Algorithmic Graph Theory and Perfect Graphs
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1 Algorithmic Graph Theory and Perfect Graphs Second Edition Martin Charles Golumbic Caesarea Rothschild Institute University of Haifa Haifa, Israel 2004 ELSEVIER.. Amsterdam - Boston - Heidelberg - London - New York - Oxford Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo
2 Foreword 2004 Foreword Preface Acknowledgments List of Symbols Corrections and Errata Xlll xv xvii xix xxi xxiii CHAPTER 1 Graph Theoretic Foundations Basic Definitions and Notations Intersection Graphs 3. Interval Graphs A Sneak Preview of the Notions Coming Up 4. Summary CHAPTER 2 The Design of Efficient Algorithms The Complexity of Computer Algorithms Data Structures 3. How to Explore a Graph 4. Transitive Tournaments and Topological Sorting vii
3 viii CHAPTER 3 Perfect Graphs The Star of the Show The Perfect Graph Theorem 3. p-critical and Partitionable Graphs 4. A Polyhedral Characterization of Perfect Graphs 5. A Polyhedral Characterization of p-critical Graphs 6. The Strong Perfect Graph Conjecture Contents CHAPTER 4 Triangulated Graphs Introduction Characterizing Triangulated Graphs 3. Recognizing Triangulated Graphs by Lexicographic Breadth-First Search 4. The Complexity of Recognizing Triangulated Graphs 5. Triangulated Graphs as Intersection Graphs 6. Triangulated Graphs Are Perfect 7. Fast Algorithms for the COLORING, CLIQUE, STABLE SET, and CLIQUE-COVER Problems on Triangulated Graphs CHAPTER 5 Comparability Graphs F-Chains and Implication Classes 105 Uniquely Partially Orderable Graphs The Number of Transitive Orientations Schemes and G- Decompositions An Algorithm for Assigning Transitive Orientations The T*-Matroid of a Graph The Complexity of Comparability Graph Recognition Coloring and Other Problems on Comparability Graphs 132
4 IX The Dimension of Partial Orders CHAPTER 6 Split Graphs An Introduction to Chapters 6-8: Interval, Permutation, and Split Graphs Characterizing Split Graphs 3. Degree Sequences and Split Graphs CHAPTER 7 Permutation Graphs Introduction Characterizing Permutation Graphs Permutation Labelings Applications Sorting a Permutation Using Queues in Parallel CHAPTER 8 Interval Graphs How It All Started Some Characterizations of Interval Graphs The Complexity of Consecutive 1 's Testing Applications of Interval Graphs Preference and Indifference Circular-Arc Graphs CHAPTER 9 Superperfect Graphs Coloring Weighted Graphs Sunernerfection An Infinite Class of Superperfect Noncomparability Graphs 209
5 4. When Does Superperfect Equal Comparability? Composition of Superperfect Graphs A Representation Using the Consecutive l's Property CHAPTER 10 Threshold Graphs The Threshold Dimension Degree Partition of Threshold Graphs 3. A Characterization Using Permutations 4. An Application to Synchronizing Parallel Processes CHAPTER 11 Not So Perfect Graphs Sorting a Permutation Using Stacks in Parallel Intersecting Chords of a Circle 3. Overlap Graphs 4. Fast Algorithms for Maximum Stable Set and Maximum Clique of These Not So Perfect Graphs 5. A Graph Theoretic Characterization of Overlap Graphs CHAPTER 12 Perfect Gaussian Elimination Perfect Elimination Matrices Symmetric Matrices Perfect Elimination Bipartite Graphs Chordal Bipartite Graphs
6 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 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, C, T, T Illustrated 275 Epilogue Index 307
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