi on 58 vertices like the one illustrated in Fig. 2 as follows. Suppose sil= u, o r ii,, si2= ub or i i b , and si3= u, or U,. Let Gi contain three copies of H with designated vertices ui, and iii,, uib and i i j b , and uicand i,. Each of cli and cZiis connected to ui, if sil= u,, to iii, if sil = ii,, to uib if si2= U b , to iiib if si2 = f i b , to ujc if si3= u,, and to di, if

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H

Fig. 1. A ‘building block’ H for graph ti

sll=ii,. Each of cgr and c41 is connected to the three designated vertices not adjacent to c l r and c Z I . Let G be the graph containing disjoint copies of GI, G 2 , .. . , G, to which we add the following vertices and edges. For each occurrence of a U h or iih with 1G h G n in distinct clauses c, and cl add four vertices of degree two as follows. Assume s,, is u h or i i h and assume sIt is u h or i i h where l ~ i < j ~andm 1 S r, t G 3. Let two of the four vertices be adjacent to ii,h and to U,h, and let the other two be adjacent to U,h and to ii,h. The graph G is illustrated in Fig. 3 for

c = (U1 V U 2 V Ug) A ( f i x

V U 2 V Uq) A (U2 V Ug V U s ) .

uzl

Note that G contains 58m vertices in G,. Further, each uiIor U, is adjacent to at most 2(m - 1) vertices in G - uE1G,, and so there are at most 6m2 52m vertices in G, and G can be constructed from C in time polynomial in m.

+

Theorem. Graph G has two DISJOINT MINIMUM DOMINATING SETS (that is, M J G ) = 0 ) if and only if U has a NOT-ALL-EQUAL 3SAT truth assignment for C. Proof. First, assume there is a NOT-ALL-EQUAL 3SAT truth assignment for

s,

‘li

‘2i

‘3i

‘4i

Fig. 2. A larger ‘building block’ ti, for ti for C, = (a, v u6 v a,)

On minimum dominating se& with minimum intersections

243

C. Let S1 consist of those ui in U that receive the value true and S2 = U - S1. Construct two vertex subsets D1 and 0 2 of V ( G ) as follows. For each uii in G if uj E S1 then place uii and the corresponding w and ~ 1 of 4 its building block H (as in Fig. 1) in D1 and place iiii and the corresponding v and x13 in 0 2 , and if ui E S2 then place uii, w and x14 in 0 2 and iiii, v and x13in D1. For example, if C = ( u l v u2 v u3)A (el v ii2 v ii4) A (u2v u3 v us) then one NOT-ALL-EQUAL 3SAT truth assignment is to let S1= { u l , u4, u s } , and the nine darkened uii and

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244

ii, in Fig. 3 are placed in D1 (with two additional vertices from each H) with the nine undarkened ui, and Uij going into 0 2 . As previously noted, every minimum dominating set for G must contain at least three or more vertices from each H. Thus y ( G ) 2 9m. Clearly D1 n 0 2 = 0 and ID11 = 1021 =9m, and it is straighforward to see that each of D1 and 0 2 dominates G. Hence, G has two DISJOINT MINIMUM DOMINATING SETS. Conversely, assume G has two DISJOINT MINIMUM DOMINATING SETS, say D1 and 0 2 . As noted, for each uij either D1 or 0 2 contains uij and two specified vertices from its H-subgraph, and the other contains ii, and two specified vertices from this same H subgraph. Suppose Uik and u j k are vertices of G with i # j (for example, u12 and u32in Fig. 3). To see that Uik and iijk cannot both be in D1 or both be in 0 2 , note that if Uik and iijk are in D 1 (and so iiik and u j k are in D2), then D1 must also contain the two vertices x and y of degree two adjacent to iiik and u;k (for example, vertices x and y in Fig. 3). Recognizing that one of the two specified vertices in D 1 from the H-subgraph of ii;k dominates u ; k , the set D1 - x - y + iiik would also be a dominating set, contradicting the minimality of D1. Consequently, if Uik and u;k are vertices of G with i # j then both are in D1 or both are in 0 2 . Therefore, the following is a well defined truth assignment for U. For each u k E U find a Uik in G , and let u k be true if Uik E D1 and false if Ujk E 0 2 . It remains only to show that each clause ci has at least one true (respectively, false) literal. If not, we can assume clause ci has three true literals. Then each of c3iand cqiis adjacent to three vertices in 0 2 , so cgi and cqiare in D1. Letting x be one of the vertices adjacent to c3i and cqirwe see that D1- cY - cqi+ x is a dominating set, contradicting the minimality of D1. Using Fig. 3 as an example, if u1 and u2 and uj are true then {G1, U 2 , ii3} c 0 2 and { u,,, u12,uI3,cgl,c,,} c D1, and D1 - cgl- c41 ii13 would be a dominating set strictly smaller than dominating set D1. It follows that U has a NOT-ALL-EQUAL 3SAT truth assignment for C. 0

+

3. A linear algorithm for determining M y ( T )

In this section we will present a linear algorithm for determining M,,(T), the minimum cardinality of the intersection of two minimum dominating sets of tree T. Section 4 will note how an algorithm for finding two minimum dominating sets whose intersection has cardinality My( T) can be derived. Without loss of generality, it will be assumed that a11 trees are rooted at some vertex which can be chosen arbitrarily. This will enable us to use recursive representations of trees. Given a rooted tree T, we will represent T by the number of nodes in T, say n, an endnode list EL = (ul, u2, . . . , un), and an The endnode list is any enumeraassociated parent list PA = (u,, u2, . . . , tion of the nodes of T in which each node precedes its parent. In the associated

