Abstract
Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P.
In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56 |Conv(B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 , ... , p n and S a set of m points contained in the interior of P, m ≤ n − 1. Then there is a convex decomposition {P 1 , ... , P n } of P such that all points from S lie on the boundaries of P 1 , ... , P n , and each P i contains a whole edge of P on its boundary.
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F. Hurtado partially supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692. C. Merino supported by CONACYT of Mexico, Proyecto 43098. J. Urrutia supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876, and MCYT BFM2003-04062. I. Ventura partially supported by Project MCYT BFM2003-04062.
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Hurtado, F., Merino, C., Oliveros, D. et al. On Polygons Enclosing Point Sets II. Graphs and Combinatorics 25, 327–339 (2009). https://doi.org/10.1007/s00373-009-0848-6
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DOI: https://doi.org/10.1007/s00373-009-0848-6