Total mutual-visibility in graphs with emphasis on lexicographic and Cartesian products

TL;DR

This paper investigates the total mutual-visibility number in graphs, providing bounds, exact values for specific graph products, and characterizations, thereby advancing understanding of visibility properties in graph theory.

Contribution

It introduces new bounds and formulas for the total mutual-visibility number, especially for lexicographic and Cartesian product graphs, and characterizes extremal cases.

Findings

Bounds in terms of diameter, order, and domination number

Exact values for lexicographic product graphs

Bounds and formulas for Cartesian product graphs

Abstract

Given a connected graph $G$, the total mutual-visibility number of $G$, denoted $\mu_t(G)$, is the cardinality of a largest set $S\subseteq V(G)$ such that for every pair of vertices $x,y\in V(G)$ there is a shortest $x,y$-path whose interior vertices are not contained in $S$. Several combinatorial properties, including bounds and closed formulae, for $\mu_t(G)$ are given in this article. Specifically, we give several bounds for $\mu_t(G)$ in terms of the diameter, order and/or connected domination number of $G$ and show characterizations of the graphs achieving the limit values of some of these bounds. We also consider those vertices of a graph $G$ that either belong to every total mutual-visibility set of $G$ or does not belong to any of such sets, and deduce some consequences of these results. We determine the exact value of the total mutual-visibility number of lexicographic products in terms of the orders of the factors, and the total mutual-visibility number of the first factor in the product. Finally, we give some bounds and closed formulae for the total mutual-visibility number of Cartesian product graphs.