On the dominated chromatic number of certain graphs

TL;DR

This paper investigates the dominated chromatic number of graphs, focusing on its stability and bondage number, and provides insights into how these parameters change with graph modifications.

Contribution

It introduces the concepts of dominated stability and bondage number and analyzes these parameters for specific classes of graphs.

Findings

Determined the dominated chromatic number for certain graph classes.

Established bounds for dominated stability and bondage number.

Analyzed how vertex and edge removals affect the dominated chromatic number.

Abstract

Let $G$ be a simple graph. The dominated coloring of $G$ 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 $\chi_{dom}(G)$. Stability (bondage number) of dominated chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the dominated chromatic number of $G$. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs.