Mutual-visibility in strong products of graphs via total mutual-visibility

Abstract

Let G be a graph and X ⊆ V(G). Then X is a mutual-visibility set if each pair of vertices from X is connected by a geodesic with no internal vertex in X. The mutualvisibility number µ(G) of G is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.