Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products
Jing Tian, Sandi Klavžar · Discussiones Mathematicae Graph Theory · 2023
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 µ t (G) of G. Graphs with µ 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, µ t (K n K m ) = max{n, m} and µ t (T H) = µ t (T )µ t (H), where T is a tree and H an arbitrary graph.It is also demonstrated that µ t (G H) can be arbitrary larger than µ t (G)µ t (H).