Minimal Delaunay triangulations of hyperbolic surfaces

11/19/2020

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by   Matthijs Ebbens, et al.

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Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the Ω(√(g)) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well.