Abstract
Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoilike 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 socalled admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoilike 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 Voronoilike cycles in an admissible bisector system, and show that the Voronoilike 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 Voronoilike 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 Voronoilike 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 Voronoilike) trees and forests in linear expected time, given the order of their leaves along a Voronoilike 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].
BibTeX  Entry
@InProceedings{papadopoulou:LIPIcs.SoCG.2023.52,
author = {Papadopoulou, Evanthia},
title = {{Abstract VoronoiLike Graphs: Extending Delaunay’s Theorem and Applications}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {52:152:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772730},
ISSN = {18688969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17902},
URN = {urn:nbn:de:0030drops179024},
doi = {10.4230/LIPIcs.SoCG.2023.52},
annote = {Keywords: Voronoilike graph, abstract Voronoi diagram, Delaunay’s theorem, Voronoi tree, lineartime randomized algorithm, constraint Delaunay triangulation}
}
Keywords: 

Voronoilike graph, abstract Voronoi diagram, Delaunay’s theorem, Voronoi tree, lineartime randomized algorithm, constraint Delaunay triangulation 
Collection: 

39th International Symposium on Computational Geometry (SoCG 2023) 
Issue Date: 

2023 
Date of publication: 

09.06.2023 