Color class dominating sets in ladder and grid graphs

Abstract

Let $G=(V, E)$ be a graph. 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 $\mathrm{G}$. The color class domination number of $\mathrm{G}$ is the minimum cardinality taken over all minimal color class dominating sets of $\mathrm{G}$ and is denoted by $\gamma_x(G)$. In this paper, we obtain $\gamma_x(G)$ for Ladder graph and Grid graph.