Super Guarding and Dark Rays in Art Galleries

MIT CompGeom Group; Hugo A. Akitaya; Erik D. Demaine; Adam Hesterberg; Anna Lubiw; Jayson Lynch; Joseph O'Rourke; Frederick Stock

arXiv · 2404.04613

Super Guarding and Dark Rays in Art Galleries

MIT CompGeom Group·Hugo A. Akitaya·Erik D. Demaine·Adam Hesterberg·Anna Lubiw·Jayson Lynch·Joseph O'Rourke·Frederick Stock

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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.

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BibTeX RIS

MIT CompGeom Group, Hugo A. Akitaya, Erik D. Demaine, Adam Hesterberg, Anna Lubiw, Jayson Lynch, Joseph O'Rourke, Frederick Stock. 2024-04-06. Super Guarding and Dark Rays in Art Galleries. https://arxiv.org/abs/2404.04613

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