Mutual visibility in hypercube-like graphs
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Let G be a graph and X⊆ 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.
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