481 research outputs found

    Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

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    We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph HH as a minor to graphs excluding HH as a topological subgraph. We prove that for a fixed HH, every graph excluding HH as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter H|H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding HH as a topological subgraph, Partial Dominating Set (find kk vertices whose closed neighborhood has maximum size) can be solved in time f(H,k)nO(1)f(H,k)\cdot n^{O(1)} time. More significantly, we show that on graphs excluding HH as a topological subgraph, Graph Isomorphism can be solved in time nf(H)n^{f(H)}. This result unifies and generalizes two previously known important polynomial-time solvable cases of Graph Isomorphism: bounded-degree graphs and HH-minor free graphs. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition

    Successor-Invariant First-Order Logic on Graphs with Excluded Topological Subgraphs

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    We show that the model-checking problem for successor-invariant first-order logic is fixed-parameter tractable on graphs with excluded topological subgraphs when parameterised by both the size of the input formula and the size of the exluded topological subgraph. Furthermore, we show that model-checking for order-invariant first-order logic is tractable on coloured posets of bounded width, parameterised by both the size of the input formula and the width of the poset. Our result for successor-invariant FO extends previous results for this logic on planar graphs (Engelmann et al., LICS 2012) and graphs with excluded minors (Eickmeyer et al., LICS 2013), further narrowing the gap between what is known for FO and what is known for successor-invariant FO. The proof uses Grohe and Marx's structure theorem for graphs with excluded topological subgraphs. For order-invariant FO we show that Gajarsk\'y et al.'s recent result for FO carries over to order-invariant FO

    Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

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    We give a fixed-parameter tractable algorithm that, given a parameter kk and two graphs G1,G2G_1,G_2, either concludes that one of these graphs has treewidth at least kk, or determines whether G1G_1 and G2G_2 are isomorphic. The running time of the algorithm on an nn-vertex graph is 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in 2O(k5logk)n52^{O(k^5\log k)}\cdot n^5 time that, for a given graph GG on nn vertices, either concludes that the treewidth of GG is at least kk, or: * finds in an isomorphic-invariant way a graph c(G)\mathfrak{c}(G) that is isomorphic to GG; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes GG together with a tree decomposition of GG of width O(k4)O(k^4). Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for G1G_1 and G2G_2 are equal.Comment: Full version of a paper presented at FOCS 201

    Induced Minor Free Graphs: Isomorphism and Clique-width

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    Given two graphs GG and HH, we say that GG contains HH as an induced minor if a graph isomorphic to HH can be obtained from GG by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on graphs that exclude a fixed graph as an induced minor. More precisely, we determine for every graph HH that Graph Isomorphism is polynomial-time solvable on HH-induced-minor-free graphs or that it is GI-complete. Additionally, we classify those graphs HH for which HH-induced-minor-free graphs have bounded clique-width. These two results complement similar dichotomies for graphs that exclude a fixed graph as an induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously appeared in the proceedings of the 41st International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2015

    Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

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    We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time npolylog(k)n^{\operatorname{polylog}(k)} where nn is the number of vertices and kk is the minimum treewidth of the given graphs and polylog(k)\operatorname{polylog}(k) is some polynomial in log(k)\operatorname{log}(k). Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur

    Deciding first-order properties of nowhere dense graphs

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    Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page
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