On minimum dominating sets with minimum intersections

245

1

&ll

Fig. 4. A tree with EL = (5,6,11,7,8,9,10,2,3,4,1) and PA = (2,3,7,3,3,4,4,1, 1 , 1).

parent list, each ui is the parent of vi in T. Note that PA has length n - 1 and not n, since the root v, has no parent. For example, the tree of Fig. 4 may be represented by n = 11, EL = ( 5 , 6 , 1 1 , 7 , 8 , 9 , 1 0 , 2 , 3 , 4 , l), and PA = (2,3,7,3,3,4,4,1,1,1). These lists can be constructed for a tree of n nodes in time O(n) (see e.g. [25,34]), so requiring them does not increase the order of execution time of our algorithm. We will also make use of the following notation. As the vertex v is reached in a left-to-right processing of the endnode list, let Tv be the subtree induced by v and all of its descendants. (Note that all of the descendants of v have already been processed since they precede v in EL.) Let u be the parent of v (which is determined using PA) and let Tu' be the subtree induced by u, the children of u that precede v in EL, and the descendants of all such children. Finally, let Tu be composed of Tu', Tv and the edge (u, v). Note that Tu does not necessarily contain all of the descendants of u since there may be children of u which appear after v in EL. See Fig. 5. The first three parameters we are interested in will be used to ensure that we

Fig. 5. Illustration of Tu, Tu, and Tu' notation

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get minimum dominating sets. Note that y ( T ) can be achieved without the use of these parameters, but we employ them here because they will be used in evaluating the minimum size of the intersection of two MDS’s later in this section. For a vertex u E V ( T ) ,define

y,(Tu)

= MIN{IDI: u E D ,

D dominates T u } ,

y,(Tu) = MIN(ID1: u @ D, D dominates T u } , y,( Tu) = M I N { ID I : u $ D , D dominates Tu - u } . That is, let y y ( T u ) be the minimum order of a dominating set of Tu which contains u, let y,(Tu) be the minimum order of a dominating set of Tu which does not contain u, and let y,( Tu) be the minimum order of a dominating set of Tu - u which does not contain u. This third parameter will be useful when u is to be dominated by its parent or by an as yet unprocessed child. Note that, since any dominating set of Tu that does not contain u is also a dominating set of Tu - u, we have that y,(Tu) G y n (Tu ) for all u E V ( T ) . Also ~ ( T u ) the , minimum number of vertices in a dominating set of Tu, can be expressed by

Y(TU)= MIN{yy(Tu), Y,(TU)).

(1)

Now, if D is a minimum dominating set of Tu and if u E D then we may write D = U U V where U is a minimum dominating set of Tu’, u E U , and V is a smallest possible vertex subset of Tv that dominates Tv - u (and may or may not dominate v, since v is dominated by u ) . The vertex v may or may not be an element of V. Thus y y ( T u )= y,(Tu’) + M I N { y y ( T v ) ,y,(Tv), y,(Tv)}. But since y,( Tv)S y,( Tv)we may write this recursive relation as Y Y ( W = Y , ( W + MIN{Y?(TV),Y r m J ) } .

(2)

Similarly it is straightforward to derive the following:

To see how these parameters should be initialized, consider a subtree consisting of a single vertex v. Then the minimum number of vertices needed to dominate the subtree using v is one; it is not possible to dominate the subtree without using v ; and zero vertices are required to dominate the subtree minus v. Thus for any endpoint v, of a tree T we may initialize y,(Tv,)= 1, y,(Tv,)= 30, and y,(Tv,) = 0. And for any internal node u, of T we may initialize yy(Tu,’) = 1, y,( Tu,’) = 30, and yIs(Tu:) = 0. After this initialization we may proceed through the endnode list evaluating equations (1) through (4) for u, where u, is the parent of the current endnode list entry v,. Once v, is reached, we can determine Y ( T )= Y(Tvn)= M I N { Y Y ( w l ) >un(Tvn)>. For example, y ( T ) = 4 for the tree of Fig. 6.

On minimum dominating sets with minimum intersections

247

Table 1

ELu

5 6 11 7 8 9 10 2 3 4 1

Y,(Tv) 1 1 1 1 1 1 1 1 2 1 5

Y,(TV) W W W

1 m W

m

1

3 2 4

YATV) 0 0 0 1 0 0

0 1 3 2 4

Fig. 6. A tree with y ( T ) = 4.

While the three y-type parameters were used for maintaining information about any one dominating set, we will now introduce six more parameters for maintaining information relating two dominating sets. More specifically, they are used for maintaining the minimum order of the intersection of two minimum dominating sets with certain additional properties. For u E V ( T ) , define

Ayy (Tu) = MIN{ ID1nD21: u E D1, u E D2, D1and D2 each dominate Tu, IDiJ= y,(Tu)}; I,,(Tu) = MIN{JD1n 4 1 : u C$ D j , Dj dominates Tu,

lDil = yn(Tu)); A,,(Tu) = MIN{IDl

n D21:u C$ D j , Dj dominates Tu - u,

lDil= Y E ( T U ) } ; I,,,(Tu) = MIN{ ID1n D21:u E D1, u r$ D2, Di dominates Tu, YY(TU),ID21= Yn(Tu)}; A,,,(Tu) = MIN{IDI n 41:u E D1, u 4 D2, D1dominates Tu, ID1)= yy(Tu),Dz dominates Tu - u, 1D21 = y,(Tu)}; A,,(Tu) = MIN{IDl n D21:u 4 Di, D1dominates Tu, Dz dominates Tu - u, IDl[= y,,(Tu), lDzl = y,(Tu)}. ID11=

That is, Ay,,( Tu) is the minimum cardinality of the intersection of two MDS’s of Tu, each of which contains u ; I,,(Tu) is the minimum cardinality of the intersection of two MDS’s of Tu, neither of which contains u ; I,, is the minimum cardinality of the intersection of two MDS’s of Tu - u, neither of which contains u ; A,,, is the minimum cardinality of the intersection of two MDS’s of Tu, one containing u and the other not containing u ; I,,, is the minimum cardinality of the intersection of an MDS of Tu containing u and an MDS of Tu - u not containing u ; and An, is the minimum cardinality of the intersection of an MDS of Tu and an MDS of Tu - u, neither of which contains u.

