On the Approximability of Graph Visibility Problems

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Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci

Published: 01 Jan 2025, Last Modified: 27 Sept 2025WALCOM 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph G of n vertices asks to find the largest set of vertices \(X\subseteq V(G)\), also called \(\mu \)-set, such that for any two vertices \(u,v\in X\), there is a shortest u, v-path P where all internal vertices of P are not in X. This means that u and v are visible w.r.t. X. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside X. The mutual-visibility problem and all its variations are known to be \(\textsf{NP}\)-complete on graphs of diameter 4.