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

Abstract

AbstractGiven a connected graph G, the total mutual-visibility number of G, denoted $$\mu _t(G)$$ μ t ( G ) , is the cardinality of a largest set $$S\subseteq V(G)$$ S ⊆ V ( G ) such that for every pair of vertices $$x,y\in V(G)$$ 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 $$\mu _t(G)$$ μ t ( G ) are given in this article. Specifically, we give several bounds for $$\mu _t(G)$$ 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.