2,651 research outputs found

    The Disjoint Domination Game

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    We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the (2:1)(2:1) biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the (a:b)(a:b) biased game for (a:b)(2:1)(a:b)\neq (2:1). For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.Comment: 18 page

    Protecting a Graph with Mobile Guards

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    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

    Measuring domination in directed graphs

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    Pareto optimality in house allocation problems

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    We study Pareto optimal matchings in the context of house allocation problems. We present an O(\sqrt{n}m) algorithm, based on Gales Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching

    Maker-Breaker domination game on trees when Staller wins

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    In the Maker-Breaker domination game played on a graph GG, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then γSMB(G)\gamma_{\rm SMB}(G) (resp., γSMB(G)\gamma_{\rm SMB}'(G)) denotes the minimum number of moves Staller needs to win. For every positive integer kk, trees TT with γSMB(T)=k\gamma_{\rm SMB}'(T)=k are characterized. Applying hypergraphs, a general upper bound on γSMB\gamma_{\rm SMB}' is proved. Let S=S(n1,,n)S = S(n_1,\dots, n_\ell) be the subdivided star obtained from the star with nn edges by subdividing its edges n11,,n1n_1-1, \ldots, n_\ell-1 times, respectively. Then γSMB(S)\gamma_{\rm SMB}'(S) is determined in all the cases except when 4\ell\ge 4 and each nin_i is even. The simplest formula is obtained when there are are at least two odd nin_is. If n1n_1 and n2n_2 are the two smallest such numbers, then γSMB(S(n1,,n))=log2(n1+n2+1)\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil. For caterpillars, exact formulas for γSMB\gamma_{\rm SMB} and for γSMB\gamma_{\rm SMB}' are established
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