Mutual-visibility Coloring of Graphs

TL;DR

This paper investigates the mutual-visibility coloring problem in graphs, proving its computational complexity and providing exact solutions for specific tree structures.

Contribution

It establishes NP-completeness of the mutual-visibility chromatic number problem and computes exact values for glued binary and t-ary trees.

Findings

Mutual-visibility chromatic number determination is NP-complete.

Exact values are found for glued binary trees.

Exact values are found for glued t-ary trees.

Abstract

The mutual-visibility chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ such that each color class is a mutual-visibility set. In this paper, we prove that determining the mutual-visibility chromatic number of a graph is NP-complete even when restricted to the class of graphs having diameter four and mutual-visibility chromatic number two. We further determine the exact value of the mutual-visibility chromatic number for glued binary trees and glued $t$-ary trees.