On the mutual visibility in Cartesian products and triangle-free graphs

Given a graph G=(V(G),E(G)) and a set P⊆V(G), the following concepts have been recently introduced: (i) two elements of P are mutually visible if there is a shortest path between them without further elements of P; (ii) P is a mutual-visibility set if its elements are pairwise mutually visible; (iii) the mutual-visibility number of G is the cardinality of any largest mutual-visibility set. In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and investigate independent mutual-visibility sets. In the very special case of the Cartesian product of two complete graphs the problem is shown to be equivalent to the well-known Zarenkiewicz's problem. We also characterize the triangle-free graphs with the mutual-visibility number equal to 3.