If $x\in V(G)$, then $S\subseteq V(G)\setminus\{x\}$ is an $x$-visibility set if for any $y\in S$ there exists a shortest $x,y$-path avoiding $S$. The $x$-visibility number $v_x(G)$ is the maximum cardinality of an $x$-visibility set, and the maximum value of $v_x(G)$ among all vertices $x$ of $G$ is the vertex visibility number ${\rm vv}(G)$ of $G$. It is proved that ${\rm vv}(G)$ is equal to the largest possible number of leaves of a shortest-path tree of $G$. Deciding whether $v_x(G) \ge k$ holds for given $G$, a vertex $x\in V(G)$, and a positive integer $k$ is NP-complete even for graphs of diameter $2$. Several general sharp lower and upper bounds on the vertex visibility number are proved. The vertex visibility number of Cartesian products is also bounded from below and above, and the exact value of the vertex visibility number is determined for square grids, square prisms, and square toruses.

翻译：若$x\in V(G)$，则$S\subseteq V(G)\setminus\{x\}$是一个$x$-可见集，当且仅当对于任意$y\in S$，均存在一条避开$S$的最短$x,y$-路径。$x$-可见数$v_x(G)$是$x$-可见集的最大基数，而$v_x(G)$在所有顶点$x\in V(G)$中的最大值称为图$G$的顶点可见数${\rm vv}(G)$。本文证明了${\rm vv}(G)$等于$G$的最短路径树可能具有的最大叶子数。对于给定的图$G$、顶点$x\in V(G)$和正整数$k$，判定是否满足$v_x(G) \ge k$是NP完全的，即使对于直径为$2$的图亦然。本文证明了顶点可见数的若干一般性尖锐下界与上界。同时，对笛卡尔积的顶点可见数给出了下界与上界的估计，并精确确定了方格网格、方棱柱和方形环面的顶点可见数值。