Algorithmic Graph Theory and Perfect Graphs Second Edition ANNALS OF DISCRETE MATHEMATICS 57 Series Editor: Peter L. HAMMER Rutgers University, Piscataway, N J, U.S.A Algorithmic Graph Theory and Perfect Graphs Second Edition Martin Charles Golumbic Caesarea Rothschild Institute University of Haifa Haifa, Israel 2OO4 ELSEVIER Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore- Sydney- Tokyo ELSEVIER B.V. ELSEVIER Inc. ELSEVIER Ltd ELSEVIER Ltd Sara Burgerhartstraat 25 525 B Street, Suite 1900 The Boulevard, Langford Lane 84 Theobalds Road P.O. Box 211, 1000 AE San Diego, CA 92101-4495 Kidlington,Oxford OX5 1GB London WC 1X 8RR Amsterdam, The Netherlands USA UK UK © 2004 Elsevier B.V. All rights reserved. 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Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice 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. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 1980 (Academic Press, ISBN 0-12-289260-7) Second edition 2004 Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library. ISBN: 0-444-51530-5 The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in Hungary. Dedicated to my parents l,'l~ ~"'f~ 1"I::I ~/~rl This Page Intentionally Left Blank Contents ... Foreword 2004 Xlll Foreword xv Preface xvii Acknowledgments xix List of Symbols xx i Corrections and Errata xxiii CHAPTER 1 Graph Theoretic Foundations 1. Basic Definitions and Notations 1 2. Intersection Graphs 9 3. Interval Graphs-A Sneak Preview of the Notions Coming Up 13 4. Summary 17 Exercises 18 Bibliography 20 CHAPTER 2 The Design of Efficient Algorithms 1. The Complexity of Computer Algorithms 22 2. Data Structures 31 3. How to Explore a Graph 37 4. Transitive Tournaments and Topological Sorting 42 Exercises 45 Bibliography 48 vii viii Contents CHAPTER 3 Perfect Graphs 1. The Star of the Show 51 2. The Perfect Graph Theorem 53 3. p-Critical and Partitionable Graphs 58 4. A Polyhedral Characterization of Perfect Graphs 62 5. A Polyhedral Characterization of p-Critical Graphs 65 6. The Strong Perfect Graph Conjecture 71 Exercises 75 Bibliography 77 CHAPTER 4 Triangulated Graphs 1. Introduction 81 2. Characterizing Triangulated Graphs 81 3. Recognizing Triangulated Graphs by Lexicographic Breadth-First Search 84 4. The Complexity of Recognizing Triangulated Graphs 87 5. Triangulated Graphs as Intersection Graphs 91 6. Triangulated Graphs Are Perfect 94 7. Fast Algorithms for the COLORING, CLIQUE, STABLE SET, and CLIQUE-COVER Problems on Triangulated Graphs 98 Exercises 100 Bibliography 102 CHAPTER 5 Comparability Graphs 1. r-Chains and Implication Classes 105 2. Uniquely Partially Orderable Graphs 109 3. The Number of Transitive Orientations 113 4. Schemes and G-Decompositions- An Algorithm for Assigning Transitive Orientations 120 5. The r*-Matroid of a Graph 124 6. The Complexity of Comparability Graph Recognition 129 7. Coloring and Other Problems on Comparability Graphs 132 Contents ix 8. The Dimension of Partial Orders 135 Exercises 139 Bibliography 1 42 CHAPTER 6 Split Graphs 1. An Introduction to Chapters 68:Interval, Permutation, and Split Graphs 149 2. Characterizing Split Graphs 149 3. Degree Sequences and Split Graphs 152 Exercises 155 Bibliography 156 CHAPTER 7 Permutation Graphs 1. Introduction 157 2. Characterizing Permutation Graphs 158 3. Permutation Labelings 160 4. Applications 162 5. Sorting a Permutation Using Queues in Parallel 164 Exercises 168 Bibliography 169 CHAPTER 8 Interval Graphs 1. How It All Started 171 2. Some Characterizations of Interval Graphs 172 3. The Complexity of Consecutive 1’s Testing 175 4. Applications of Interval Graphs 181 5. Preference and Indifference 185 6. Circular- Arc Graphs 188 Exercises 193 Bibliography 197 CHAPTER 9 Superperfect Graphs 1. Coloring Weighted Graphs 203 2. Superperfection 206 3. An Infinite Class of Superperfect Noncomparability Graphs 209 X Contents 4. When Does Superperfect Equal Comparability? 