Dominated chromatic number of some operations on a graph

TL;DR

This paper investigates how the dominated chromatic number of a graph changes when the graph undergoes various vertex and edge operations, providing insights into graph coloring stability under modifications.

Contribution

It introduces new results on the behavior of the dominated chromatic number under different graph operations, expanding understanding of graph coloring dynamics.

Findings

Determined how vertex operations affect $oxed{ ext{dominated chromatic number}}$.

Analyzed the impact of edge modifications on $oxed{ ext{dominated chromatic number}}$.

Provided bounds and exact values for specific graph classes after operations.

Abstract

Let $G$ be a simple graph. The dominated coloring of a graph $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $\chi_{dom}(G)$. In this paper, we examine the effects on $\chi_{dom}(G)$ when $G$ is modified by operations on vertex and edge of $G$.