Some bounds on the p-domination number in trees

Let p be a positive integer andG= (V ,E) a graph. A subset S of V is a p-dominating set if every vertex of V − S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p − 1. The minimum cardinality of a p-dominating set a ofG is the p-domination number p(G) and the maximum cardinality of a p-dependent set ofG is the p-dependence number p(G). For every positive integer p 2, we show that for a bipartite graphG, p(G) is bounded above by (|V | + |Yp|)/2, where Yp is the set of vertices of G of degree at most p− 1, and for every tree T, p(T ) is bounded below by p−1(T ). Moreover, we characterize the trees achieving equality in each bound. © 2006 Elsevier B.V. All rights reserved.