Effects of Some Operations on Domination Chromatic Number in Graphs

TL;DR

This paper investigates how various graph operations like removal, contraction, subdivision, and cycle extension influence the domination chromatic number, providing insights into its behavior under structural modifications.

Contribution

It introduces the effects of common graph operations on the domination chromatic number, expanding understanding of this parameter's sensitivity to graph transformations.

Findings

Vertex removal can increase or decrease $ ext{χ}_{dd}(G)$ depending on the graph.

Edge contraction impacts the domination chromatic number in specific ways.

Cycle extension alters $ ext{χ}_{dd}(G)$, with effects depending on cycle length.

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.