Isolate super domination in graphs

Abstract

A dominating set $D$ of $V(G)$ in a graph $G=(V,E)$ is called super dominating set if for every $v \in V(G)-D$, there exists an external private neighbour of $v$ with respect to $v \in V(G)-D$. A super dominating set $D$ of a graph $G$ is said to be an "Isolate Super Dominating set"(ISD-set) of $G$ if $\langle D \rangle$ has at least one isolated vertex. The Isolate Super Domination number(ISD-number), denoted by $\gamma_{0,sp} (G)$, is the minimum cardinality of an isolate super dominating set of $G$. In this paper, we initiate s study on this parameter. We obtain basic properties of isolate super dominating sets in graphs. Also we present upper and lower bounds for the isolate super domination number.