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D. L. Grinstead, P.J. Slater

To determine the initial conditions for the A parameters, again consider a subtree consisting only of the vertex v. Any MDS of the subtree consists of exactly the vertex v, so Ayy(Tv)= 1. It is not possible to dominate the subtree without using v, so A,,(Tu) = Ay,(Tv)= A,,(Tv) = 0s. No vertices are necessary to dominate Tv - v , so A,,(Tv) = 0. Since {v} is a MDS of Tv and 0 is a MDS of Tv - v, Ay,(Tv) = 0. We therefore initialize all endpoints of a tree T in this manner. Also, for internal nodes u, Ayy(Tu’)= 1, A,,(Tu‘) = Ayfl(Tu‘) = A,,(Tu’) = 00, and A,,( Tu’) = Ay,(Tu’) = 0. After this initialization, proceed through EL starting with v and evaluating each of equations (5) through (10) for uiwhere v, is the current element of EL. Once the vertex u, is reached we are ready to determine M y . First, since Ay,(Tv,), A,,(Tv,), and A,fi(Tv,) consider sets which are only required to dominate Tv, - v,, they are not used in choosing My.Second, we must insure that only minimum dominating sets of T = Tv are considered, so if yy(Tvfl)< y,(Tv,) then MJT) = Ayy(Tv,);if yy(Tv,) > y,(Tv,) then M y ( T )= A,,(Tv,); and = min(Ayy(Tv,), A Y n ( T ~ ,A,,(Tv,)). ), if yY(Tv,) = y,(Tv,) then My(T) The y-recurrences previously given as formulae ( 2 ) , (3) and (4) appear in the following algorithm marked as lines ( 2 ) , (3) and (4). The A-recurrences appear in lines marked (5)-(10). Verification of these recurrences is somewhat straightforward. We explain only one of these, namely AY, (line (8) in the algorithm). In the algorithm, we write Ay,(u) for Ayn(Tu), Y,(u) for y,(Tu), etc. The parameter A,,,(u) represents the minimum cardinality of the intersection of yy-set of Tu (that is, a set D1 which dominates Tu, contains u, and has cardinality y,(Tu)) and a y,-set of Tu (that is, a set D2 which dominates Tu, does not contain u, and has cardinality y,( Tu)). To formulate the appropriate recurrences, we consider the intersection of such sets D, and D2 with the vertices of Tv, specifically with the vertex v and its immediate children. This leads to six possibilities: (1) We first suppose v E D1 and v E D2. This can only happen if yy(Tu) = yy(Tu’) + y,,(Tv) and y,(Tu) = y,(Tu’) + yy(Tv). In such a case, Ay,(Tu) s Ay,(Tu’) + AyY(Tu).Thus in the algorithm, if the two y conditions are met we set a variable D 1 equal to Ayy(Tv)+ Ay,(Tu’). If not both y conditions are met then this case must be excluded, so we set D l : = w . Variables 0 2 through 0 6 are similarly defined in cases (2) through (6) below. Noting that 0 3 always equals w , we then set Ay,(Tu) := MIN(D1, 0 2 , 04, D5, 0 6 ) in line (8) of the algorithm. (2) Suppose ZJ is not an element of D1 or D2 but v is dominated by a vertex in each of D, and 4 other than by u. In this case, 0 2 equals A,,(Tv) + Ay,(Tu’) or 0 2 equals w . (3) Suppose ZJ is not an element of D1 or D2and is not necessarily dominated by a vertex other than u in each set. These conditions will not quarantee that v is dominated, thus 0 3 would always be infinity. (4) Suppose v is an element of D , and v is not an element of D2 but a child of ZJ

On minimum dominating sets with minimum intersections

249

is an element of D2. In this case, 0 4 = Ay,(Tu’) + Ay,(Tv)or 0 4 = m. (If the role of v is reversed in the two sets, the situation will be subsumed in case (3.) (5) Suppose v is not an element of 0, (and neither are any of its children) and v is an element of D2. In this case, 05=Ay,(Tu’)+ Ay,(Tv) or 0 5 = w . (If the role of v is reversed in the two sets, the situation is excluded since the conditions will not guarantee that v is dominated.) (6) Suppose v is not an element of D1 or D2 and that v is not necessarily dominated by a vertex other than u in 0,. In this case, 0 6 = An,(Tv) + Ay,(Tu’) or 0 6 = m. (If the role of v is reversed in the two sets, the situation is excluded since the conditions will not quarantee that v is dominated in 4.) The other A recurrences can be similarly justified and the algorithm follows.

