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Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs

Authors Henry Adams , Sushovan Majhi , Fedor Manin , Žiga Virk , Nicolò Zava

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Topology
  • Mathematics of computing → Graph theory
Keywords
  • Gromov-Hausdorff distance
  • distortion
  • connectedness
  • Borsuk-Ulam theorem

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Abstract

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.

Cite As Get BibTex

Henry Adams, Sushovan Majhi, Fedor Manin, Žiga Virk, and Nicolò Zava. Lower Bounding the Gromov-Hausdorff Distance in Metric Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.3

Author Details

Henry Adams
  • University of Florida, Gainesville, FL, USA
Sushovan Majhi
  • George Washington University, Washington, DC, USA
Fedor Manin
  • University of Toronto, Canada
Žiga Virk
  • University of Ljubljana, Slovenia
  • Institute IMFM, Ljubljana, Slovenia
Nicolò Zava
  • Institute of Science and Technology, Klosterneuburg, Austria

Funding

  • Adams, Henry: Simons Foundation Travel Support for Mathematicians.
  • Virk, Žiga: Slovene research agency grant P1-0292.
  • Zava, Nicolò: FWF Grant, Project number I4245-N35.

References

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