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Lecture Notes in Logic, 47

Descriptive Complexity, Canonisation, and Definable Graph Structure Theory


Martin Grohe





Year: 2017
Price:$155.00 ISBN-13: 9781107014527
554 pages. Hardcover.
(when ordering, include the discount code ASL2016 to receive the 25% ASL member discount)

Descriptive complexity theory establishes a connection between the computational complexity of algorithmic problems (the computational resources required to solve the problems) and their descriptive complexity (the language resources required to describe the problems). This groundbreaking book approaches descriptive complexity from the angle of modern structural graph theory, specifically graph minor theory. It develops a 'definable structure theory' concerned with the logical definability of graph theoretic concepts such as tree decompositions and embeddings. The first part starts with an introduction to the background, from logic, complexity, and graph theory, and develops the theory up to first applications in descriptive complexity theory and graph isomorphism testing. It may serve as the basis for a graduate-level course. The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

Table of Contents
Chapter 1 Introduction
Part I The basic theory
Chapter 2. Background from graph theory and logic
Chapter 3. Descriptive complexity
Chapter 4. Treelike decompositions
Chapter 5. Definable decompositions
Chapter 6. Graphs of bounded tree width
Chapter 7. Ordered treelike decompositions
chapter 8. 3-Connected components
Chapter 9. Graphs embeddable in a surface
Part II Definable decompositions of graphs with excluded minors
Chapter 10. Quasi-4-connected components
Chapter 11. K5-minor free graphs
Chapter 12. Completions of pre-decompositions
Chapter 13. Almost planar graphs
Chapter 14. Almost planar completions
Chapter 15. Almost embeddable graphs
Chapter 16. Decompositions of almost embeddable graphs
Chapter 17. Graphs with excluded minors
Chapter 18. Bits and pieces
Appendix Robertson and Seymour's version of the local structure theorem
Symbol index


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