423 research outputs found

    Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

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    The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity 2O(k)O(n)2^{O(k)} O(n). We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA '95 / JGAA '99)

    A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

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    Given a graph GG cellularly embedded on a surface Σ\Sigma of genus gg, a cut graph is a subgraph of GG such that cutting Σ\Sigma along GG yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any ε>0\varepsilon >0, we show how to compute a (1+ε)(1+ \varepsilon) approximation of the shortest cut graph in time f(ε,g)n3f(\varepsilon, g)n^3. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

    JSJ decompositions of Quadratic Baumslag-Solitar groups

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    Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math. Helv. 78 (2003) pp 740-751) that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester's results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.Comment: 20 pages, 2 figures. Several corrections and improvements from referee's report. Imprtant changes in Definition 5.1, and the proof of Theorem 5.5 (previously 5.4). Lemma 5.4 was adde

    Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves

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    Let Mg,[n]{\cal M}_{g,[n]}, for 2g−2+n>02g-2+n>0, be the D-M moduli stack of smooth curves of genus gg labeled by nn unordered distinct points. The main result of the paper is that a finite, connected \'etale cover {\cal M}^\l of Mg,[n]{\cal M}_{g,[n]}, defined over a sub-pp-adic field kk, is "almost" anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let \pi_1({\cal M}^\l_{\ol{k}}) be the geometric algebraic fundamental group of {\cal M}^\l and let {Out}^*(\pi_1({\cal M}^\l_{\ol{k}})) be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of {\cal M}^\l (this is the "∗\ast-condition" motivating the "almost" above). Let us denote by {Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})) the subgroup consisting of elements which commute with the natural action of the absolute Galois group GkG_k of kk. Let us assume, moreover, that the generic point of the D-M stack {\cal M}^\l has a trivial automorphisms group. Then, there is a natural isomorphism: {Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})). This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-pp-adic fields.Comment: This paper has been withdrawn because of a flaw in the paper "Profinite Teichm\"uller theory" of the first author, on which this paper built o

    Finite rigid sets in sphere complexes

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    A subcomplex X≤CX\leq \mathcal{C} of a simplicial complex is rigid if every locally injective, simplicial map X→CX\to\mathcal{C} is the restriction of an automorphism of C\mathcal{C}. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite rigid sets. The Hatcher sphere complex is an analog of the curve complex for isotopy classes of essential spheres in a connect sum of nn copies of S1×S2S^1\times S^2. We show that there is an exhaustion of the sphere complex by finite rigid sets for all n≥3n\ge 3 and that when n=2n=2 the sphere complex does not have finite rigid sets.Comment: 13 pages, 5 figure
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