Algorithm MGAMMA (T) ( * for finding the minimum cardinality of the intersection of two minimum dominating sets for a tree * )

For i : =1 to n do yy(i) := 1; y,(i) := 03; ys(i) := 0; y(i) := 1; A y y ( i ) := 1; A&) := 00; A&&) := 0; A y , ( i ) : = m ; A y , ( i ) : = O ; A,&):=m;

( * COMPUTING Ayy * ) If Yy New = Yy(PA(i)) + Yy(EL(i)) then A 1 := Ayy(EL(i)) else A1 := 00 If yy New = yy(PA(i))+ y,(EL(i)) then A 3 := A,,(EL(i)) else A3 := 00 If yy New = yy(PA(i)) + yy(EL(i)) and yy New = yy(PA(i)) + y,(EL(i)) then A5 := Ay,(EL(i)) else A5 := w Ayy(PA(i)):= Ayy(PA(i)) MIN(A1, A3, A5)

+

( * COMPUTING A,, * ) If y, New= y,(PA(i))+ yy(EL(i)) then B 1 : = A,,(PA(i)) + Ayy(EL(i)) else Bl:=w

D . L . Grinstead, P.J. Slater

250

If Yn New= Y,(PA(I')) + rn(ELG))

+ A,,(EL(i))

then B2 := A,,(PA(i)) else B2 := 00

If y, New = y,(PA(i)) + y,(EL(Z)) and y, New = y,(PA(i)) then B4 := A,,(PA(i)) + Ay,(EL(i)) else B4 := 00 A,,(PA(i)) := MIN(B1, B2, B4)

+ y,(EL(i)) (6)

( * COMPUTING A,i,i * ) If y, New = y,,(PA(i)) y,(EL(i)) then C l := Ayy(EL(i)) else C1:= 00

+

+ y,(EL(i)) then C2 := An,,(EL(i)) else C2 := c~

If y, New = y,(PA(i))

If y, New = y,(PA(i)) + y,(EL(i)) and yfi New = y,(PA(i)) then C4 := Ay,(EL(i)) else C4 := 00

+ y,(EL(i))

A,i,(PA(Z)) := A,j,j(PA(Z)) + MIN(C1, C2, C4)

( * COMPUTING A,, * ) If yy New = y,(PA(i)) + y,(EL(i)) and A, New = y,(PA(i)) then 0 1 := Ayy(EL(i))t-A,,(PA(i)) else D1:= 30

(7)

+ y,(EL(i))

If yy New = y,(PA(i))+ y,(EL(z)) and y, New = y,(PA(i))+ y,(EL(i)) then 0 2 := A,,(EL(i)) else 0 2 := 00

+ Ayn(PA(i))

+ y,(EL(i)) and y, New = yn(PA(i))+ y,(EL(i)) then 0 4 := Ay,(PA(i)) + Ayn(EL(i)) else 0 4 := 00 If yy New = y,(PA(i)) + yfi(EL(i))and y, New = y,(PA(i)) + y,(EL(i)) then D5 := A,,(PA(i)) + A,,(EL(i)) else D5 := 03 If yy New = y,(PA(i))

+ y,(EL(i)) and y, New = y,(PA(i)) then 0 6 := A,,(EL(i)) + A,,(PA(i)) else 0 6 := 00 &(PA(;)) := MIN(D1, 0 2 , 0 4 , D5, 0 6 )

If yy New = y,(PA(i))

+ yn(EL(i))

( * COMPUTING A,, * ) If yy New = y,(PA(i))+ y,(EL(i)) and yii New = y,(PA(i))+ y,(EL(i)) then E l := Ayy(EL(i)) else El := 00

On minimum dominating sets with minimum intersections

If y,, New = y,,(PA(i)) + yn(EL(i)) and y, New = y,(PA(i)) then E 2 := Ann(EL(i))

251

+ y,(EL(i))

ELSE E 2 := a, If y,, New = y,,(PA(i)) + y,,(EL(i)) and y, New = y,(PA(i)) then E 4 := Ayn(EL(i)) else E 4 := 03 If yy New = y,,(PA(i)) + y,(EL(i)) and y, New = y,(PA(i)) then E 5 := Ayz(EL(i)) else E5 := If y,, New = y,,(PA(i)) + y,(EL(i)) and y, New = y,(PA(i)) then E6 := Ani(EL(i)) else E6:=03 A,,,(PA(i)) = AYi(PA(z))

+ y,,(EL(i)) + yy(EL(i)) + y,,(EL(i))

+ MIN(E1, E2, E4, E5, E6)

( * COMPUTING A,,, * ) If yn New = y,(PA(i)) y,,(EL(i)) and y, New = yi(PA(i)) then F1 := Ayy(EL(i))+ A,,(PA(i)) else F1:= 03

+

If yn New = y,(PA(i)) + y,,(EL(i)) and y, New = y,(PA(i)) then F 2 := A,,,,(EL(i)) + A,,,(PA(i)) else F2 := 03 If yn New = y,(PA(i)) + y,,(EL(i)) and y, New = y,(PA(i)) then F 4 := A,,,,(EL(i))+ A,,(PA(i)) else F 4 := If yn New = y,,(PA(i)) + y,,(EL(i)) and y, New = y,(PA(i)) then F4B := A,,,(EL(i)) + A,,,(PA(i)) else F4B := m An,(PA(i)) := MIN(F1, F2, F4, F4B) yy(PA(i)):= yyNew; y,,(PA(i)) := y,,New; y,(PA(i)) := y,,New; {end for loop}

(9)

+ y,,(EL(i)) + y,,(EL(i))

+ yn(EL(i))

+ y,,(EL(i))

252

D. L. Grinstead, P.J. SIater

Table 2. Parameters I for tree of Fig. 6 showing that M,(T)

5 6

30

7

1 1 1 I

a

1

53

9 10 2 3 4 1

1 1

11

1 1 1 3

3)

a

1 3)

a

1

2 2 2

0 0 0

m

1 0

0

0 0 1 2 2 2

30

m m

m

00

0 0 0 2

= MIN{3,2,2} = 2

0 0 0 0 0 0 0 0 0 0 2

X 3)

r

1 il)

m r

1 2 2 2

For the tree of Fig. 6, Table 2 represents the values of all of the A parameters and hence M y ( T )= 2.

