The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Tur\'an problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size.

翻译：图$G$中的互可见性问题旨在寻找最大顶点集$S\subseteq V(G)$的基数，使得对于任意两个顶点$x,y\in S$，存在一条最短$x,y$-路径$P$，且$P$的所有内部顶点均不在$S$中。这也被称为$x,y$关于$S$可见，或简称为$S$-可见。基于对$S$内部和/或外部顶点可见性性质的扩展，该问题存在多种变体，分别称为完全可见、外部可见和双重互可见性问题。本文聚焦于研究直径二图中对应的四个可见性参数，通过给出这些参数的上界和/或闭合公式展开分析。两个完全图的笛卡尔积中的互可见性问题等价于著名的Zarankievicz问题的一个实例。本文进一步研究了两个完全图笛卡尔积的对偶可见性与外部可见性问题，以及这些图的直积中的所有互可见性问题。此外，我们还研究了完全图与完全二分图的线图中的所有互可见性问题。作为该研究的结论，我们揭示了上述问题与经典Turán问题若干实例之间的关联。最后，我们对余图及若干非平凡的最小尺寸直径二图的可视性问题进行了分析。