General bounds on limited broadcast domination

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded by a constant k . The minimum cost of such a dominating broadcast is the k -broadcast dominating number. We present a uni ed upper bound on this parameter for any value of k in terms of both k and the order of the graph. For the speci c case of the 2-broadcast dominating number, we show that this bound is tight for graphs as large as desired. We also study the family of caterpillars, providing a smaller upper bound, which is attained by a set of such graphs with unbounded order.Preprin

On perfect and quasiperfect dominations in graphs

A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Postprint (published version