Unique isolate dom-chromatic number in graphs

Abstract

An unique isolate dom-coloring set(UIDC-set) $D$ of a graph $G$ is a Dom-coloring set(DC-set) $D\subseteq V(G)$ such that $\langle D\rangle$ has exactly one isolated vertex. The minimum cardinality of a minimal UIDC-set is called unique isolate dom-chromatic number(UIDC-number) of $G$, it is denoted by $\gamma_{0,dc}^U(G)$. In this paper, we introduce and study some properties of UIDC-set and we give UIDC-number of some families of graphs.