Dominator color class dominating sets on fire cracker, gear and flower graphs

Abstract

Let $G=(V, E)$ be a graph. Let $\mathscr{C}=\left\{\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3 \ldots \mathscr{C}_\chi\right\}$ be a proper coloring of $G . \mathscr{C}$ is called a dominator color class dominating set if each vertex $v$ in $G$ is dominated by a color class $C_i \in \mathscr{C}$ and each color class $C_i \in \mathscr{C}$ 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 $G$ and is denoted by $\gamma_\chi^d(G)$. In this paper, we obtain $\gamma_\chi^d(G)$ for Fire cracker graph, Gear graph, Flower graph and Sunflower graph.