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) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies.

翻译：令$G$为一个图，且$X\subseteq V(G)$。若存在一条最短$u,v$-路径，其所有内部顶点均不属于$X$，则称$G$中的顶点$x$与$y$是$X$-可见的。集合$X$称为$G$的互可见集，如果$X$中任意两个顶点都是$X$-可见的；而$X$称为全互可见集，如果$V(G)$中任意两个顶点都是$X$-可见的。最大互可见集（或全互可见集）的基数称为$G$的互可见数（或全互可见数），记作$\mu(G)$（或$\mu_t(G)$）。已知计算$\mu(G)$和$\mu_t(G)$均为NP完全问题。本文研究了超立方体类网络（即超立方体、立方体连接环与蝶形网络）中的（全）互可见性。关于$\mu(G)$的计算，我们对超立方体与立方体连接环给出了近似算法，并为蝶形网络给出了精确公式。关于$\mu_t(G)$的计算（该问题在文献中已针对超立方体展开研究），我们为立方体连接环与蝶形网络提供了精确公式。