On the mutual visibility in Cartesian products and triangle-free graphs

TL;DR

This paper studies the mutual visibility concept in graphs, focusing on Cartesian products and triangle-free graphs, introducing new types of mutual-visibility sets and characterizing specific graph classes.

Contribution

It extends the mutual-visibility framework to Cartesian products, introduces independent mutual-visibility sets, and characterizes triangle-free graphs with a mutual-visibility number of three.

Findings

Mutual-visibility in Cartesian products is analyzed and related to Zarenkiewicz's problem.

Characterization of triangle-free graphs with mutual-visibility number 3.

Introduction of independent mutual-visibility sets in graph analysis.

Abstract

Given a graph $G=(V(G), E(G))$ and a set $P\subseteq V(G)$, the following concepts have been recently introduced: $(i)$ two elements of $P$ are \emph{mutually visible} if there is a shortest path between them without further elements of $P$; $(ii)$ $P$ is a \emph{mutual-visibility set} if its elements are pairwise mutually visible; $(iii)$ the \emph{mutual-visibility number} of $G$ is the size of any largest mutual-visibility set. % In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and investigate independent mutual-visibility sets. In the very special case of the Cartesian product of two complete graphs the problem is shown to be equivalent to the well-known Zarenkiewicz's problem. We also characterize the triangle-free graphs with the mutual-visibility number equal to $3$.