Abstract
By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that \({B\cup R}\) is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is bounded from above by a polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.
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Radoslav Fulek, Balázs Keszegh and Filip Morić gratefully acknowledge support from Swiss National Science Foundation, Grant No. 200021-125287/1. Partially supported by grant OTKA NK 78439.
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Fulek, R., Keszegh, B., Morić, F. et al. On Polygons Excluding Point Sets. Graphs and Combinatorics 29, 1741–1753 (2013). https://doi.org/10.1007/s00373-012-1221-8
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DOI: https://doi.org/10.1007/s00373-012-1221-8