Dominator Coloring Changing and Stable Graphs upon Vertex Removal

A dominator coloring is a proper vertex coloring of a graph \(G\) such that each vertex is adjacent to all the vertices of at least one color class or either alone in its color class. The minimum cardinality among all dominator coloring of \(G\) is a dominator chromatic number of \(G\), denoted by \(X_d(G)\). On removal of a vertex the dominator chromatic number may increase or decrease or remain unaltered. In this paper, we have characterized nontrivial trees for which dominator chromatic number is stable.