Dispersive Vertex Guarding for Simple and Non-Simple Polygons
<|reference_start|>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 $\ell$; 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 $2$.<|reference_end|>
@article{fekete2024dispersive,
    title={Dispersive Vertex Guarding for Simple and Non-Simple Polygons},
    author={S'andor P. Fekete and Joseph S. B. Mitchell and Christian Rieck and
  Christian Scheffer and Christiane Schmidt},
    journal={arXiv preprint arXiv:2406.05861},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.05861},
    primaryClass={cs.CG}
}