Effects of Some Operations on Domination Chromatic Number in Graphs

Abstract

For a simple graph $G$, a domination coloring of $G$ is a proper vertex coloring such that every vertex of $G$ dominates at least one color class, and every color class is dominated by at least one vertex. The domination chromatic number, denoted by ${\chi }_{dd}(G)$, is minimum number of colors among all domination colorings of $G$. In this paper, we discuss the effects of some typical operations on ${\chi }_{dd}(G)$, such as vertex (edge) removal, vertex (edge) contraction, edge subdivision, and cycle extending.