Isolate dom-chromatic number in graphs

Abstract

An isolate dom-coloring set(IDC-set) $D$ of a graph $G$ is a dom-coloring set(DC-set) $D\subseteq V(G)$ such that $\langle D\rangle$ has at least one isolated vertex. An IDC-set is minimal if no proper subset of $D$ is an IDC-set. The minimum cardinality of a minimal isolate DC-set is called IDC-number of $G$, it is denoted by $\gamma_{0,dc}(G)$. In this paper, we study some properties of IDC-set and we give IDC-number of some families of graphs.