An algorithm for the dominator chromatic number of a tree

A dominator coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that it is a proper coloring and every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of $G$ is called the dominator chromatic number of $G$ . In this paper, we give a polynomial time algorithm computing the dominator chromatic number for every nontrivial tree.