On graphs with equal domination and 2-domination numbers

Abstract

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