A study on unique strong isolate semitotal domination in graphs

Abstract

A set $S$ of vertices in a graph $G$ is said to be a unique strong isolate semitotal dominating set of $G$ if it is an isolated dominating set of $G$ in which there exists exactly one vertex $v\in D$ such that $N_2(v)\cap D=\phi$, where $N_2(v)=\{x/d(x,v)\leq 2$ and $x \neq v\}$ and $N_2(u)\cap D\neq \phi$ for every $u(\neq v)\in D$ and $|D-\{v\}|>1$. In this paper, we obtained some bounds for USISTD-set. Further, we studied some basic properties of USISTD-set. Also the USISTD number for disconnected graphs are obtained.