Mutual visibility in hypercube-like graphs

Abstract

Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $μ(G)$ (resp. $μ_t(G)$) of $G$. It is known that computing $μ(G)$ is an NP-complete problem, as well as $μ_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $μ(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $μ_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies.