Guaranteeing Hausdorff-Gromov-Hausdorff Equality for Metric Graphs

We prove the equality of Hausdorff and Gromov-Hausdorff distance $d_{GH}(G,U)=d_H(G,U)$ for a metric graph $G$ and a subset $U\subseteq G$, given that the Hausdorff distance between $G$ and $U$ is attained not just near the leaves of $G$ and the Gromov-Hausdorff distance is not too big compared to the graph’s systole.