Mutual-visibility problems in Kneser and Johnson graphs

Abstract

Let $G$ be a connected graph and $\mathcal{X}\subseteq \mathcal{V}(\mathcal{G})$. By definition, two vertices $u$ and $v$ are $\mathcal{X}$-visible in $G$ if there exists a shortest $u,v$-path with all internal vertices being outside of the set $\mathcal{X}$. The largest size of $\mathcal{X}$ such that any two vertices of $G$ (resp. any two vertices from $\mathcal{X}$) are $\mathcal{X}$-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 mutual-visibility 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.