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 The second part is more advanced and mainly devoted to the proof of a single, previously unpublished theorem: properties of graphs with excluded minors are decidable in polynomial time if, and only if, they are definable in fixed-point logic with counting.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

531 | |

### Other editions - View all

Descriptive Complexity, Canonisation, and Definable Graph Structure Theory Martin Grohe Limited preview - 2017 |

### Common terms and phrases

3-connected graph 3-hinge admits IFP-definable ordered algorithm bd(D binary relation branch vertices canonisation captures PTIME CLAIM class of graphs closed disk complete graph component of G connected component contains contradiction Corollary d-scheme decomposition of G defines an ordered definition denote descriptive complexity edge endvertices facial cycle formula Furthermore graph embedded graph G grid Hence homeomorphic IFP-definable ordered treelike IFP-formula IFP+C implies induced subgraph induction internally disjoint isomorphism Lemma Let G Let H Let Q let t e V(D Lifting Lemma linear order logic noncontractible Note NP(t od-scheme ordered completion ordered treelike decompositions p-arrangement parametrised path planar graphs polynomial pre-decomposition proof of Lemma prove PTIME quasi-4-connected r-node satisfied segment simple closed curve simple curve ſº(t Suppose surface t-structures torso transduction tree decomposition tree width tuple vertex set