An introduction of total dominator color class total dominating sets in graphs

Abstract

Let $G$ be a finite, undirected and connected graph with minimum degree at least one. In this paper we define a new graph parameter called total dominator color class total domination number of $G$. A proper coloring $\mathcal{C}$ of $G$ is said to be a total dominator color class total dominating set of $G$ if each vertex properly dominates a color class in $\mathcal{C}$ and each color class in $\mathcal{C}$ is properly dominated by a vertex in $\mathrm{V}(\mathrm{G})$. A total dominator color class total dominating set $D$ of $G$ is a minimal total dominator color class total dominating set if no proper subset of $D$ is a total dominator color class total dominating set of $G$ . The total dominator color class total domination number is the minimum cardinality taken over all minimal total dominator color class total dominating sets in $G$ and is denoted by $\gamma_{\chi}^{t d}(G)$. Here we obtain the characterization of lower and upper bounds of $\gamma_{\chi}^{t d}(G)$, and total dominator color class total domination number of Path graph, Cycle graph and Banana graph.