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.
On the reconstruction of geodesic subspaces ofR n.International Journal of Computational Geometry & Applications, 32(01n02):91–117, 2022
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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.