212 5. Composition of Superperfect Graphs 214 6. A Representation Using the Consecutive 1’s Property 215 Exercises 218 Bibliography 218 CHAPTER 10 Th reshoId Graphs 1. The Threshold Dimension 219 2. Degree Partition of Threshold Graphs 223 3. A Characterization Using Permutations 221 4. An Application to Synchronizing Parallel Processes 229 Exercises 23 1 Bibliography 234 CHAPTER 11 Not So Perfect Graphs 1. Sorting a Permutation Using Stacks in Parallel 235 2. Intersecting Chords of a Circle 231 3. Overlap Graphs 242 4. Fast Algorithms for Maximum Stable Set and Maximum Clique of These Not So Perfect Graphs 244 5. A Graph Theoretic Characterization of Overlap Graphs 248 Exercises 25 1 Bibliography 253 CHAPTER 12 Perfect Gaussian Elimination 1. Perfect Elimination Matrices 254 2. Symmetric Matrices 256 3. Perfect Elimination Bipartite Graphs 259 4. Chordal Bipartite Graphs 26 1 Exercises 264 Bibliography 266 Contents xi Appendix A. A Small Collection of NP-complete Problems 269 B. An Algorithm for Set Union, Intersection, Difference, and Symmetric Difference of Two Subsets 270 C. Topological Sorting: An Example of Algorithm 2.4 27 1 D. An Illustration of the Decomposition Algorithm 273 E. The Properties P.E.B., C.B., (P.E.B.)', (C .B.) ' Illustrated 273 F. The Properties c, r,T, T Illustrated 275 Epilogue 2004 277 Index 307 This Page Intentionally Left Blank Foreword 2004: the Annals edition The publication of this new edition of Algorithmic Graph Theory and Perfect Graphs marks twenty three years since its first appearance. My original motivation for writing the book was to collect and unify the topic to act as a spring board for researchers, and especially graduate students, to pursue new directions of investigation. The ensuing years have been an amazingly fruitful period of research in this area. To my great satisfaction, the number of relevant journal articles in the literature has grown tenfold. I can hardly express my admiration to all these authors for creating a success story for algorithmic graph theory far beyond my own imagination. The world of perfect graphs has grown to include over 200 special graph classes. The Venn diagrams that I used to show some of the inclusions between classes in the First Generation, for example Figure 9.9 (on page 212), have yielded to Hasse diagrams for the Second Generation, like the one from Golumbic and Trenk [2004] reprinted in Figure 13.3 at the end of this edition. Perhaps the most important new development in the theory of perfect graphs is the recent proof of the Strong Perfect Graph Conjecture by Chudnovsky, Robertson, Seymour and Thomas, announced in May 2002. News of this was im- mediately passed on to Claude Berge, who sadly passed away on June 30, 2002. On the algorithmic side, many of the problems which were open in 1980 have subsequently been settled, and algorithms on new classes of perfect graphs have been studied. For example, tolerance graphs generalize both interval graphs and permutation graphs, and coloring tolerance graphs in polynomial time is important in solving scheduling problems where a measure of flexibility or tolerance is allowed for sharing or relinquishing resources when total exclusivity prevents a solution. At the end of this new edition, I have added a short chapter called XIII xiv Foreword Epilogue 2004 in which I survey a few of my favorite results and research directions from the Second Generation. Its intension is to whet the appetite. Six books have appeared recently which cover advanced research in this area.