Variety of mutual-visibility problems 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.