Descriptive complexity, canonisation, and definable graph structure theory

Description

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 The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.  Read more... Abstract: This groundbreaking, yet accessible book contains original results on the interaction between graph theory and computational complexity using methods from finite model theory. As well as a wealth of new, previously unpublished results, the author also gives an account of the established results in the area.  Read more... Content: 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.