Geodesic
Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1277-1291

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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).
Keywords: mutual-visibility set, total mutual-visibility set, bypass vertex, Cartesian product of graphs, tree 
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     title = {Graphs with total mutual-visibility number zero and total mutual-visibility in {Cartesian} products},
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Tian, Jing; Klavžar, Sandi. Graphs with total mutual-visibility number zero and total mutual-visibility in Cartesian products. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1277-1291. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a2/