26,221 research outputs found

    Strong geodetic problem on Cartesian products of graphs

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    The strong geodetic problem is a recent variation of the geodetic problem. For a graph GG, its strong geodetic number sg(G){\rm sg}(G) is the cardinality of a smallest vertex subset SS, such that each vertex of GG lies on a fixed shortest path between a pair of vertices from SS. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for sg(G □ H){\rm sg}(G \,\square\, H) is determined, as well as exact values for Km □ KnK_m \,\square\, K_n, K1,k □ PlK_{1, k} \,\square\, P_l, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.Comment: 18 pages, 9 figure

    The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

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    We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V(G), and the following terminology. Two vertices u,v is an element of V(G) are strongly resolved by a vertex w is an element of V(G), if there is a shortest w-v path containing u or a shortest w-u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S subset of V is an SSMG for F, if such set S is a strong metric generator for every graph G is an element of F. The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F, and is denoted by Sds(F). The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sds(F) is described. That is, it is proved that computing Sds(F) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature

    Partitioning the vertex set of GG to make G □ HG\,\Box\, H an efficient open domination graph

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    A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs GG for which the Cartesian product Gâ–ˇHG \Box H is an efficient open domination graph when HH is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of V(G)V(G). For the class of trees when HH is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products Gâ–ˇHG \Box H when HH is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure
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