Dispersive Vertex Guarding for Simple and Non-Simple Polygons

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 $\mathrm{\ell }$; 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 $\mathrm{\ell }$. 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 $2$.