## Deciding First-Order Properties of Locally Tree-Decomposable Graphs (1999)

Venue: | In Proc. 26th ICALP |

Citations: | 74 - 12 self |

### BibTeX

@INPROCEEDINGS{Frick99decidingfirst-order,

author = {Markus Frick and Martin Grohe},

title = {Deciding First-Order Properties of Locally Tree-Decomposable Graphs},

booktitle = {In Proc. 26th ICALP},

year = {1999},

pages = {105--135}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We introduce the concept of a class of graphs being locally tree-decomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We show that for each locally tree-decomposable class C of graphs and for each property ' of graphs that is denable in rst-order logic, there is a linear time algorithm deciding whether a given graph G 2 C has property '. 1 Introduction It is an important task in the theory of algorithms to nd feasible instances of otherwise intractable algorithmic problems. A notion that has turned out to be extremely useful in this context is that of tree-width of a graph. 3-Colorability, Hamiltonicity, and many other NP-complete properties of graphs can be decided in linear time when restricted to graphs whose tree-width is bounded by a xed constant (see [Bod97] for a survey). Courcelle [Cou90] proved a meta-theorem, which easily implies numer...

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Citation Context ...meter tractability gives some evidence that the answer is no, as has been independently proved by Downey, Fellows, Taylor [11] and Papadimitriou, Yannakakis [20] (deciding first-order properties is AW=-=[1]-=--complete). Theorem 1.2 shows that deciding first-order properties of structures in a class of bounded local tree-width is fixed-parameter tractable. Furthermore, it has been used in [16] to show that... |

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Citation Context ...tes the number of vertices of the input graph). However, on planar graphs k-DOMINATING SET can be solved in time O(11 k n), and on graphs of valence at most l, it can be solved in time O((l + 1) k n) =-=[10]-=-. Unfortunately, the analogue of Courcelle’s theorem does not hold for planar graphs or classes of bounded valence; 3-COLORABILITY is a monadic second-order definable property that remains NP-complete... |

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Citation Context ...e and ϕ a property definable in first-order logic. Then there is a linear time algorithm deciding whether a given structure A ∈ C has property ϕ. It may be worth mentioning that in the terminology of =-=[27]-=-, our result can be rephrased as follows: When restricted to a locally tree-decomposable class of structures, the data complexity of first-order logic is in linear time. Examples of first-order defina... |

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Citation Context ...or planar graphs or classes of bounded valence; 3-COLORABILITY is a monadic second-order definable property that remains NP-complete when restricted to the class of planar graphs of valence at most 4 =-=[18]-=-. Instead of monadic second-order, we study the complexity of first-order definable properties. A preliminary version appeared in Proceedings of the 26th International Colloquium on Automata, Language... |

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Citation Context ...e algorithm that decides whether a given structure of valence at most l has this property. An observation that has been used for various algorithms on planar graphs (essentially it goes back to Baker =-=[5]-=-) is that there is a bound on the tree-width of a planar graph only depending on its diameter. A different way to see this is that a local neighborhood of a vertex in a planar graph has tree-width bou... |

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Citation Context ... width of a tree-decomposition (T , (Bt)t∈T) is max{|Bt| | t ∈ T } − 1. The tree-width tw(A) of A is the minimal width of a tree-decomposition of A. We occasionally use the following simple fact (cf. =-=[24]-=-). Lemma 4.1. Let w ≥ 1 and τ a vocabulary. Then there is a constant c such that for every τ-structure A of tree-width at most w we have ||A|| ≤ c|A|. Bodlaender [6] proved that for each w ≥ 1 there i... |

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Citation Context ...AMILTONICITY, and many other NP-complete properties of graphs can be decided in linear time when restricted to graphs whose tree-width is bounded by a fixed constant (see [7] for a survey). Courcelle =-=[8]-=- proved a meta-theorem, which easily implies numerous results of the abovementioned type: Let w ≥ 1 and ϕ be a property of graphs that is definable in monadic second-order logic. Then ϕ can be decided... |

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Citation Context ..., though. Example 6.10. We construct a class C of graphs of bounded local tree-width such that for every constant c there is a graph G ∈ C with ||G|| ≥ c|G|. We use the following theorem due to Erdös =-=[14]-=-: For all g, k ≥ 1 there exists a graph of girth greater than g and chromatic number greater than k. Remember that the girth g(G) of a graph G is the length of the shortest cycle in G and the chromati... |

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Citation Context ... know, such a transformation may blow up the formula size by a nonelementary factor. For conjunctive queries, we can avoid Gaifman’s theorem and instead use algorithmic techniques similar to those in =-=[13]-=-. With these techniques, the dependence on the formula size can be reduced to a singly exponential factor, which seems acceptable because usually in practice we have to 13sevaluate small formulas (que... |

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Citation Context ...operties is AW[1]-complete). Theorem 1.2 shows that deciding first-order properties of structures in a class of bounded local tree-width is fixed-parameter tractable. Furthermore, it has been used in =-=[16]-=- to show that for every class C of graphs such that there is some graph that is not a minor of any graph in C, deciding first-order properties of graphs in C is fixed-parameter tractable. The proofs o... |