Descriptive Complexity, Canonisation, and Definable Graph Structure Theory

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Cambridge University Press, Aug 17, 2017 - Computers - 554 pages
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 The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.
 

Contents

Background from graph theory and logic
14
Descriptive complexity
40
Treelike decompositions
94
Definable decompositions
123
Graphs of bounded tree width
148
Ordered treelike decompositions
155
3connected components
176
Graphs embeddable in a surface
189
Almost planar graphs
301
Almost planar completions
361
Almostembeddable graphs
393
Decompositions of almostembeddable graphs
438
Graphs with excluded minors
487
Bits and pieces
502
Appendix A Robertson and Seymours version of the Local
518
Symbol index
531

Quasi4connected components
232
K5minorfree graphs
272

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