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

Let G be a graph and X subset of 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 mu ( 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. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).