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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| Vydáno v: | Discrete mathematics and theoretical computer science Ročník 26:2; číslo Combinatorics |
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| Hlavní autoři: | , , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Discrete Mathematics & Theoretical Computer Science
01.01.2024
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| Témata: | |
| ISSN: | 1365-8050, 1365-8050 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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. |
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| ISSN: | 1365-8050 1365-8050 |
| DOI: | 10.46298/dmtcs.12443 |