Mutual Visibility in Graphs

TL;DR

This paper introduces the mutual-visibility number in graphs, explores its properties, computational complexity, and behavior on various graph classes, establishing foundational results for this new graph invariant.

Contribution

It defines the mutual-visibility number, proves NP-completeness of the related decision problem, and analyzes this invariant on specific graph classes.

Findings

Mutual-visibility problem is NP-complete.

Checking mutual-visibility sets can be done in polynomial time.

Mutual-visibility numbers are characterized for certain graph classes.

Abstract

Let $G=(V,E)$ be a graph and $P\subseteq V$ a set of points. Two points are mutually visible if there is a shortest path between them without further points. $P$ is a mutual-visibility set if its points are pairwise mutually visible. The mutual-visibility number of $G$ is the size of any largest mutual-visibility set. In this paper we start the study about this new invariant and the mutual-visibility sets in undirected graphs. We introduce the mutual-visibility problem which asks to find a mutual-visibility set with a size larger than a given number. We show that this problem is NP-complete, whereas, to check whether a given set of points is a mutual-visibility set is solvable in polynomial time. Then we study mutual-visibility sets and mutual-visibility numbers on special classes of graphs, such as block graphs, trees, grids, tori, complete bipartite graphs, cographs. We also provide some relations of the mutual-visibility number of a graph with other invariants.