Perfect domination edge subdivision critical and stable Graphs

Abstract

Let $G$ be a graph. A subset $S$ of vertices in a graph $G$ is a perfect dominating set if every vertex in $V \backslash S$ is adjacent to exactly one vertex in $S$. A graph is perfect domination edge subdivision critical if the subdivision of an arbitrary edge increases the perfect domination number. On the other hand, a graph is perfect domination edge subdivision stable if the subdivision of an arbitrary edge leaves the perfect domination number unchanged. In this paper, we initiate the study of perfect domination critical and stable graphs upon edge subdivision. We discuss some graphs which are perfect domination critical and stable.