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arXiv · 2406.05861

Dispersive Vertex Guarding for Simple and Non-Simple Polygons

Sándor P. Fekete·Joseph S. B. Mitchell·Christian Rieck·Christian Scheffer·Christiane Schmidt
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Abstract

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 $2$.

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Sándor P. Fekete, Joseph S. B. Mitchell, Christian Rieck, Christian Scheffer, Christiane Schmidt. 2024-06-09. Dispersive Vertex Guarding for Simple and Non-Simple Polygons. https://arxiv.org/abs/2406.05861

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