4,541 research outputs found

    Heredity for generalized power domination

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    In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for γ_p,k(G−e)\gamma\_{p,k}(G-e), γ_p,k(G/e)\gamma\_{p,k}(G/e) and for γ_p,k(G−v)\gamma\_{p,k}(G-v) in terms of γ_p,k(G)\gamma\_{p,k}(G), and give examples for which these bounds are tight. We characterize all graphs for which γ_p,k(G−e)=γ_p,k(G)+1\gamma\_{p,k}(G-e) = \gamma\_{p,k}(G)+1 for any edge ee. We also consider the behaviour of the propagation radius of graphs by similar modifications.Comment: Discrete Mathematics and Theoretical Computer Science, 201

    Maximum Number of Minimum Dominating and Minimum Total Dominating Sets

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    Given a connected graph with domination (or total domination) number \gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of dominating and total dominating sets of size \gamma. An exact answer is provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page

    Location-domination in line graphs

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    A set DD of vertices of a graph GG is locating if every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)∩D≠N(v)∩DN(u) \cap D \neq N(v) \cap D, where N(u)N(u) denotes the open neighborhood of uu. If DD is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of GG. A graph GG is twin-free if every two distinct vertices of GG have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph GG without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure
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