The edge labeling of higher order Voronoi diagrams

TL;DR

This paper introduces an edge labeling method for higher order Voronoi diagrams, exploring their structural properties and identifying configurations that cannot occur for small k, using elementary geometric arguments.

Contribution

It presents a new edge labeling approach for order-k Voronoi diagrams and extends the understanding of their structural properties beyond previous work.

Findings

V_k(S) has a small orientable cycle and path double cover

Identifies configurations impossible in V_k(S) for small k

Provides a systematic study of properties using elementary geometry

Abstract

We present an edge labeling of order-$k$ Voronoi diagrams, $V_k(S)$, of point sets $S$ in the plane, and study properties of the regions defined by them. Among them, we show that $V_k(S)$ has a small orientable cycle and path double cover, and we identify configurations that cannot appear in $V_k(S)$ for small values of $k$. This paper also contains a systematic study of well-known and new properties of $V_k(S)$, all whose proofs only rely on elementary geometric arguments in the plane. The maybe most comprehensive study of structural properties of $V_k(S)$ was done by D.T. Lee (On k-nearest neighbor Voronoi diagrams in the plane) in 1982. Our work reviews and extends the list of properties of higher order Voronoi diagrams.