Lower bounding the Gromov–Hausdorff distance in metric graphs LIPIcs Adams, Henry Majhi, Sushovan Manin, Fedor Virk, Ziga Zava, Nicolò ; https://orcid.org/0000-0001-8686-1888 Gromov–Hausdorff distance distortion connectedness Borsuk–Ulam theorem ddc:500 Let G be a finite, connected metric graph and let X be a subset of G. 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 = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/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: G → X with too small distortion contradicts the connectedness of G. Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2026 info:eu-repo/semantics/conferenceObject doc-type:conferenceObject conferenceObject http://purl.org/coar/resource_type/c_5794 https://research-explorer.ista.ac.at/record/22003 https://research-explorer.ista.ac.at/download/22003/22115 Adams H, Majhi S, Manin F, Virk Z, Zava N. Lower bounding the Gromov–Hausdorff distance in metric graphs. In: <i>42nd International Symposium on Computational Geometry</i>. Vol 367. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2026. doi:<a href="https://doi.org/10.4230/LIPIcs.SoCG.2026.3">10.4230/LIPIcs.SoCG.2026.3</a> eng info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SoCG.2026.3 info:eu-repo/semantics/altIdentifier/e-issn/1868-8969 info:eu-repo/semantics/altIdentifier/isbn/9783959774185 info:eu-repo/semantics/altIdentifier/arxiv/2411.09182 info:eu-repo/semantics/openAccess