110,799 research outputs found

    Note on Perfect Forests

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    A spanning subgraph FF of a graph GG is called perfect if FF is a forest, the degree dF(x)d_F(x) of each vertex xx in FF is odd, and each tree of FF is an induced subgraph of GG. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph GG contains a perfect forest if and only if GG has an even number of vertices

    Note on Perfect Forests in Digraphs

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    A spanning subgraph FF of a graph GG is called {\em perfect} if FF is a forest, the degree dF(x)d_F(x) of each vertex xx in FF is odd, and each tree of FF is an induced subgraph of GG. Alex Scott (Graphs \& Combin., 2001) proved that every connected graph GG contains a perfect forest if and only if GG has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP-hard, for the three others this problem is polynomial-time solvable. Moreover, every digraph with only one strong component contains a directed forest of each of these three generalization types. One of our results extends Scott's theorem to digraphs in a non-trivial way

    Characterizing perfect recall using next-step temporal operators in S5 and sub-S5 Epistemic Temporal Logic

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    We review the notion of perfect recall in the literature on interpreted systems, game theory, and epistemic logic. In the context of Epistemic Temporal Logic (ETL), we give a (to our knowledge) novel frame condition for perfect recall, which is local and can straightforwardly be translated to a defining formula in a language that only has next-step temporal operators. This frame condition also gives rise to a complete axiomatization for S5 ETL frames with perfect recall. We then consider how to extend and consolidate the notion of perfect recall in sub-S5 settings, where the various notions discussed are no longer equivalent

    Fine-grained dichotomies for the Tutte plane and Boolean #CSP

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    Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1y=1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given nn-vertex graph cannot be determined in time exp(o(n))exp(o(n)) unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with nn vertices cannot be computed in time exp(o(n))exp(o(n)) unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.Comment: 16 pages, 1 figur

    Burning a Graph is Hard

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    Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is \textbf{NP}-complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path-forests if the number of arms and components, respectively, are fixed.Comment: 20 Pages, 4 figures, presented at GRASTA-MAC 2015 (October 19-23rd, 2015, Montr\'eal, Canada
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