Vietoris–Rips complexes of metric spaces near a metric graph

For a sufficiently small scale β >0 , the Vietoris–Rips complex ℛ_β (S) of a metric space S with a small Gromov–Hausdorff distance to a closed Riemannian manifold M has been already known to recover M up to homotopy type. While the qualitative result is remarkable and generalizes naturally to the recovery of spaces beyond Riemannian manifolds—such as geodesic metric spaces with a positive convexity radius—the generality comes at a cost. Although the scale parameter β is known to depend only on the geometric properties of the geodesic space, how to quantitatively choose such a β for a given geodesic space is still elusive. In this work, we focus on the topological recovery of a special type of geodesic space, called a metric graph. For an abstract metric graph 𝒢 and a (sample) metric space S with a small Gromov–Hausdorff distance to it, we provide a description of β based on the convexity radius of 𝒢 in order for ℛ_β (S) to be homotopy equivalent to 𝒢 . Our investigation also extends to the study of the Vietoris–Rips complexes of a Euclidean subset S⊂ℝ^d with a small Hausdorff distance to an embedded metric graph 𝒢⊂ℝ^d . From the pairwise Euclidean distances of points of S , we introduce a family (parametrized by ε ) of path-based Vietoris–Rips complexes ℛ^ε _β (S) for a scale β >0 . Based on the convexity radius and distortion of the embedding of 𝒢 , we show how to choose a suitable parameter ε and a scale β such that ℛ^ε _β (S) is homotopy equivalent to 𝒢 .