2-Point set domination in separable graphs

Abstract

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.