Introduction of color class dominating sets in graphs

Abstract

Let $G=(V, E)$ be a graph. In this paper, we define a new graph parameter called color class domination number of $G$. A color class dominating set of $G$ is a proper coloring $\mathscr{C}$ of $G$ with the extra property that every color class in $\mathscr{C}$ is dominated by a vertex in $G$. A color class dominating set is said to be a minimal color class dominating set if no proper subset of $\mathscr{C}$ 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 $\gamma_\chi(G)$ for Path graph, Cycle graph, Helm graph, Flower graph, Sunflower graph, Gear graph and Sunlet graph.