Perfect Graphs

Edited by

Jorge L. Ramirez Alfonsm University of Bonn, Germany

Bruce A. Reed CNRS, Paris, France

JOHN WILEY & SONS, LTD Chichester • New York • Weinheim • Brisbane • Singapore • Toronto

Contents

List of Contributors xiii

Preface xv

Acknowledgements xxi

1 Origins and Genesis

C. Berge and J.L. Ramirez Alfonsin 1 1.1 Perfection 1 1.2 Communication Theory 1 1.3 The Perfect Graph Conjecture 3 1.4 Shannon's Capacity 7 1.5 Translation of the Halle-Wittenberg Proceedings 7 1.6 Indian Report 10 References 11

2 From Conjecture to Theorem Bruce A. Reed 13 2.1 Gallai's Graphs 14 2.2 The Perfect Graph Theorem 16 2.3 Some Polyhedral Consequences 18 2.4 A Stronger Theorem 22 References 23

3 A Translation of Gallai's Paper: 'Transitiv Orientierbare Graphen' Frederic Maffray and Myriam Preissmann 25 Translators' Foreword 25 3.1 Introduction and Results 26 3.2 The Proofs of Theorems (3.1.2), (3.1.5) and (3.1.6) 34 3.3 The proofs of (3.1.8) and (3.1.9) 38 3.4 The Proof of (3.1.16) 40 3.5 The Proof of (3.1.17) 44 3.6 Determination of All Irreducible Graphs 47 3.7 Determination of the Irreducible Graphs 56 References 65

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4 Even Pairs Hazel Everett, Celina M. H. de Figueiredo, Claudia Linhares Sales, Frederic Maffray, Oscar Porto and Bruce A. Reed 67 4.1 Introduction 67 4.2 Even Pairs and Perfect Graphs 70 4.3 Perfectly Contractile Graphs 72

4.3.1 Weakly Triangulated Graphs 72 4.3.2 Meyniel Graphs 73 4.3.3 Perfectly Orderable Graphs 75 4.3.4 Other Classes of Perfectly Contractile Graphs 77 4.3.5 Graphs that Might Be Perfectly Contractile 78 4.3.6 Forbidden Subgraphs in Perfectly Contractile Graphs 78

4.4 Quasi-parity Graphs 80 4.4.1 Characterization of QP and SQP Graphs 82

4.5 Recent Progress 83 4.5.1 Planar Graphs 84 4.5.2 Claw-Free Graphs 85 4.5.3 Bull-Free Graphs 86 4.5.4 Diamond-Free Graphs 87

4.6 Odd Pairs 88 References 89

5 The Pj-Structure of Perfect Graphs Stefan Hougardy 93 5.1 Introduction 93 5.2 P4-Structure: Basics, Isomorphisms and Recognition 94 5.3 Modules, /i-Sets, Split Graphs and Unique Pj-Structure 96 5.4 The Semi-Strong Perfect Graph Theorem 101 5.5 The Structure of the Pt-Isomorphism Classes 102 5.6 Recognizing P4-Structure 105 5.7 The Pi-Structure of Minimally Imperfect Graphs 106 5.8 The Partner Structure and Other Generalizations 107 5.9 Рз-Structure 108 References 110

6 Forbidding Holes and Antiholes Ryan Hayward and Bruce A. Reed 113 6.1 Introduction 113 6.2 Graphs with No Holes 114 6.3 Graphs with No Discs 116 6.4 Graphs with No Long Holes 120 6.5 Balanced Matrices 121 6.6 Bipartite Graphs with No Hole of Length 4fc + 2 122

6.6.1 Some Preliminary Remarks 124 6.6.2 Clean Holes 125 6.6.3 A Recognition Algorithm 126 6.6.4 A Fresh Look at Cleanliness 127 6.6.5 Recognizing Balanceable Graphs 129

