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 number@c"p(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number @c"1(G) is the usual domination number@c(G). This definition immediately leads to the inequality @c(G)=<@c"2(G). In this paper we present some sufficient as well as some necessary conditions for graphs G with the property that @c"2(G)=@c(G). In particular, we characterize all cactus graphs H with @c"2(H)=@c(H).