Abstract
We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.
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Research partially supported by the FWF (Austrian Fonds zur Förderung der Wissenschaftlichen Forschung) under grant S09205-N12, FSP Industrial Geometry.
Research partially supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692.
Research supported by the Deutsche Forschungsgemeinschaft within the European graduate program “Combinatorics, Geometry, and Computation” (No. GRK 588/2).
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Aichholzer, O., Huemer, C., Kappes, S. et al. Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles. Graphs and Combinatorics 23, 481–507 (2007). https://doi.org/10.1007/s00373-007-0752-x
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DOI: https://doi.org/10.1007/s00373-007-0752-x