4. Further extensions The algorithm presented in Section 3 determines the parameter M y ( T ) , the minimum cardinality of the intersection of two MDS’s of tree T. Two minimum dominating sets whose intersection has such cardinality can be obtained by one additional scan (this time right-to-left rather than left-to-right) of the endpoint list. (See Grinstead [20].)We will not fully present the procedure here, but only mention that in order to find the sets some additional information is collected in the first scan. For example, in the right-to-left scan of the endpoint list, the root of the tree is the first to be processed. As each vertex is reached, we will decide whether or not to use the vertex in D 1 (the first MDS) and whether or not to use it in 0 2 (the second MDS). If, in either set, it is ever decided not to use a vertex which has not already been dominated by its parent, we must have some information about its children as to which one will ‘cost’ the least in terms of the size of the MDS and in terms of the cardinality of the intersection of D 1 and 0 2 . A more complicated, but still linear, algorithm for finding the minimum cardinality of the intersection of two minimum dominating sets in a series-parallel graph will appear in Grinstead [20]. Since series parallel graphs have two terminals, this algorithm must consider nine y-type parameters (rather than three for the tree case) and forty-five A-type parameters (as opposed to six for the tree case). Since there are three basic ways of connecting two series-parallel subgraphs, each of the nine y-type and forty-five A-type parameters must have three subcases. Finally, as previously noted, we view this minimum intersection MDS problem as a prototype of many possible problems involving such sets as dominating and/or independent vertex sets, with details concerning various problems to appear in Grinstead [20].

On minimum dominating sets with minimum intersections

253

5. Addendum As indicated in the introduction, much work is being done in the developing theory of polynomial/linear algorithms for graph theoretic problems. Some quite recent work is concerned with predicting the nature of problems for which there will exist linear time algorithms on recursive families of graphs. Such work includes that of Bern, Lawler and Wong [7], Bodlaender [lo], Borie, Parker and Tovey [Ill, Mahajan and Peters [27], and Seese [31]. We also note that results on M J G ) for series-parallel graphs appear in Grinstead and Slater [21].

References [l] S. Arnborg and A. Proskurowski, Linear-time algorithms for NP-hard problems restricted to partial k-trees, Report No. TRITA-NA-8404, The Royal Insitute of Technology, Sweden, 1984. [2] D.W. Bange, A.E. Barkauskas and P.J. Slater, A constructive characterization of trees with two disjoint minimum dominating sets, in: Proceedings Ninth S.E. Conference on Combinatorics, Graph Theory and Computing (Utilitas Mathematica, Winnipeg, 1978) 101-112. [3] D.W. Bange, A.E. Barkauskas and P.J. Slater, Disjoint dominating sets in trees, Sandia Laboratories Report, SAND 78-10875, 1978. [4] D.W. Bange, A.E. Barkauskas and P.J. Slater, Efficient dominating sets in graphs, in: R.D. Ringeisen, and F.S. Roberts, eds., Applications of Discrete Mathematics (SIAM, Philadelphia, PA, 1988) 189-199. [5] D.W. Bange, A.E. Barkauskas and P.J. Slater, Disjoint domination algorithms for trees, presented at the First Clemson University/ONR Mini-Conference on Discrete Mathematics, October, 1986. [6] M.W. Bern, E.L. Lawler and A.L. Wong, Why certain subgraph computations require only linear time, in: Proceedings 26th Annual Symposium on the Foundations of Computer Science (Portland, OR, 1985) 117-125. [7] M.W. Bern, E.L. Lawler and A.L. Wong, Linear time computation of optimal subgraphs of decomposable graphs, J. Algorithms 8 (1987) 216-235. [8] T. Beyer, Proskurowski, S. Hedetniemi and S. Mitchell, Independent domination in trees, in: Proceedings Eight S.E. Conference on Combinatorics, Graph Theory and Computing (Utilitas Mathematica, Winnipeg, 1977) 231-328. [9] H.L. Bodlaender, Polynomial algorithms for chromatic index and graph isomorphism on partial k-trees, Tech. Rept. RUU-CS-87-17, Dept. of Computer Science, Univ. of Utrecht, Netherlands, October 1987. [ 101 H.L. Bodlaender, Dynamic programming on graphs with bounded tree-width, Ph.D. Thesis, MIT/LCS/TR-394, Massachusetts Institute of Technology, June 1987; and Tech. Rept. RUU-CS-87-22, Dept. Computer Science, Univ. Utrecht, Netherlands, November 1987. [ l l ] R.B. Borie, R. Gary Parker and Craig A. Tovey, Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursively constructed graph families, preprint, Georgia Institute of Technology, July 1988. [12] E.J. Cockayne, S. Goodman and S.T. Hedetniemi, A linear algorithm for the domination number of a tree, Inform. Process. Lett. 4 (1975) 41-44. [13] E.J. Cockayne and S.T. Hedetniemi, Optimal domination in graphs, IEEE Trans. Circuits and Systems CAS-22 (1975) 855-857. [14] G.J. Chang and G.L. Nemhauser, R-domination on block graphs, Oper. Res. Lett. 1 (1982) 214-218. [15] C.J. Colbourn, P.J. Slater and L.K. Stewart, Locating-dominating sets in seriesparallel

