On graphs with equal domination and 2-domination numbers

Let G be a simple graph, and let p be a positive integer. A subset D⊆V(G) is a p-dominating set of the graph G, if every vertex v∈V(G)-D is adjacent to at least p vertices in D. The p-domination numberγ_p(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ₁(G) is the usual domination numberγ(G). This definition immediately leads to the inequality γ(G)?γ₂(G).In this paper we present some sufficient as well as some necessary conditions for graphs G with the property that γ₂(G)=γ(G). In particular, we characterize all cactus graphs H with γ₂(H)=γ(H).