Isolate perfect dom-chromatic number in graphs

Abstract

An isolate perfect dom-coloring set(IPDC-set) $D$ of a graph $G$ is a dom-coloring set $D\subseteq V(G)$ such that $\langle D\rangle$ has at least one isolated vertex with $|N(r)\cap D|=1, \forall r\in V(G)-D$. The minimum cardinality of an IPDC-set is called isolate perfect dom-coloring number(IPDC-number) of $G$, it is denoted by $\gamma_{0,p,dc}(G)$. In this paper, we introduce and study some properties of IPDC-set. Also, we give IPDC-number of paths and some path related graphs.