CONTENTS ix

6.7 Graphs without Even Holes 130 6.7.1 The Decomposition Theorem 131

6.8 /^-Perfect Graphs 132 6.9 Graphs without Odd Holes 133 References 135

7 Perfectly Orderable Graphs: A Survey Chinh T. Hoäng 139 7.1 Introduction 139 7.2 Classical Graphs 140

7.2.1 Triangulated Graphs 141 7.2.2 Chordal Bipartite Graphs and Gamma-Free

Matrices 141 7.2.3 Comparability Graphs 141 7.2.4 Interval Graphs 142

7.3 Minimal Nonperfectly Orderable Graphs 142 7.4 Orientations 144

7.4.1 Six Classes of Perfectly Orderable Graphs 144 7.4.2 Opposition Orientation 146

7.5 Generalizations of Triangulated Graphs 147 7.5.1 Quasi-triangulated Graphs 147 7.5.2 Brittle Graphs 148 7.5.3 Quasi-brittle Graphs 149 7.5.4 C-orientation 149

7.6 Generalizations of Complements of Chordal Bipartite Graphs 149 7.6.1 Claw-Free Graphs 149 7.6.2 P5-Free Weakly Triangulated Graphs 150 7.6.3 Bull-Free Perfectly Orderable Graphs 151 7.6.4 D-Graphs 151

7.7 Other Classes of Perfectly Orderable Graphs 152 7.7.1 Complements of Tolerance Graphs 152 7.7.2 Strongly Perfectly Orderable Graphs 152 7.7.3 Charming Graphs 153 7.7.4 Nice Graphs 153 7.7.5 Bipartable Graphs 153 7.7.6 Intersection and Union of Two Threshold Graphs 153 7.7.7 P4 Composition 156

7.8 Vertex Orderings 156 7.8.1 Lexicographic Breadth-First Search 157 7.8.2 Maximal Cardinality Search 158 7.8.3 Orders by Degrees 158

7.9 Generalizations of Perfectly Orderable Graphs 159 7.9.1 Quasi-perfectly Orderable Graphs 160 7.9.2 Properly Orderable Graphs 160 7.9.3 Perfectly Orientable Graphs 161

7.10 Optimizing Perfectly Ordered Graphs 161 References 163

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8 Cutsets in Perfect and Minimal Imperfect Graphs Irena Rusu 167 8.1 Introduction 167 8.2 How Did It Start? 168 8.3 Main Results on Minimal Imperfect Graphs 169 8.4 Applications: Star Cutsets 172 8.5 Applications: Clique and Multipartite Cutsets 174 8.6 Applications: Stable Cutsets 176 8.7 Two (Resolved) Conjectures 178 8.8 The Connectivity of Minimal Imperfect Graphs 179 8.9 Some (More) Problems 181 References 181

9 Some Aspects of Minimal Imperfect Graphs Myriam Preissmann and Andrds Sebo 185 9.1 Introduction 185

9.1.1 Definitions and Notation 186 9.1.2 Classifications of the Results 188 9.1.3 Polyhedra 189 9.1.4 Testing Imperfectness and Partitionability 189

9.2 Imperfect and Partitionable Graphs 191 9.2.1 Basic properties 191 9.2.2 Small Certificate for Imperfectness 193 9.2.3 Small Certificates for Partitionable Graphs 194

9.3 Properties 195 9.3.1 Partitionable and Minimal Imperfect Graphs 195 9.3.2 Genuine properties 197

9.4 Constructions 207 9.4.1 Generalities 207 9.4.2 Partitionable Graphs with Circular Symmetry 209 9.4.3 Adding a Critical Clique 210

References 212

10 Graph Imperfection and Channel Assignment Colin McDiarmid 215 10.1 Introduction 215 10.2 The Imperfection Ratio 217 10.3 Alternative Definitions 219 10.4 Further Results and Questions 220

