36,684 research outputs found

    A note on the packing of two copies of some trees into their third power

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    AbstractIt is proved in [1] that if a tree T of order n is not a star, then there exists an edge-disjoint placement of two copies of this tree into its fourth power.In this paper, we prove the packing of some trees into their third power

    Large induced subgraphs via triangulations and CMSO

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    We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

    Structurally Parameterized d-Scattered Set

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    In dd-Scattered Set we are given an (edge-weighted) graph and are asked to select at least kk vertices, so that the distance between any pair is at least dd, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any d≥2d\ge2, an O∗(dtw)O^*(d^{\textrm{tw}})-time algorithm, where tw\textrm{tw} is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - dd-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if kk is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A 2O(td2)2^{O(\textrm{td}^2)}-time algorithm parameterized by tree-depth (td\textrm{td}), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter ϵ>0\epsilon > 0, runs in time O∗((tw/ϵ)O(tw))O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})}) and returns a d/(1+ϵ)d/(1+\epsilon)-scattered set of size kk, if a dd-scattered set of the same size exists

    Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

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    We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Σ2\Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

    Extremal Infinite Graph Theory

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    We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
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