Strongly 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 $V(G)-D$. 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. An Isolate Super Dominating Set is said to be a "Strongly Isolate Super Dominating Set"(SISD-set), if there exists $a \in D$ such that $N_2(a) \cap D= \emptyset$, where $N_2(a)=\{b:d(a,b) \leq 2$ and $a \neq b\}$. The strongly isolate super domination number(SISD-number), denoted by $\gamma^S_{0,sp} (G)$, is the minimum cardinality of strongly isolate super dominating set of $G$. In this paper, we initiate a study on this parameter. We obtain basic properties of strongly isolate super dominating sets in graphs. Also we present upper and lower bounds for the strongly isolate super domination number.