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$P$满足$V(P)\cap X \subseteq \{u,v\}$，则称顶点$u$和$v$是$X$-可见的。若$X$中任意两个顶点均为$X$-可见，则称$X$为互可见集。$G$的互可见数定义为$G$的最大互可见集的基数，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，引入了多种互可见性问题，由此得到全互可见数、对偶互可见数和外互可见数。我们首先证明这些图不变量彼此之间以及与经典互可见数存在关联，进而证明这三个新引入的互可见问题在计算上具有困难性。基于此结论，我们计算或界定了若干图类（如网格图和环面图）中这些参数的值。最后，通过对比特定图族中这些参数的数值结果，进行了参数间的交叉比较分析。