On pseudo-convex partitions of a planar point set
Aichholzer et al. [O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, C.D. Tóth, Decompositions, partitions, and coverings with convex polygons and pseudo-triangles, Graphs and Combinatorics 23 (2007) 481–507] introduced the notion of pseudo-convex partitioning of planar point sets and proved that t...
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| Published in: | Discrete mathematics Vol. 313; no. 21; pp. 2401 - 2408 |
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01.11.2013
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| Abstract | Aichholzer et al. [O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, C.D. Tóth, Decompositions, partitions, and coverings with convex polygons and pseudo-triangles, Graphs and Combinatorics 23 (2007) 481–507] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number ψ(n) satisfies 34⌊n4⌋≤ψ(n)≤⌈n4⌉. In this paper we prove that ψ(13)=3, which improves the upper bound on ψ(n) to ⌈3n13⌉, thus answering a question posed by Aichholzer et al. in the same paper. |
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| AbstractList | Aichholzer et al. [O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, C.D. TA3th, Decompositions, partitions, and coverings with convex polygons and pseudo-triangles, Graphs and Combinatorics 23 (2007) 481a507] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number I(n)I(n) satisfies 34an4aaOI(n)aOan4a. In this paper we prove that I(13)=3I(13)=3, which improves the upper bound on I(n)I(n) to a3n13a, thus answering a question posed by Aichholzer et al. in the same paper. Aichholzer et al. [O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, C.D. Tóth, Decompositions, partitions, and coverings with convex polygons and pseudo-triangles, Graphs and Combinatorics 23 (2007) 481–507] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number ψ(n) satisfies 34⌊n4⌋≤ψ(n)≤⌈n4⌉. In this paper we prove that ψ(13)=3, which improves the upper bound on ψ(n) to ⌈3n13⌉, thus answering a question posed by Aichholzer et al. in the same paper. |
| Author | Das, Sandip Bhattacharya, Bhaswar B. |
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| Keywords | Discrete geometry Partition Convex hull Empty convex polygons Pseudo-triangles |
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| SubjectTerms | Combinatorial analysis Convex hull Coverings Discrete geometry Empty convex polygons Graphs Mathematical analysis Partition Partitioning Partitions Polygons Pseudo-triangles Upper bounds |
| Title | On pseudo-convex partitions of a planar point set |
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