On polynomials associated to Voronoi diagrams of point sets and crossing numbers
Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclo...
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Discrete Mathematics & Theoretical Computer Science
2024-11-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | http://dmtcs.episciences.org/12443/pdf |
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| author | Mercè Claverol Andrea de las Heras-Parrilla David Flores-Peñaloza Clemens Huemer David Orden |
| author_facet | Mercè Claverol Andrea de las Heras-Parrilla David Flores-Peñaloza Clemens Huemer David Orden |
| author_sort | Mercè Claverol |
| collection | DOAJ |
| description | Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials. |
| format | Article |
| id | doaj-art-df5da669aede463bbd7dd513169f11cd |
| institution | OA Journals |
| issn | 1365-8050 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj-art-df5da669aede463bbd7dd513169f11cd2025-08-20T01:49:33ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502024-11-01vol. 26:2Combinatorics10.46298/dmtcs.1244312443On polynomials associated to Voronoi diagrams of point sets and crossing numbersMercè ClaverolAndrea de las Heras-ParrillaDavid Flores-PeñalozaClemens HuemerDavid OrdenThree polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.http://dmtcs.episciences.org/12443/pdfmathematics - combinatoricscomputer science - computational geometrycomputer science - discrete mathematics |
| spellingShingle | Mercè Claverol Andrea de las Heras-Parrilla David Flores-Peñaloza Clemens Huemer David Orden On polynomials associated to Voronoi diagrams of point sets and crossing numbers Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics computer science - computational geometry computer science - discrete mathematics |
| title | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| title_full | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| title_fullStr | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| title_full_unstemmed | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| title_short | On polynomials associated to Voronoi diagrams of point sets and crossing numbers |
| title_sort | on polynomials associated to voronoi diagrams of point sets and crossing numbers |
| topic | mathematics - combinatorics computer science - computational geometry computer science - discrete mathematics |
| url | http://dmtcs.episciences.org/12443/pdf |
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