Mutual visibility in graphs

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 se...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 419; p. 126850
Main Author: Di Stefano, Gabriele
Format: Journal Article
Language:English
Published: Elsevier Inc 15.04.2022
Subjects:
Computational complexity
Graph classes
Graph invariant
Mutual visibility
Mutual visibility
Graph classes
Graph invariant
Computational complexity
ISSN:0096-3003
Online Access:Get full text
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Summary: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.
ISSN:0096-3003
DOI:10.1016/j.amc.2021.126850