Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products

TL;DR

This paper studies the total mutual-visibility number in graphs, characterizes graphs with zero such number, and explores its behavior in Cartesian products, providing bounds and exact values for specific cases.

Contribution

It characterizes graphs with zero total mutual-visibility number and derives bounds and exact formulas for this number in Cartesian product graphs.

Findings

Graphs with zero total mutual-visibility are characterized.

Exact total mutual-visibility numbers are determined for Cartesian products of complete graphs and trees.

The total mutual-visibility number can be arbitrarily larger than the product of individual numbers.

Abstract

If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $\mu_{\rm t}(G)$ of $G$. Graphs with $\mu_{\rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $\mu_{\rm t}(K_n\,\square\, K_m) = \max\{n,m\}$ and $\mu_{\rm t}(T\,\square\, H) = \mu_{\rm t}(T)\mu_{\rm t}(H)$, where $T$ is a tree and $H$ an arbitrary graph. It is also demonstrated that $\mu_{\rm t}(G\,\square\, H)$ can be arbitrary larger than $\mu_{\rm t}(G)\mu_{\rm t}(H)$.