Dominator Chromatic Number of Derived Graph of Some Graphs

Abstract

In a graph $G$, the distance $d(u, v)$ between a pair of vertices $u$ and $v$ is the length of a shortest path joining them. The Derived graph, $G^{\dag} $ is the graph whose vertices are same as the vertices of $G$ and two vertices in $G^{\dag} $ are adjacent if and only if the distance between them in $G$ is two. In this paper we obtain the exact value for $\chi_d$ for Derived Graph of Path, Cycle, Sunlet graph, Bistar graph and Triple Star graph respectively.