6,100 research outputs found

    Bondage number of grid graphs

    Full text link
    The bondage number b(G)b(G) of a nonempty graph GG is the cardinality of a smallest set of edges whose removal from GG results in a graph with domination number greater than the domination number of GG. Here we study the bondage number of some grid-like graphs. In this sense, we obtain some bounds or exact values of the bondage number of some strong product and direct product of two paths.Comment: 13 pages. Discrete Applied Mathematics, 201

    The bondage number of graphs on topological surfaces and Teschner's conjecture

    Get PDF
    The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner's Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201

    The Algorithmic Complexity of Bondage and Reinforcement Problems in bipartite graphs

    Full text link
    Let G=(V,E)G=(V,E) be a graph. A subset D⊆VD\subseteq V is a dominating set if every vertex not in DD is adjacent to a vertex in DD. The domination number of GG, denoted by γ(G)\gamma(G), is the smallest cardinality of a dominating set of GG. The bondage number of a nonempty graph GG is the smallest number of edges whose removal from GG results in a graph with domination number larger than γ(G)\gamma(G). The reinforcement number of GG is the smallest number of edges whose addition to GG results in a graph with smaller domination number than γ(G)\gamma(G). In 2012, Hu and Xu proved that the decision problems for the bondage, the total bondage, the reinforcement and the total reinforcement numbers are all NP-hard in general graphs. In this paper, we improve these results to bipartite graphs.Comment: 13 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1109.1657; and text overlap with arXiv:1204.4010 by other author

    The bondage number of random graphs

    Get PDF
    A dominating set of a graph is a subset DD of its vertices such that every vertex not in DD is adjacent to at least one member of DD. The domination number of a graph GG is the number of vertices in a smallest dominating set of GG. The bondage number of a nonempty graph GG is the size of a smallest set of edges whose removal from GG results in a graph with domination number greater than the domination number of GG. In this note, we study the bondage number of binomial random graph G(n,p)G(n,p). We obtain a lower bound that matches the order of the trivial upper bound. As a side product, we give a one-point concentration result for the domination number of G(n,p)G(n,p) under certain restrictions
    • …
    corecore