Variety of mutual-visibility problems in hypercubes

TL;DR

This paper investigates various mutual-visibility problems in hypercubes, defining different types of visibility sets and numbers, and presents new results on their properties and sizes.

Contribution

It introduces and analyzes the total, outer, and dual mutual-visibility numbers in hypercubes, expanding understanding of visibility concepts in graph theory.

Findings

Determined bounds for mutual-visibility numbers in hypercubes.

Characterized the structure of maximum mutual-visibility sets.

Compared different types of mutual-visibility sets in hypercubes.

Abstract

Let $G$ be a graph and $M \subseteq V(G)$. Vertices $x, y \in M$ are $M$-visible if there exists a shortest $x,y$-path of $G$ that does not pass through any vertex of $M \setminus \{x, y \}$. We say that $M$ is a mutual-visibility set if each pair of vertices of $M$ is $M$-visible, while the size of any largest mutual-visibility set of $G$ is the mutual-visibility number of $G$. If some additional combinations for pairs of vertices $x, y$ are required to be $M$-visible, we obtain the total (every $x,y \in V(G)$ are $M$-visible), the outer (every $x \in M$ and every $y \in V(G) \setminus M$ are $M$-visible), and the dual (every $x,y \in V(G) \setminus M$ are $M$-visible) mutual-visibility set of $G$. The cardinalities of the largest of the above defined sets are known as the total, the outer, and the dual mutual-visibility number of $G$, respectively. We present results on the variety of mutual-visibility problems in hypercubes.