# More on the total dominator chromatic number of a graph

## TL;DR

This paper investigates the total dominator chromatic number of graphs, focusing on the neighborhood corona and r-gluing operations, and explores the stability and bondage number related to vertex and edge removal effects.

## Contribution

It introduces new bounds and properties for the total dominator chromatic number in complex graph operations and analyzes stability and bondage numbers for specific graph classes.

## Key findings

- Derived bounds for TDC number of neighborhood corona graphs.
- Analyzed TDC number of r-gluing of graphs.
- Studied stability and bondage number for certain graphs.

## Abstract

Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $\chi_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of $r$-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the TDC-number of $G$. We study the stability and bondage number of certatin graphs.