Work

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Year: 2024

Type: article

Abstract: 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.

Source: Discrete Mathematics & Theoretical Computer Science

DOI: 10.46298/dmtcs.12443

Authors: Mercè Claverol, Andrea de las Heras-Parrilla, David Flores-Peñaloza, Clemens Huemer, David Orden

Institutions: Universitat Politècnica de Catalunya, Universidad Nacional Autónoma de México, Universidad de Alcalá

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Agencia Estatal de Investigación