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：设$X$是图$G$的顶点子集，若存在一条最短$u,v$-路$P$使得$V(P)\cap X \subseteq \{u,v\}$，则称顶点$u$和$v$为$X$-可见的。若$X$中任意两个顶点均为$X$-可见的，则称$X$为互视集。$G$的互视数是$G$的最大互视集的基数，该概念已有研究。本文基于对自然顶点对需满足$X$-可见性的不同要求，引入了多种互视问题，由此得到总互视数、对偶互视数和外互视数。我们首先证明这些图不变量相互关联，并与经典互视数存在联系；继而证明三个新引入的互视问题在计算上具有困难性。基于这一结果，我们对若干图类（如网格图和环面图）计算或界定了它们的取值。最后，通过特定图族的计算结果，对上述参数值进行了对比分析。