If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.

翻译：如果 $X$ 是图 $G$ 的顶点子集，则当存在一个最短的 $u,v$-路径 $P$ 使得 $V(P)\cap X \subseteq \{u,v\}$ 时，称顶点 $u$ 和 $v$ 是 $X$-可见的。如果 $X$ 中任意两个顶点都是 $X$-可见的，则称 $X$ 是一个互不遮挡点集。$G$ 的互不遮挡数是 $G$ 的最大互不遮挡集的基数，并已被研究。在本文中，提出了一些基于 $X$ 中的哪些顶点应该是 $X$-可见的自然对中的互不遮挡问题。这产生了总的、对偶的和外部的互不遮挡数。我们首先展示这些图不变量如何相互关联以及它们与经典互不遮挡数的关系，然后证明了这三个新增的互不遮挡问题的计算难度。根据这个结果，我们计算或估计了几个图形类的值，其中包括网格图和环面图。我们通过为一些特定族群进行的计算，呈现了这些参数值之间的比较。