Combinatorial properties of abstract Voronoi diagrams
Abstract Voronoi diagrams are defined by a system of bisecting curves in the plane, rather than by the concept of distance. So far, their investigation was based on the assumption that each subfamily of the given family of curves yields a partition of the plane into connected Voronoi regions [4,5,9]. Here we prove these conditions to be equivalent with some simple combinatorial properties of the curves that need only be verified for subfamilies of size 3. Using the simpler characterization, we are able to show that all singularities resulting from bisecting curves that share a curve segment or touch one another can be resolved by deforming the curves in a suitable way. In the new system, any two curves properly cross whereever they intersect, but each subfamily still yields the same Voronoi diagram as before, after possibly contracting some edges.
Key wordsAbstract Voronoi diagram bisector Voronoi diagram
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