Minimal 2-point set dominating sets of a graph

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 induced subgraph 〈 S ∪ T 〉 is connected. In this paper we characterize minimal 2-psd sets for a general graph. Based on the structure we examine 2-psd sets in a separable graph and discuss the criterion for a 2-psd set to be minimal.