254

D.L. Grinstead, P.J. Slater

networks, Proceedings 16th Annual Conference on Numerical Mathematics and Computing, Winnipeg, 1986, Congr. Numer. 56 (1987) 135-162. [I61 A.K. Dewdney, Fast Turing reductions between problems in NP, Chapter 4: Reductions between NP-complete problems, Report #71, Dept. Computer Science, Univ. Western Ontario, 1981. [17] M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoret. Comput. Sci. 1 (1976) 237-267. (181 M.R. Garey, D.S. Johnson and R.E. Tarjan, 1976, unpublished. [19] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979). [20] D.L. Grinstead, Algoritmic templates and multiset problems in graphs, Ph.D Thesis, Univ. of Alabama in Huntsville, 1989. [21] D.L. Grinstead and P.J. Slater, On the minimum intersection of minimum dominating sets in seris-parallel graphs, 1988, submitted for publication. [22] D.S. Johnson, The NP-completeness column: An ongoing guide, J. Algorithms 6 (1985) 434-451. [23] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J. W. Thatcher, eds, Complexity of Computer Computations (Plenum, New York, 1972) 85-103. 124) T. Kikuno, N. Yoshida and Y. Kakuda, A linear algorithm for the domination number of a series-parallel graph, Discrete Appl. Math. 5 (1983) 299-311. 1251 D.E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms (AddisonWesley, Reading, MA, 1968) 334-338. [26] E.L. Lawler and P.J. Slater, A linear time algorithm for finding an optimal dominating subforest of a tree, in: Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, MI, 1984 (Wiley, New York, 1985) 501-506. I271 S. Mahajan and J.G. Peters, Algorithms for regular properties in recursive graphs, in: Twenty-fifth Annual Allerton Conference on Communications, Control, and Computing, 1987, 14-23. [28] J . Pfaff, R. Laskar and S.T. Hedetniemi, Linear algorithms for independent domination and total domination in seriesparallel graphs, Congr. Numer. 45 (1985) 71-82. [29] N. Robertson and P.D. Seymour, Graph minors 11: Algorithmic aspects of tree-width, J. Algorithms 7 (1986) 309-322. [30] T.J. Schaefer, The complexity of satisfiability problems, in: Proceedings 10th Annual ACM Synposium on Theory of Computing (Association for Computing Machinery, New York, 1978) 2 16-226. [31) D. Seese, Tree-partite graphs and the complexity of algorithms, Tech. Rept. P-MATH-08/86, Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut fur Mathematik, Berlin, 1986. [32] P.J. Slater, R-domination in graphs, J . Assoc. Comput. Mach. 23 (1976) 446-450. [33] K. Takamizawa, T. Nishizeki and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. Assoc. Comput. Mach. 29 (1982) 623-641. (341 R.E. Tarjan, Depth-first search and linear graph algorithms, SIAM J . Comput. 1 (1972) 146- 160. [35] T.V. Wimer, Linear algorithms on k-terminal graphs, Ph.D. Dissertation, Computer Science Department, Clemson University, 1987.

Part V. Bibliography

This Page Intentionally Left Blank

Discrete Mathematics 86 (1990) 257-277 North-Holland

257

BIBLIOGRAPHY ON DOMINATION IN GRAPHS AND SOME BASIC DEFINITIONS OF DOMINATION PARAMETERS S.T. HEDETNIEMI and R.C. LASKAR Departments of Computer Science and Mathematical Sciences, Clemson University, Clemson, SC 29631, USA

Received 2 December 1988

Introduction The following bibliography on Domination in Graphs has been compiled over the past six years at Clemson University, where we regularly maintain a computer data base on this topic. Several people have been especially helpful in keeping this bibliography up-to-date and we would like to thank them: E.J. Cockayne, Victoria, British Columbia; P.J. Slater, Huntsville, Alabama; Maciej Sysio, Wroclaw, Poland; Bohdan Zelinka, Liberec, Czechoslovakia; E. Sampathkumar, Dharwad, India; A. Brandstadt, Rostock, GDR; and Peter Hammer, New Brunswick, New Jersey. This bibliography essentially starts with the graph theory texts of Konig (1950), Berge (1958) and Ore (1962). Although a few research papers on domination were published between 1958 and 1975, a survey paper by Cockayne and Hedetniemi (1975) served to focus attention on the subject sufficiently to ‘get the ball rolling’. By 1988 the domination bibliography included well over 300 citations, about one-third of which are concerned with algorithms for computing various domination numbers for special classes of graphs. In our view, the rapid growth in the number of domination papers is attributable largely to three factors: (i) the diversity of applications to both real-world and other mathematical ‘covering’ or ‘location’ problems; (ii) the wide variety of domination parameters that can be defined; (iii) the NP-completeness of the basic domination problem, its close and ‘natural’ relationships to other NP-complete problems, and the subsequent interest in finding polynomial time solutions to domination problems in special classes of graphs. Thus we expect that this bibliography will continue to grow at a steady rate. As far as we know, only four survey papers have been written on domination in graphs: Cockayne and Hedetniemi, 1975; 0012-365X/90/$03.50 @ 1990-Elsevier Science Publishers B .V. (North-Holland)

