Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products

Abstract

If G is a graph and X V (G), then X is a total mutual-visibility set if every pair of vertices x and y of G admits a shortest x, y-path P with V (P ) X {x, y}. The cardinality of a largest total mutual-visibility set of G is the total mutual-visibility number t (G) of G. Graphs with t (G) = 0 are characterized as the graphs in which every vertex is the central vertex of a convex P 3 . The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, t (K n K m ) = max{n, m} and t (T H) = t (T ) t (H), where T is a tree and H an arbitrary graph. It is also demonstrated that t (G H) can be arbitrary larger than t (G) t (H).