Super Guarding and Dark Rays in Art Galleries
Computational Geometry
2025-09-18 v3 Metric Geometry
Abstract
We explore an Art Gallery variant where each point of a polygon must be seen by k guards, and guards cannot see through other guards. Surprisingly, even covering convex polygons under this variant is not straightforward. For example, covering every point in a triangle k=4 times (a 4-cover) requires 5 guards, and achieving a 10-cover requires 12 guards. Our main result is tight bounds on k-covering a convex polygon of n vertices, for all k and n. The proofs of both upper and lower bounds are nontrivial. We also obtain bounds for simple polygons, leaving tight bounds an open problem.
Keywords
Cite
@article{arxiv.2404.04613,
title = {Super Guarding and Dark Rays in Art Galleries},
author = {MIT CompGeom Group and Hugo A. Akitaya and Erik D. Demaine and Adam Hesterberg and Anna Lubiw and Jayson Lynch and Joseph O'Rourke and Frederick Stock},
journal= {arXiv preprint arXiv:2404.04613},
year = {2025}
}
Comments
23 pages, 16 figures, 9 references. v3 incorporates referee suggestions