We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon $\mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at least $1$, and a rational number $\ell$; decide whether there is a set of vertex guards such that $\mathcal{P}$ is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least $\ell$. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2$. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least $2$. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most $2$.

翻译：我们研究带有顶点守卫的分散式艺术画廊问题：给定一个多边形 $\mathcal{P}$，其顶点间的成对测地线欧几里得距离至少为 $1$，以及一个有理数 $\ell$；判断是否存在一组顶点守卫使得 $\mathcal{P}$ 被覆盖，且任意两个守卫之间的最小测地线欧几里得距离（即所谓的分散距离）至少为 $\ell$。我们证明，判断一个带孔多边形是否存在一组分散距离为 $2$ 的顶点守卫是 NP-完全的。另一方面，我们提供了一种算法，能够在简单多边形中放置分散距离至少为 $2$ 的顶点守卫。这一结果是紧的，因为存在某些简单多边形，其中任何顶点守卫集的分散距离至多为 $2$。