Dominator color class dominating sets in triangular ladder and mobius ladder graphs

Abstract

Let $G=(V,E)$ be a graph. Let $\C= \{ \C_1, \C_2, \dots, \C_{\chi}\}$ be a proper coloring of $G$. $\C$ is called dominator color class dominating set of $G$ if each vertex $v$ in $G$ is dominated by a color class $\C_i\in \C$ and each $\C_i\in \C$ is dominated by a vertex $v$ in $G$. The dominator color class domination number is the minimum cardinality taken over all dominator color class dominating sets in $G$ is denoted by $\gamma_\chi^{d}(G)$. In this paper we obtain $\gamma_\chi^{d}(G)$ for triangular ladder graph and mobius ladder graph.