Total Mutual-Visibility in Graphs with Emphasis on Lexicographic and Cartesian Products

Given a connected graph G , the total mutual-visibility number of G , denoted μ _t(G) , is the cardinality of a largest set S⊆ V(G) such that for every pair of vertices x,y∈ V(G) there is a shortest x , y -path whose interior vertices are not contained in S . Several combinatorial properties, including bounds and closed formulae, for μ _t(G) are given in this article. Specifically, we give several bounds for μ _t(G) in terms of the diameter, order and/or connected domination number of G and show characterizations of the graphs achieving the limit values of some of these bounds. We also consider those vertices of a graph G that either belong to every total mutual-visibility set of G or does not belong to any of such sets, and deduce some consequences of these results. We determine the exact value of the total mutual-visibility number of lexicographic products in terms of the orders of the factors, and the total mutual-visibility number of the first factor in the product. Finally, we give some bounds and closed formulae for the total mutual-visibility number of Cartesian product graphs.