23,843 research outputs found

    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

    On Minimum Maximal Distance-k Matchings

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    We study the computational complexity of several problems connected with finding a maximal distance-kk matching of minimum cardinality or minimum weight in a given graph. We introduce the class of kk-equimatchable graphs which is an edge analogue of kk-equipackable graphs. We prove that the recognition of kk-equimatchable graphs is co-NP-complete for any fixed k2k \ge 2. We provide a simple characterization for the class of strongly chordal graphs with equal kk-packing and kk-domination numbers. We also prove that for any fixed integer 1\ell \ge 1 the problem of finding a minimum weight maximal distance-22\ell matching and the problem of finding a minimum weight (21)(2 \ell - 1)-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of δlnV(G)\delta \ln |V(G)| unless P=NP\mathrm{P} = \mathrm{NP}, where δ\delta is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

    A characterization of trees with equal 2-domination and 2-independence numbers

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    A set SS of vertices in a graph GG is a 22-dominating set if every vertex of GG not in SS is adjacent to at least two vertices in SS, and SS is a 22-independent set if every vertex in SS is adjacent to at most one vertex of SS. The 22-domination number γ2(G)\gamma_2(G) is the minimum cardinality of a 22-dominating set in GG, and the 22-independence number α2(G)\alpha_2(G) is the maximum cardinality of a 22-independent set in GG. Chellali and Meddah [{\it Trees with equal 22-domination and 22-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal 22-domination and 22-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum 22-dominating and maximum 22-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.Comment: 17 pages, 4 figure

    Rainbow domination and related problems on some classes of perfect graphs

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    Let kNk \in \mathbb{N} and let GG be a graph. A function f:V(G)2[k]f: V(G) \rightarrow 2^{[k]} is a rainbow function if, for every vertex xx with f(x)=f(x)=\emptyset, f(N(x))=[k]f(N(x)) =[k]. The rainbow domination number γkr(G)\gamma_{kr}(G) is the minimum of xV(G)f(x)\sum_{x \in V(G)} |f(x)| over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

    Degree Sequence Index Strategy

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    We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the kk-independence and the kk-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and K1,rK_{1,r}-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
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