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Perfect Graphs
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Perfect Graphs
Hardback ISBN: 9780471489702
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Taking a fresh approach to graph theory, this book surveys the latest research articles, highlighting the new directions and seminal results. It also emphasizes the links the subject has to other areas of mathematics and its applications. In particular, the links between perfect graphs and frequency assignment for telecommunications are discussed.
The theory of perfect graphs was born out of a conjecture about graph colouring made by Claude Berge in 1960. That conjecture remains unsolved, but has generated an important area of research in combinatorics. This book: Includes an introduction by Claude Berge, the founder of perfect graph theory Discusses the most recent developments in the field of perfect graph theory Provides a thorough historical overview of the subject Internationally respected authors highlight the new directions, seminal results and the links the field has with other subjects Discusses how semidefinite programming evolved out of perfect graph theory The early developments of the theory are included to lay the groundwork for the later chapters. The most recent developments of perfect graph theory are discussed in detail, highlighting seminal results, new directions, and links to other areas of mathematics and their applications. These applications include frequency assignment for telecommunication systems, integer programming and optimisation.
ISBN  471489700 
ISBN13  9780471489702 
Publisher  John Wiley & Sons Ltd 
Format  Hardback 
Publication date  14/09/2001 
Pages  386 
Weight (grammes)  820 
Published in  United Kingdom 
Height (mm)  250 
Width (mm)  179 
List of Contributors.
Preface.
Acknowledgements.
1. Origins and Genesis (C. Berge and J.L. Ramirez Alfonsin).
Perfection.
Communication Theory.
The Perfect Graph Conjecture.
Shannon's Capacity.
Translation of the HalleWittenberg Proceedings.
Indian Report.
References.
2. From Conjecture to Theorem (Bruce A Reed).
Gallai's Graphs.
The Perfect Graph Theorem.
Some Polyhedral Consequences.
A Stronger Theorem.
References.
3. A Translation of Gallai's Paper: "Transitiv Orientierbare Graphen" (Frederic Maffray and Myriam Preissmann).
Introduction and Results.
The Proofs of Theorems (3.12), (3.15) and 3.16).
The Proofs of (3.18) and (3.19).
The Proofs of (3.1.16).
The Proofs of (3.1.17).
Determination of all Irreducible Graphs.
Determination of the Irreducible Graphs.
References.
4. Even Pairs (Hazel Everett et al).
Introduction.
Even Pairs and Perfect Graphs.
Perfectly Contractile Graphs.
Quasiparity Graphs.
Recent Progress.
Odd Pairs.
References.
5. The P4Structure of Perfect Graphs (Stefan Hougardy).
Introduction.
P4Stucture: Basics, Isomorphisms and Recognition.
Modules, hSets, Split Graphs and Unique P4Structure.
The SemiStrong perfect Graph Theorem.
The Structure of the P4Isomorphism Classes.
Recognizing P4Structure.
The P4Structure of Minimally Imperfect Graphs.
The Partner Structure and Other Generalizations.
P3Structure.
References.
6. Forbidding Holes and Antiholes (Ryan Hayward and Bruce A. Reed).
Introduction.
Graphs with No Holes.
Graphs with No Discs.
Graphs with No Long Holes.
Balanced Matrices.
Bipartitie Graphs with No Hole of Length 4k + 2.
Graphs without Even Holes.
Perfect Graphs.
Graphs without Odd Holes.
References.
7. Perfectly Orderable Graphs: A Survey (Chinh T Hoang).
Introduction.
Classical Graphs.
Minimal Nonperfectly Orderable Graphs.
Orientations.
Generalizations of Triangulated Graphs.
Generalizations of Complements of Chordal Bipartitie Graphs.
Other Classes of Perfectly Orderable Graphs.
Vertex Orderings.
Generalizations of Perfectly Orderable Graphs.
Optimizing Perfectly Ordered Graphs.
References.
8. Cutsets in Perfect and Minimal Imperfect Graphs (Irena Rusu).
Introduction.
How Did It Start?
Main Results on Minimal Imperfect Graphs.
Applications: Star Cutsets.
Applications: Clique and Multipartite Cutsets.
Applications: Stable Cutsets.
Two (Resolved) Conjectures.
The Connectivity of Minimal Imperfect Graphs.
Some (More) Problems.
References.
9. Some Aspects of Minimal Imperfect Graphs (Myriam Preissmann and Andras Sebo).
Introduction.
Imperfect and Partitionable Graphs.
Properties.
Constructions.
References.
10. Graph Imperfection and Channel Assignment (Colin McDiarmid).
Introduction.
The Imperfection Ratio.
An Alternative Definition.
Further Results and Questions.
background on Channel Assignment.
References.
11. A Gentle Introduction to Semidefinite Programming (Bruce A. Reed).
Introduction.
The Ellipsoid Method.
Solving Semidefinite Programs.
Randomized Rounding and Derandomization.
Approximating MAXCUT.
Approximating Bandwidth.
Graph Colouring.
12. The Theta Body.
References.
The Theta Body and Imperfection (F.B. Shepherd).
Background and Overview.
Optimization, Convexity and Geometry.
The Theta Body.
Partitionable Graphs.
Perfect Graph Characterizations and a Continuous Perfect Graph Conjecture.
References.
13. Perfect Graphs and Graph Entropy (Gabor Simonyi).
Introduction.
The InformationTheoretic Interpretation.
Some Basic Properties.
Structural Theorems: Relation to Perfectness.
Applications.
Generalizations.
Graph Capacities and Other Related Functionals.
References.
14 A Bibliography on Perfect Graphs (Vaek Chvatal).
Index.