Minimal Delaunay triangulations of hyperbolic surfaces

TL;DR

This paper investigates the minimal number of vertices needed for Delaunay triangulations on hyperbolic surfaces, establishing bounds and optimal constructions related to the genus of the surface.

Contribution

It provides bounds on the number of vertices for Delaunay triangulations of hyperbolic surfaces and constructs examples showing these bounds are tight.

Findings

Every hyperbolic surface of genus g has a Delaunay triangulation with O(g) vertices.

Constructs hyperbolic surfaces where the O(g) bound is optimal.

Shows the lower bound of Ω(√g) vertices is tight for hyperbolic surfaces.

Abstract

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 $\Omega(\sqrt{g})$ 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.