Lower Bounding the Gromov--Hausdorff distance in Metric Graphs

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 $d_{\text{gh}}(G,X)=d_{\text{h}}(G,X)$. When the metric graph is the circle $G=S^1$ with circumference $2\pi$, a recent study established the equality $d_{\text{gh}}(S^1,X)=d_{\text{h}}(S^1,X)$ whenever $d_{\text{gh}}(S^1,X)<\frac{\pi }{6}$. Our results relax this hypothesis to $d_{\text{gh}}(S^1,X)<\frac{\pi }{3}$, and furthermore, we show that the constant $\frac{\pi }{3}$ is the best possible. We lower bound the Gromov–Hausdorff distance $d_{\text{gh}}(G,X)$ by the Hausdorff distance $d_{\text{h}}(G,X)$ via a simple topological obstruction: the existence of a possibly discontinuous function $f\text{\hspace{0.17em}⁣}:G\to X$ with too small distortion contradicts the connectedness of $G$.