19,193 research outputs found

    3-Factor-criticality in double domination edge critical graphs

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    A vertex subset SS of a graph GG is a double dominating set of GG if ∣N[v]∩S∣≥2|N[v]\cap S|\geq 2 for each vertex vv of GG, where N[v]N[v] is the set of the vertex vv and vertices adjacent to vv. The double domination number of GG, denoted by γ×2(G)\gamma_{\times 2}(G), is the cardinality of a smallest double dominating set of GG. A graph GG is said to be double domination edge critical if γ×2(G+e)<γ×2(G)\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G) for any edge e∉Ee \notin E. A double domination edge critical graph GG with γ×2(G)=k\gamma_{\times 2}(G)=k is called kk-γ×2(G)\gamma_{\times 2}(G)-critical. A graph GG is rr-factor-critical if G−SG-S has a perfect matching for each set SS of rr vertices in GG. In this paper we show that GG is 3-factor-critical if GG is a 3-connected claw-free 44-γ×2(G)\gamma_{\times 2}(G)-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

    Zero forcing in iterated line digraphs

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    Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications

    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
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