Lower Bounding the Gromov--Hausdorff distance in Metric Graphs

TL;DR

This paper establishes conditions under which the Gromov--Hausdorff distance equals the Hausdorff distance in metric graphs, extending previous results for circles and providing optimal bounds.

Contribution

It generalizes the equality condition between Gromov--Hausdorff and Hausdorff distances for metric graphs, especially circles, and introduces a topological obstruction for lower bounds.

Findings

Equality holds when the subset is sufficiently dense in the metric graph.

The bound for circles is improved from π/6 to π/3, which is shown to be optimal.

A topological obstruction is identified that provides a lower bound for the Gromov--Hausdorff distance.

Abstract

Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_\gh(G,X)=d_\h(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2\pi$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\frac{\pi}{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\frac{\pi}{3}$, and furthermore, we show that the constant $\frac{\pi}{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_\gh(G,X)$ by the Hausdorff distance $d_\h(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$.