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Super Guarding and Dark Rays in Art Galleries (2404.04613v2)
Published 6 Apr 2024 in cs.CG and math.MG
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.
- The art gallery problem is ∃ℝℝ\exists\mathbb{R}∃ blackboard_R-complete. J. ACM, 69(1), 2021.
- K𝐾Kitalic_K-guarding polygons on the plane. In Proc. 6th Canad. Conf. Comput. Geom., pages 381–386, 1994.
- On k𝑘kitalic_k-guarding polygons. In Proc. 25th Canad. Conf. Comput. Geom., 2013.
- Discrete and Computational Geometry. Princeton University Press, Princeton, NJ, 2011.
- 2-coloring point guards in a k𝑘kitalic_k-guarded polygon. In 40th European Workshop Comput. Geom., 2024. Abstract.
- Stephen Fisk. A short proof of Chvátal’s watchman theorem. J. Combin. Theory Ser. B, 24:374, 1978.
- Super guarding and dark rays in art galleries. In Proc. 25th Canad. Conf. Comput. Geom., pages 51–62, August 2013.
- Joseph O’Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, New York, NY, 1987. http://cs.smith.edu/~jorourke/books/ArtGalleryTheorems/.
- Ihsan Salleh. K𝐾Kitalic_K-vertex guarding simple polygons. Comput. Geom. Theory Appl, 42(4):352–361, 2009.
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