A NOTE ON THE p- DOMINATION NUMBER OF TREES

Let p be a positive integer and G = (V(G); E(G)) a graph. A p-dominating set of G is a subset S of V(G) such that every vertex not in S is dominated by at least p vertices in S. The p-domination number gamma(p)(G) is the minimum cardinality among the p-dominating sets of G. Let T be a tree with order n >= 2 and p >= 2 a positive integer. A vertex of V(T) is a p-leaf if it has degree at most p - 1, while a p-support vertex is a vertex of degree at least p adjacent to a p-leaf. In this note, we show that gamma(p)(T) >= (n + vertical bar L-p(T)vertical bar - vertical bar S-p(T)vertical bar)/2, where L-p(T) and S-p(T) are the sets of p-leaves and p-support vertices of T, respectively. Moreover, we characterize all trees attaining this lower bound.