Guaranteeing Hausdorff-Gromov-Hausdorff Equality for Metric Graphs

Paul Schott

Guaranteeing Hausdorff-Gromov-Hausdorff Equality for Metric Graphs

Metric Geometry 2026-07-29 v1

Authors: Paul Schott

Abstract

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.

Keywords

Hausdorff distance Gromov-Hausdorff distance metric graph systole equality of distances

Cite

@article{arxiv.2607.26810,
  title  = {Guaranteeing Hausdorff-Gromov-Hausdorff Equality for Metric Graphs},
  author = {Paul Schott},
  journal= {arXiv preprint arXiv:2607.26810},
  year   = {2026}
}