Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications

Author Evanthia Papadopoulou

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Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland


I would like to thank Kolja Junginger for preliminary discussions, numerous figures and comments on earlier versions of this work. I would also like to thank Franz Aurenhammer for valuable suggestions, and Elena Arseneva for constructive comments and figures.

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Evanthia Papadopoulou. Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex v is called locally Voronoi, if v and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay’s theorem on (locally) empty circles. Further, we define Voronoi-like cycles in an admissible bisector system, and show that the Voronoi-like graph induced by such a cycle C is a unique tree (or a forest, if C is unbounded). In the special case where C is the boundary of an abstract Voronoi region, the induced Voronoi-like graph can be computed in expected linear time following the technique of [Junginger and Papadopoulou SOCG'18]. Otherwise, within the same time, the algorithm constructs the Voronoi-like graph of a cycle C′ on the same set (or subset) of sites, which may equal C or be enclosed by C. Overall, the technique computes abstract Voronoi (or Voronoi-like) trees and forests in linear expected time, given the order of their leaves along a Voronoi-like cycle. We show a direct application in updating a constraint Delaunay triangulation in linear expected time, after the insertion of a new segment constraint, simplifying upon the result of [Shewchuk and Brown CGTA 2015].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Voronoi-like graph
  • abstract Voronoi diagram
  • Delaunay’s theorem
  • Voronoi tree
  • linear-time randomized algorithm
  • constraint Delaunay triangulation


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