Variety of mutual-visibility problems in hypercubes

Abstract

Let G be a graph and $M\subseteq V\left(G\right)$. 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 $x,y\in V\left(G\right)$ are M-visible), the outer (every $x\in M$ and every $y\in V\left(G\right)\setminus M$ are M-visible), and the dual (every $x,y\in V\left(G\right)\setminus M$ 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.