For planar Euclidean graphs, a path-metric Vietoris-Rips complex of a sufficiently dense sample is homotopy equivalent to the graph, and its shadow is a Hausdorff-close reconstruction, under explicit scale conditions.
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Vietoris--Rips Shadow for Euclidean Graph Reconstruction
For planar Euclidean graphs, a path-metric Vietoris-Rips complex of a sufficiently dense sample is homotopy equivalent to the graph, and its shadow is a Hausdorff-close reconstruction, under explicit scale conditions.