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 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.