Dispersive Vertex Guarding for Simple and Non-Simple Polygons
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arXiv
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| Format: | Preprint |
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2024
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| author | Fekete, Sándor P. Mitchell, Joseph S. B. Rieck, Christian Scheffer, Christian Schmidt, Christiane |
| author_facet | Fekete, Sándor P. Mitchell, Joseph S. B. Rieck, Christian Scheffer, Christian Schmidt, Christiane |
| contents | 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 $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dispersive Vertex Guarding for Simple and Non-Simple Polygons Fekete, Sándor P. Mitchell, Joseph S. B. Rieck, Christian Scheffer, Christian Schmidt, Christiane Computational Geometry F.2.2 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 $2$. |
| title | Dispersive Vertex Guarding for Simple and Non-Simple Polygons |
| topic | Computational Geometry F.2.2 |
| url | https://arxiv.org/abs/2406.05861 |