Mutual-visibility sets in Cartesian products of paths and cycles

TL;DR

This paper investigates the mutual-visibility problem in Cartesian products of paths and cycles, providing optimal solutions and complete characterizations for most cases, advancing understanding of visibility sets in these graph classes.

Contribution

It offers new solutions and complete characterizations for the mutual-visibility problem in Cartesian products of paths and cycles, a previously less understood area.

Findings

Optimal mutual-visibility sets identified for most Cartesian products of a cycle and a path.

Complete solutions provided for the mutual-visibility problem in Cartesian products of two cycles.

Advances in understanding visibility properties in complex graph structures.

Abstract

For a given graph $G$, the mutual-visibility problem asks for the largest set of vertices $M \subseteq V(G)$ with the property that for any pair of vertices $u,v \in M$ there exists a shortest $u,v$-path of $G$ that does not pass through any other vertex in $M$. The mutual-visibility problem for Cartesian products of a cycle and a path, as well as for Cartesian products of two cycles, is considered. Optimal solutions are provided for the majority of Cartesian products of a cycle and a path, while for the other family of graphs, the problem is completely solved.