Total dominator color class total dominating sets in Dutch windmill graph and coconut tree graph

Abstract

Let $G$ be a finite, undirected and connected graph with minimum degree at least one. 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 $\gamma_{\chi}^{t d}(G)$ for dutch windmill graph and coconut tree graph.