Mutual-visibility problems in Kneser and Johnson graphs

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Let G be a connected graph and X ⊆ V (G). By definition, two vertices u and v are X -visible in G if there exists a shortest u, v-path with all internal vertices being outside of the set X . The largest size of X such that any two vertices of G (resp. any two vertices from X ) are X -visible is the total mutual-visibility number (resp. the mutual-visibility number) of G. In this paper, we determine the total mutual-visibility number of Kneser graphs, bipartite Kneser graphs, and Johnson graphs. The formulas proved for Kneser, and bipartite Kneser graphs are related to the size of transversal-critical uniform hypergraphs, while the total mutualvisibility number of Johnson graphs is equal to a hypergraph Turán number. Exact values or estimations for the mutual-visibility number over these graph classes are also established.