The edge labeling of higher order Voronoi diagrams

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.