Abstract
We show that (A), 14 points on the boundary of a Jordan curve, and (B), 16 points in convex position encircled by a Jordan curve, cannot be shattered by interior visibility domains. This means that there always exists a subset of the given points, for which no point of the curve’s interior domain sees all points of the subset and no point of its complement. As a consequence, we obtain a new result on guarding art galleries. If each point of the art gallery sees at least an r-th part of the gallery’s boundary, then the art gallery can be covered by 13 ·C ·r logr guards placed on the boundary. Here, C is the constant from the ε-net theorem.
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Gilbers, A., Klein, R. (2009). New Results on Visibility in Simple Polygons. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_29
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DOI: https://doi.org/10.1007/978-3-642-03367-4_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
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