Mutual and total mutual visibility in hypercube-like graphs

Abstract

Let G be a graph and $X\subseteq V\left(G\right)$. Then, vertices x and y of G are X-visible if there exists a shortest $x,y$-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 $V\left(G\right)$ 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) $\mu \left(G\right)$ (resp. ${\mu }_t\left(G\right)$) of G. It is known that computing $\mu \left(G\right)$ is an NP-complete problem, as well as ${\mu }_t\left(G\right)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, Fibonacci cubes, cube-connected cycles, and butterflies). Concerning computing $\mu \left(G\right)$, we provide approximation algorithms for hypercubes, Fibonacci cubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing ${\mu }_t\left(G\right)$ (in the literature, already studied in hypercubes), whereas we obtain exact formulae for both cube-connected cycles and butterflies.