The p-Bondage Number of Trees

Let p be a positive integer and G =( V, E) be a simple graph. A p- dominating set of G is a subset D subset of V such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with Delta(G) >= p is the minimum cardinality among all sets of edges B subset of E for which gamma p(G - B) > gamma(G). For any integer p >= 2 and tree T with Delta(T) >= p, this paper shows that 1 <= b(p)(T) <= (T) - p + 1, and characterizes all trees achieving the equalities.