Reeb Graph of Sample Thickenings

Reeb Graph of Sample Thickenings H˚avard Bakke Bjerkevik∗ Nello Blaser† Lars M. Salbu‡ December 10, 2025 Abstract We consider the Reeb graph of a thickening of points sampled from an unknown space. Our main contribution is a framework to transfer reconstruction results similar to the well-known work of Niyogi, Smale, and Weinberger to the setting of Reeb graphs. To this end, we first generalize and study the interleaving distances for Reeb graphs. We find that many of the results previously established for constructible spaces also hold for general topological spaces. We use this to show that under certain conditions for topological spaces with real-valued Lipschitz maps, the Reeb graph of a sample thickening approximates the Reeb graph of the underlying space. Finally, we provide an algorithm for computing the Reeb graph of a sample thickening. 1 Introduction Collecting data through experiments typically does not give complete information about the system. Instead we get a finite set of data points sampled from a larger space, and we want to study properties of this underlying space. Geometric reconstruction concerns the problem of recovering topological information, like homology/homotopy groups [12,18,19,27] or even the homotopy type [1,3,4,27,33,37,44], of an unknown space by considering a finite set of sampled points. Assumptions on the underlying space are needed to ensure reconstruction, usually based on geometric properties like the (local) (µ−)reach [2,17,28], distortion [30], convexity radius [32] or weak feature size [18]. Additionally, the samples need to be dense and well-distributed, and are only sometimes allowed sample noise (e.g. [3]). A classical way of describing shapes is with Reeb graphs [9]. Given an R-space, namely a topological space X with a continuous function f : X →R, the Reeb graph Rb(X, f) is the quotient space constructed from X by identifying points in the same connected component of level sets f −1(a) for a ∈R. Introduced by Reeb in 1946 [39], it has found applications in diverse fields, ranging from computer graphics (see survey [9]) to neuroscience (e.g. [40,41]). We consider the problem of approximating the Reeb graph of an unknown space from a sample and give a frame- work to transfer reconstruction results to this setting. In particular, we look at results that recover the homotopy type by constructing a larger space that deformation retracts to the unknown space, typically by thickening the point cloud [3,37,44]. To measure approximation quality, we need to compare Reeb graphs. Reeb distances include the bottleneck [20], interleaving [8,24], functional distortion and contortion distances [5,6], the Reeb radius [23], and the universal distance [7]. A lot of previous work compares these distances [5,6,8,10,16] which are often defined for special Reeb graphs. For general R-spaces, we use the interleaving distance with connected components (also discussed in [24] where they mainly consider path components). This distance has a general Reeb stability result (Theorem 3.5), that yields approximation results without any extra assumption. With additional assumptions, our methods also works for other distances (Theorem 4.5). Prior work on approximating Reeb graphs from samples includes that the Reeb graph of the Vietoris-Rips complex of a dense sample of a smooth compact manifold M can approximate the Reeb graph Rb(M, f) where f is level-set-tame Lipschitz [25, Thm. 4.7]. Mapper [43] can be seen as a discretized approximation of the Reeb graph. In particular, in [34, Cor. 6] and [11, Cor. 1] they show that the geometric and enhanced Mapper, respectively, are close in interleaving distance to the Reeb graph for constructable R-spaces. Moreover, in [15, Thm. 7] they show that Mapper is close in bottleneck distance to the Reeb graph of Morse-type functions on spaces with positive reach and convexity radius. ∗Department of Mathematics & Statistics, University at Albany, SUNY, USA; hbjerkevik@albany.edu †Department of Informatics, University of Bergen, Norway; nello.blaser@uib.no ‡Department of Informatics, University of Bergen, Norway; lars.salbu@uib.no 1 arXiv:2512.08159v1 [cs.CG] 9 Dec 2025 On the computational front, an early contribution was [42] where they found the Reeb graph of Morse functions on triangulated 2-manifolds in O(n2), where n is the number of triangles. It was later improved to O(n log n) in [21]. For the more general case of PL functions on simplicial complexes, randomized [31] and later deterministic [38] O(m log m) algorithms have been suggested, where m is the size of the simplicial 2-skeleton. 1.1 Contributions Our contributions are as follows: 1. In Theorem 3.5 we show that for continuous functions f1, f2 : X →R, the interleaving distance between Reeb graphs Rb(X, f1) and Rb(X, f2) is bounded by ∥f1 −f2∥∞. This generalizes results from [24], where this is shown for constructible R spaces, to general R-spaces. This result is interesting in itself, and nece

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