@inproceedings{22003,
  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.},
  author       = {Adams, Henry and Majhi, Sushovan and Manin, Fedor and Virk, Ziga and Zava, Nicolò},
  booktitle    = {42nd International Symposium on Computational Geometry},
  isbn         = {9783959774185},
  issn         = {1868-8969},
  keywords     = {Gromov–Hausdorff distance, distortion, connectedness, Borsuk–Ulam theorem},
  location     = {New Brunswick, NJ, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Lower bounding the Gromov–Hausdorff distance in metric graphs}},
  doi          = {10.4230/LIPIcs.SoCG.2026.3},
  volume       = {367},
  year         = {2026},
}

@inproceedings{22918,
  abstract     = {Given a weighted digraph G, a (t,g,μ)-DAG cover is a collection of g dominating DAGs D_1,… ,D_g such that all distances are approximately preserved: for every pair (u,v) of vertices, min_id_{D_i}(u,v) ≤ t⋅ d_G(u,v), and the total number of non-G edges is bounded by |(∪_i D_i)⧵ G| ≤ μ. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points. 
We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a (1,2,Õ(n⋅tw))-Steiner DAG cover. For planar digraphs we provide a (1+ε,2,Õ_ε(n))-Steiner DAG cover.
We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth 1 with stretch t < 2 and sub-quadratic number of extra edges requires Ω(log n) DAGs.},
  author       = {Bhore, Sujoy and Chang, Hsien Chih and Conroy, Jonathan and Filtser, Arnold and Oh, Eunjin and Wein, Nicole and Zheng, Da Wei},
  booktitle    = {34th Annual European Symposium on Algorithms},
  isbn         = {9783959774451},
  issn         = {1868-8969},
  keywords     = {Directed graphs, DAG (directed acyclic graphs), distortion, metric embeddings, planar graph, treewidth},
  location     = {L’Aquila, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{DAG covers for structured graphs: The Steiner point effect}},
  doi          = {10.4230/LIPIcs.ESA.2026.94},
  volume       = {388},
  year         = {2026},
}