k-Isolate perfect domination in graphs

Abstract

A dominating set $D$ of a graph $G$ is said to be a $k$-isolate perfect dominating set($k$IPDS) of $G$ if $\langle D\rangle$ has exactly $k$ isolated vertices and $D$ is a perfect dominating set of $G$. In this paper, we investigate the basic properties of $k$IPDS and provide the $k$IPD number of cycles, paths, complete $a$-partite graphs and some special graphs.