10.4.1 List Colouring 220 10.4.2 Triangle-Free Planar Graphs 221 10.4.3 Imperfection and Odd Holes 221 10.4.4 Dilation Ratio 222 10.4.5 Graph Entropy 223 10.4.6 Random Graphs 224 10.4.7 Extremal behaviour 225 10.4.8 Hardness 225 10.4.9 Lexicographic Products 225 10.4.10 Values of imp(G) 225

CONTENTS xi

10.4.11 Holes and Antiholes 226 10.4.12 Binary Imperfection Ratio 226 10.4.13 Circular Arc Graphs 226 10.4.14 Unit Disc Graphs 227

10.5 Background on Channel Assignment 227 References 229

11 A Gentle Introduction to Semi-definite Programming Bruce A. Reed 233 11.1 Introduction 233 11.2 The Ellipsoid Method 235

11.2.1 How the Algorithm Works 237 11.2.2 Solving LPs 239

11.3 Solving Semi-definite Programs 241 11.4 Randomized Rounding and Derandomization 244

11.4.1 The Method of Conditional Expectation for MAXSAT 245 11.4.2 Randomized Rounding 246 11.4.3 A Combined Approach 247 11.4.4 Random Projection 248

11.5 Approximating MAXCUT 248 11.6 Approximating Bandwidth 250

11.6.1 The First Algorithm 250 11.6.2 A More Sophisticated Algorithm 252

11.7 Graph Colouring 252 11.7.1 Colouring Perfect Graphs 252 11.7.2 Colouring 3-Colourable Graphs 254

11.8 The Theta Body 256 References 258

12 The Theta Body and Imperfection F.B. Shepherd 261 12.1 Background and Overview 261

12.1.1 Perfection and Integer Programming 261 12.1.2 Imperfection and Partitionability 262 12.1.3 Overview 263 12.1.4 Partitionability and Branch and Cut Methods for Packing

Problems 266 12.2 Optimization, Convexity and Geometry 267

12.2.1 Convexity and Encoding Conventions 268 12.2.2 Optimization over a Convex Body: Consequences of the

Ellipsoid Method 269 12.2.3 The Discovery Problem 270

12.3 The Theta Body 272 12.3.1 Three Convex Bodies 272 12.3.2 A Semi-definite Formulation 273 12.3.3 Algorithms for the Theta Body 275 12.3.4 Additive Gap Guarantees and the Protrusion of the

Theta Body 276

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12.4 Partitionable Graphs 280 12.4.1 A Characterization 280 12.4.2 A Recognition Algorithm 283

12.5 Perfect Graph Characterizations and a Continuous Perfect Graph Conjecture 285

References 289

13 Perfect Graphs and Graph Entropy Gabor Simonyi 293 13.1 Introduction 293 13.2 The Information-Theoretic Interpretation 295 13.3 Some Basic Properties 297

13.3.1 Monotonicity 297 13.3.2 Sub-additivity 298 13.3.3 Additivity of substitution 298

13.4 Structural Theorems: Relation to Perfectness 300 13.4.1 Additivity for Complementary Pairs 300 13.4.2 Imperfection Ratio 302 13.4.3 Additivity for Arbitrary Pairs 304 13.4.4 Sub- and Supermodular Pairs 306 13.4.5 Weak Additivity and Normality 307

13.5 Applications 309 13.5.1 Kahn and Kim's Application for Sorting 309 13.5.2 About Other Applications 310

13.6 Generalizations 312 13.6.1 Hypergraph Entropy 312 13.6.2 Additivity for Complementary Uniform Hypergraphs 312 13.6.3 Entropy of Convex Corners 315 13.6.4 Job Scheduling Application 315

13.7 Graph Capacities and Other Related Functionals 316 13.7.1 Shannon Capacity 316 13.7.2 Sperner Capacity 319 13.7.3 Probabilistic Loväsz Function 321 13.7.4 Co-entropy 323

References 325

14 A Bibliography on Perfect Graphs Vasek Chvdtal 329

Index 359