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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2509.17485 |
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| _version_ | 1866908552167161856 |
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| author | Tejel, Javier |
| author_facet | Tejel, Javier |
| contents | In the paper ``Lower bounds on the number of crossing-free subgraphs of $K_N$'' (Computational Geometry 16 (2000), 211-221), it is shown that a double chain of $n$ points in the plane admits at least $Ω(4.642126305^n)$ polygonizations, and it is claimed that it admits at most $O(5.61^n)$ polygonizations. In this note, we provide a proof of this last result. The proof is based on counting non-crossing path partitions for points in the plane in convex position, where a non-crossing path partition consists of a set of paths connecting the points such that no two edges cross and isolated points are allowed.
We prove that a set of $n$ points in the plane in convex position admits $\mathcal{O}^*(5.610718614^{n})$ non-crossing path partitions and a double chain of $n$ points in the plane admits at least $Ω(7.164102920^n)$ non-crossing path partitions. If isolated points are not allowed, we also show that there are $\mathcal{O}^*(4.610718614^n)$ non-crossing path partitions for $n$ points in the plane in convex position and at least $Ω(6.164492582^n)$ non-crossing path partitions in a double chain of $n$ points in the plane. In addition, using a particular family of non-crossing path partitions for points in convex position, we provide an alternative proof for the result that a double chain of $n$ points admits at least $Ω(4.642126305^n)$ polygonizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on non-crossing path partitions in the plane Tejel, Javier Computational Geometry In the paper ``Lower bounds on the number of crossing-free subgraphs of $K_N$'' (Computational Geometry 16 (2000), 211-221), it is shown that a double chain of $n$ points in the plane admits at least $Ω(4.642126305^n)$ polygonizations, and it is claimed that it admits at most $O(5.61^n)$ polygonizations. In this note, we provide a proof of this last result. The proof is based on counting non-crossing path partitions for points in the plane in convex position, where a non-crossing path partition consists of a set of paths connecting the points such that no two edges cross and isolated points are allowed. We prove that a set of $n$ points in the plane in convex position admits $\mathcal{O}^*(5.610718614^{n})$ non-crossing path partitions and a double chain of $n$ points in the plane admits at least $Ω(7.164102920^n)$ non-crossing path partitions. If isolated points are not allowed, we also show that there are $\mathcal{O}^*(4.610718614^n)$ non-crossing path partitions for $n$ points in the plane in convex position and at least $Ω(6.164492582^n)$ non-crossing path partitions in a double chain of $n$ points in the plane. In addition, using a particular family of non-crossing path partitions for points in convex position, we provide an alternative proof for the result that a double chain of $n$ points admits at least $Ω(4.642126305^n)$ polygonizations. |
| title | A note on non-crossing path partitions in the plane |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2509.17485 |