258

S. T. Hedetniemi, R. C. Laskar

Cockayne, 1978; Laskar and Walikar, 1980; Hedetniemi, Laskar and Pfaff, 1986 (a survey of irredundance in graphs). However, survey information can be found in a variety of Ph.D. dissertations in which domination in graphs is a central topic, including those of: S.L. Mitchell, University of Virginia, 1977; H.B. Walikar, Karnatak University, 1980; M. Farber, Rutgers University, 1981; G.J. Chang, Cornell University, 1982; P. Blitch, University of South Carolina, 1983; J. Pfaff, Clemson University, 1984; M. Blidia, University of Paris, 1984; P.S. Neeralagi, Karnatak University, 1985; L.K. Stewart, University of Toronto, 1985; M.R. Fellows, University of California at San Diego, 1985; K. Peters, Clemson University, 1986; L.L. Kelleher, Northeastern University, 1987; T.V. Wimer, Clemson University, 1987; G.A. Domke, Clemson University, 1988; T.V. Venkatachalam, Karnatak University, 1988; D.L. Grinstead, University of Albama in Huntsville, 1989; E.O. Hare, Clemson University, 1989; F. Maffray, University of Paris; Jacob, University of Paris. Although such a task grows increasingly difficult because of the growth of this literature, a ‘current’ survey paper or research monograph would be most welcome at this time, either on domination theory in graphs, or on the algorithmic aspects of domination in graphs. The authors would like to apologize for the relatively large number of entries which appear either as ‘manuscript’, ‘technical report’, or ‘to appear’. We have found it difficult to maintain the ‘current’ status of all of these papers, some of which may never be published. We would like to request that any and all information concerning additions, updates, corrections or suggested deletions from this bibliography be forwarded to us at Clemson University. Finally, we present the following definitions of various types of domination, not only to assist the reader in understanding the titles of these papers, but to delimit the scope of this bibliography. In particular, this bibliography does not include the many papers which have been published on: (i) maximal independent sets of vertices in graphs (which are necessarily minimal dominating sets) or maximal cliques in graphs (which are minimal dominating sets in the complement of a graph). [The reader is referred to P.L. Hammer and M.A. Hujter at Rutgers University, who maintain a comprehensive

Bibliography on dominations in graphs

259

bibliography of papers on independent sets and cliques in graphs.]; or (ii) facility location problems.

Basic definitions of domination parameters Let G = (V, E) be a graph.

N ( u ) : The open neighborhood of a vertex u is the set of vertices adjacent to u. N[u]: The closed neighborhood of a vertex u = { u } U N ( u ) . N ( S ) : The open neighborhood of a set S of vertices is the set of vertices adjacent to any vertex in S . N[S]: The closed neighborhood of a set S of vertices = N ( S ) U {S}. A set S E V is a dominating set if N [ S ] = V. A set S E V is a total dominating set if N ( S ) = V. A set S G V is a connected dominating set if S is a dominating set and the subgraph ( S ) induced by S is connected. A set S c V is an efficient dominating set if for every v E V - S, ( N [ v ]f l SJ= 1. A set F E E is an edge dominating set if every edge not in F has a vertex in common with at least one edge in F. A set S E V is a vertex-cover if every edge contains at least one vertex in S. A set F E E is an edge-cover if every vertex is incident with at least one edge in F. A set S c_ V is irredundant if for every vertex v in S, N [ v ]- N [ S - {v}] # 8. A set S c V is a neighborhood set if the union of the induced subgraphs of all the closed neighborhoods of the vertices of S is G , i.e. Uxes ( N [ x ] )= G. A vertex v and an edge ( u , w ) strongly dominate each other if (i) v = u or v = w or (ii) both (v, u ) and (v, w ) are edges in G. A vertex v and an edge (u, w ) weakly dominates each other if (i) v = u or v = w or (ii) (v, u ) or (v, w ) is an edge in G. A set S c V is independent if no two vertices in S are adjacent.

The domination number is the minimum cardinality of a dominating set. The upper domination number is the maximum cardinality of a minimal dominating set The edge-domination number is the minimum cardinality of an edgedominating set. The total dorninarion number is the minimum cardinality of a total dominating set. The connected dominution number is the minimum cardinality of a connected dominating set. The efficient domination number is the minimum cardinality of an efficient dominating set. The independent domination number is the minimum cardinality of a dominating set which is independent.

260

S. T. Hedetoiem', R. C. Laskar

& , ( C ) : The independence number is the maximum cardinality of an independent set of vertices. yo,(G): The vertex-edge weak domination number is the minimum cardinality of a set of vertices that weakly dominates E ( C ) . ylo(C): The edge-vertex weak domination number is the minimum cardinality of a set of edges that weakly dominates V ( C ) . Sy,,( G): The vertex-edge strong domination number is the minimum cardinality of a set of vertices that strongly dominates E ( G ) . Sy,,(G): The edge-vertex strong domination number is the minimum cardinality of a set of edges that strongly dominates V ( G ) . The pendant edge number is the maximum cardinality of a set of pendant edges in a spanning forest of C . The endvertex number is the maximum cardinality of a set of endvertices in a spanning tree of a connected graph G. The irredundance number is the minimum cardinality of a maximal irredundant set of vertices. The upper irredundance number is the maximum cardinality of an irredundant set of vertices. The neighborhood number is the minimum cardinality of a neighborhood set. Note that no(C)= Syo,(C). The domatic number is the maximum order of a partition of V ( G )into sets of vertices such that each such subset is a dominating set.

Bibliography H.L. Abbott and A.C. Liu, Bounds for the covering number of a graph, Discrete Math. 25 (3) (1979) 281-284. B.D. Acharya, The strong domination number of a graph and related concepts, J. Math. Phys. Sci. 14 (5) (1980) 471-475. B.D. Acharya and H.B. Walikar, Bounds on the size of a graph having unique minimum dominating set, manuscript, undated. B.D. Acharya and H.B. Walikar, Indominable graphs cannot be characterized by a finite family of forbidden subgraphs, manuscript, undated. B.D. Acharya and H.B. Walikar, On graphs having unique minimum dominating sets, Graph Theory Newsletter 8 (5) (1979) 1. W. Ahrens, Mathematische Unterhaltungen und Spiele (Leipzig, 1901).

