Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914685195911168 |
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| author | Löffler, Maarten Mchedlidze, Tamara Orden, David Tkadlec, Josef Wulms, Jules |
| author_facet | Löffler, Maarten Mchedlidze, Tamara Orden, David Tkadlec, Josef Wulms, Jules |
| contents | A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart pseudo-triangulations ($k$-DPTs), that is, pseudo-triangulations in which exactly $k$ faces are darts and all other faces are triangles. We study the flip graph for such pseudo-triangulations, in which a flip exchanges the diagonals of a pseudo-quadrilatral. Our results are as follows. We prove that the flip graph of $1$-DPTs is generally not connected, and show how to compute its connected components. Furthermore, for $k$-DPTs on a point configuration called the double chain we analyze the structure of the flip graph on a more fine-grained level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4 Löffler, Maarten Mchedlidze, Tamara Orden, David Tkadlec, Josef Wulms, Jules Computational Geometry A pseudo-triangle is a simple polygon with exactly three convex vertices, and all other vertices (if any) are distributed on three concave chains. A pseudo-triangulation~$\mathcal{T}$ of a point set~$P$ in~$\mathbb{R}^2$ is a partitioning of the convex hull of~$P$ into pseudo-triangles, such that the union of the vertices of the pseudo-triangles is exactly~$P$. We call a size-4 pseudo-triangle a dart. For a fixed $k\geq 1$, we study $k$-dart pseudo-triangulations ($k$-DPTs), that is, pseudo-triangulations in which exactly $k$ faces are darts and all other faces are triangles. We study the flip graph for such pseudo-triangulations, in which a flip exchanges the diagonals of a pseudo-quadrilatral. Our results are as follows. We prove that the flip graph of $1$-DPTs is generally not connected, and show how to compute its connected components. Furthermore, for $k$-DPTs on a point configuration called the double chain we analyze the structure of the flip graph on a more fine-grained level. |
| title | Flip Graphs of Pseudo-Triangulations With Face Degree at Most 4 |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2402.12357 |