Martin GroheCambridge University Press, 8/17/2017EAN 9781107014527, ISBN10: 1107014522Hardcover, 554 pages, 23.5 x 16 x 3.6 cmLanguage: EnglishDescriptive 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 The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.1. IntroductionPart I. The Basic Theory2. Background from graph theory and logic3. Descriptive complexity4. Treelike decompositions5. Definable decompositions6. Graphs of bounded tree width7. Ordered treelike decompositions8. 3-Connected components9. Graphs embeddable in a surfacePart II. Definable Decompositions of Graphs with Excluded Minors10. Quasi-4-connected components11. K5-minor free graphs12. Completions of pre-decompositions13. Almost planar graphs14. Almost planar completions15. Almost embeddable graphs16. Decompositions of almost embeddable graphs17. Graphs with excluded minors18. Bits and piecesAppendix. Robertson and Seymour's version of the local structure theoremReferencesSymbol indexIndex.
