666 research outputs found

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

    Full text link
    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

    Computational Geometry Column 42

    Get PDF
    A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

    On the smallest snarks with oddness 4 and connectivity 2

    Get PDF
    A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked that there are exactly two such graphs of that order. However, this remark is incorrect as -- using an exhaustive computer search -- we show that there are in fact three snarks with oddness 4 on 28 vertices. In this note we present the missing snark and also determine all snarks with oddness 4 up to 34 vertices.Comment: 5 page
    corecore