2-point set domination number of a cactus

Abstract

A set D ⊆ V(G) is a 2-point set dominating set (2-psd set) of a graph G if for any subset S ⊆ V-D, there exists a non-empty subset T ⊆ D containing at most two vertices such that the subgraph <S ∪ T> induced by S ∪ T is connected. The 2-point set domination number of G, denoted by γ_2ps(G), is the minimum cardinality of a 2-psd set of G. In this paper we give the 2-psd number γ_2ps(G) for cactus. Also we give an alternate proof for γ_2ps(G) of unicyclic graphs which also reflects the structure of minimum 2-psd sets of unicyclic graphs.