98,087 research outputs found

    On Topological Minors in Random Simplicial Complexes

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    For random graphs, the containment problem considers the probability that a binomial random graph G(n,p)G(n,p) contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at p=1/np=1/n. We consider a natural analogue of this question for higher-dimensional random complexes Xk(n,p)X^k(n,p), first studied by Cohen, Costa, Farber and Kappeler for k=2k=2. Improving previous results, we show that p=Θ(1/n)p=\Theta(1/\sqrt{n}) is the (coarse) threshold for containing a subdivision of any fixed complete 22-complex. For higher dimensions k>2k>2, we get that p=O(n1/k)p=O(n^{-1/k}) is an upper bound for the threshold probability of containing a subdivision of a fixed kk-dimensional complex.Comment: 15 page

    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

    Semi-Transitive Orientations and Word-Representable Graphs

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    A graph G=(V,E)G=(V,E) is a \emph{word-representable graph} if there exists a word WW over the alphabet VV such that letters xx and yy alternate in WW if and only if (x,y)E(x,y)\in E for each xyx\neq y. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of GG is the minimum kk such that GG is a representable by a word, where each letter occurs kk times; such a kk exists for any word-representable graph. We show that the representation number of a word-representable graph on nn vertices is at most 2n2n, while there exist graphs for which it is n/2n/2.Comment: arXiv admin note: text overlap with arXiv:0810.031
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