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Mutual visibility in graphs

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  • Di Stefano, Gabriele

Abstract

Let G=(V,E) be a graph and P⊆V a set of points. Two points are mutually visible if there is a shortest path between them without further points. P is a mutual-visibility set if its points are pairwise mutually visible. The mutual-visibility number of G is the size of any largest mutual-visibility set. In this paper we start the study about this new invariant and the mutual-visibility sets in undirected graphs. We introduce the Mutual-Visibility problem which asks to find a mutual-visibility set with a size larger than a given number. We show that this problem is NP-complete, whereas, to check whether a given set of points is a mutual-visibility set is solvable in polynomial time. Then we study mutual-visibility sets and mutual-visibility numbers on special classes of graphs, such as block graphs, trees, grids, tori, complete bipartite graphs, cographs. We also provide some relations of the mutual-visibility number of a graph with other invariants.

Suggested Citation

  • Di Stefano, Gabriele, 2022. "Mutual visibility in graphs," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    • Handle: RePEc:eee:apmaco:v:419:y:2022:i:c:s0096300321009334
    DOI: 10.1016/j.amc.2021.126850
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    References listed on IDEAS

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    1. Anand, Bijo S. & Chandran S. V., Ullas & Changat, Manoj & Klavžar, Sandi & Thomas, Elias John, 2019. "Characterization of general position sets and its applications to cographs and bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 84-89.
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    Cited by:

    1. Manuel, Paul & Brešar, Boštjan & Klavžar, Sandi, 2023. "Geodesic packing in graphs," Applied Mathematics and Computation, Elsevier, vol. 459(C).
    2. Brešar, Boštjan & Yero, Ismael G., 2024. "Lower (total) mutual-visibility number in graphs," Applied Mathematics and Computation, Elsevier, vol. 465(C).

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