Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

This groundbreaking, yet accessible book explores the interaction between graph theory and computational complexity using methods from finite model theory.

Martin Grohe (Author)

9781107014527, Cambridge University Press

Hardback, published 17 August 2017

554 pages, 60 b/w illus.
23.5 x 16 x 3.6 cm, 0.88 kg

'The book is divided evenly into two parts. Part I gives background and definitions of the main notions, and makes the book self-contained. Many results from descriptive complexity theory, and the author's earlier results, are clearly presented. Part II is devoted to the main theorem about graphs with excluded minors. The book ends with a symbol index and an index.' Pascal Michel, Mathematical Reviews

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 The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.

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

Subject Areas: Mathematical theory of computation [UYA], Algorithms & data structures [UMB], Combinatorics & graph theory [PBV], Discrete mathematics [PBD], Mathematical logic [PBCD]