On the p-reinforcement and the complexity

Let G = (V, E) be a graph and p be a positive integer. A subset S ⊆ V is called a p-dominating set if each vertex not in S has at least p neighbors in S. The p-domination number γp(G) is the size of a smallest p-dominating set of G. The p-reinforcement number rp(G) is the smallest number of edges whose addition to G results in a graph G ′ with γp(G ′) < γp(G). In this paper, we give an original study on the p-reinforcement, determine rp(G) for some graphs such as paths, cycles and complete t-partite graphs, and establish some upper bounds on rp(G). In particular, we show that the decision problem on rp(G) is NP-hard for a general graph G and a fixed integer p ≥ 2.