Uniform approximation of metrics by graphs

TL;DR

This paper investigates how certain metric spaces, including the Euclidean plane and Gromov hyperbolic spaces with bounded geometry, can be approximated by uniform graphs with bounded degrees and edge lengths.

Contribution

It establishes that these important classes of metric spaces are approximable by uniform graphs, expanding understanding of metric space graph approximations.

Findings

Euclidean plane is approximable by uniform graphs

Gromov hyperbolic geodesic spaces with bounded geometry are approximable

Open problems related to metric space approximations are posed

Abstract

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one within a finite Gromov-Hausdorff distance. We show that the Euclidean plane and Gromov hyperbolic geodesic spaces with bounded geometry are approximable by uniform graphs, and pose a number of open problems.