M. Aigner, Some theorems on coverings, Studia Sci. Math. Hungar. 5 (1970) 303-315 R.B. Allan and R. Laskar, On domination and independent domination numbers of a graph, Discrete Math. 23 (1978) 73-76.

R.B. Allan and R. Laskar, On domination and some related topics in graph theory, in: Proceedings Ninth S.E. Conference on Combinatorics, Graph Theory and Computing (Utilitas Mathematica, Winnipeg, 1978) 43-58. R.B. Allan, R. Laskar and S.T. Hedetniemi, A note on total domination, Discrete Math. 49 (1984) 7-13.

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261

M. Anciaux-Mundeleer and P. Hansen, O n kernels in strongly connected graphs, Networks 7 (3) (1977) 263-266. B. Andreas, Graphs with unique maximal clumpings, J. Graph Theory 2 (1) (1978) 19-24. V.I. Amautov, Estimation of the exterior stability number of a graph by means of the minimal degree of the vertices (Russian) Prikl. Mat. i Programmirovanie Vyp. 11 (1974) 3-8, 126.

V.I. Arnautov, The exterior stability number of a graph, Diskret. Analiz 20 (1972) 3-8. S. Arnborg, Efficient algorithms for combinatorial decomposability - a survey, BIT 25 (1985) 2-23.

problems on

graphs

with

bounded

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B. Zelinka, Total domatic number of cacti, Math. Slovaca 38 (3) (1988) 207-214.

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Discrete Mathematics 86 (1990) 279-280 North-Holland

279

AUTHOR INDEX Volume 86 (1990)

Berge, C. and P. Duchet, Recent problems and results about kernels in directed graphs (1-3) 27- 31 BollobBs, B., E.J. Cockayne and C.M. Mynhardt, On generalised minimal domination parameters for paths (1-3) 89- 97 Brigham, R.C. and R. Dutton, Factor domination in graphs (1-3) 127-136 Chvital, V. and W.J. Cook, The discipline number of a graph (1-3) 191-198 Clark, B.N., C.J. Colbourn and D.S. Johnson, Unit disk graphs (1-3) 165-177 Cockayne, E.J., Chessboard domination problems (1-3) 13- 20 Cockayne, E.J., see Bollobh, B. (1-3) 89- 97 Colbourn, C.J., see Clark, B.N. (1-3) 165-177 Colbourn, C.J. and L.K. Stewart, Permutation graphs: connected domination and Steiner trees (1-3) 179-189 Cook, W.J., see ChvBtal, V. (1-3) 191-198 Corneil, D.G. and L.K. Stewart, Dominating sets in perfect graphs (1-3) 145-164 Cozzens, M.B. and L.L. Kelleher, Dominating cliques in graphs (1-3) 101-116 Duchet, P., see Berge, C. Dutton, R., see Brigham, R.C.

(1-3) 27- 31 (1-3) 127-136

Fink, J.F., M.S. Jacobson, L.F. Kinch and J. Roberts, The bondage number of a (1-3) 47- 57 graph Grinstead, C.M., B. Hahne and D. Van Stone, On the queen domination problem (1-3) 21- 26 Grinstead, D.L. and P.J. Slater, On minimum dominating sets with minimum intersection (1-3) 239-254 Hahne B., see Grinstead, C.M. (1-3) 21- 26 Hedetniemi, S.T. and R.C. Laskar, Introduction (1-3) 3- 9 Hedetniemi, S.T. and R.C. Laskar, Bibliography on domination in graphs and some basic definitions of domination parameters (1-3) 257-277 Jacobson, M.S., see Fink, J.F. (1-3) 47- 57 Jacobson, M.S. and K. Peters, Chordal graphs and upper irredundance, upper (1-3) 59- 69 domination and independence Johnson, D.S., see Clark, B.N. (1-3) 165-177 Kelleher, L.L., see Cozzens, M.B. (1-3) 101-116 Kinch, L.F., see Fink, J.F. (1-3) 47- 57 Kratsch, D., Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs (1-3) 225-238 Laskar, R.C., see Hedetniemi, S.T. Laskar, R.C., see Hedetniemi, S.T.

(1-3) 3- 9 (1-3) 257-277

McHugh, J. and Y. Perl, Best location of service centers in a tree-like network under budget constraints (1-3) 199-214 Mynhardt, C.M., see Bollobis, B. (1-3) 89- 97 Perl, Y., see McHugh, J.

(1-3) 199-214

280

Author index

Peters, K., see Jacobson, M.S.

(1-3) 59- 69

Rall, D . , Domatically critical and domatically full graphs Roberts, J., see Fink, J.F.

(1-3) 81- 87 (1-3) 47- 57

Sampathkumar, E., The least point covering and domination numbers of a graph Skowronska, M. and M.M. Syslo, Dominating cycles in Halin graphs Slater, P.J., see Grinstead, D.L. Stewart, L.K., see Corneil, D.G. Stewart, L.K., see Colbourn, C.J. Sumner, D.P., Critical concepts in domination Syslo, M.M., see Skowronska, M.

(1-3) (1-3) (1-3) (1-3) (1-3) (1-3) (1-3)

Tuza, Z., Covering all cliques of a graph

(1-3) 117-126

Van Stone, D., see Grinstead, C.M.

(1-3) 21- 26

Zelinka, B . , Regular totally domatically full graphs

(1-3) 71- 79

137-142 215-224 239-254 145-164 179-189 33- 46 215-224