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

2022/12/14 by Tian, Jing, Klavžar, Sandi · 6 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.07193

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 μ_{\rm t}(G) of G. Graphs with μ_{\rm t}(G) = 0 are characterized as the graphs in which no vertex is the central vertex of a convex P₃. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, μ_{\rm t}(K_n \square K_m) = max\n,m\ and μ_{\rm t}(T \square H) = μ_{\rm t}(T)μ_{\rm t}(H), where T is a tree and H an arbitrary graph. It is also demonstrated that μ_{\rm t}(G \square H) can be arbitrary larger than μ_{\rm t}(G)μ_{\rm t}(H).

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