Color class dominating sets in path and cycle related graphs

Abstract

Let $G = (V, E)$ be a graph. A color class dominating set of $G$ is a proper coloring $\C$ of $G$ with extra property that every color class in $\C$ is dominated by a vertex in $G$. A color class dominating set is said to be minimal color class dominating set if no proper subset of $\C$ is a color class dominating set of $G$. The color class domination number of $G$ is the minimum cardinality taken over all minimal color class dominating sets of $G$ and is denoted by $\gamma_\chi(G)$. Here we also obtain $\gamma_\chi(G)$ of Double fan graph, splitting graph, bright sun graph and comb graph.