A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function

We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3.

The Reeb Graph Edit Distance is Universal

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.

Probabilistic convergence and stability of random mapper graphs

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $${\mathbb {X}}$$ X equipped with a continuous function $$f: {\mathbb {X}}\rightarrow \mathbb {R}$$ f : X → R . We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line $$\mathbb {R}$$ R . We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $$({\mathbb {X}}, f)$$ ( X , f ) when it is applied to points randomly sampled from a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of $$({\mathbb {X}}, f)$$ ( X , f ) , a constructible $$\mathbb {R}$$ R -space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $$({\mathbb {X}},f)$$ ( X , f ) to the mapper of a super-level set of a probability density function concentrated on $$({\mathbb {X}}, f)$$ ( X , f ) . Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.

An Efficient Data Retrieval Parallel Reeb Graph Algorithm

The Reeb graph of a scalar function that is defined on a domain gives a topologically meaningful summary of that domain. Reeb graphs have been shown in the past decade to be of great importance in geometric processing, image processing, computer graphics, and computational topology. The demand for analyzing large data sets has increased in the last decade. Hence, the parallelization of topological computations needs to be more fully considered. We propose a parallel augmented Reeb graph algorithm on triangulated meshes with and without a boundary. That is, in addition to our parallel algorithm for computing a Reeb graph, we describe a method for extracting the original manifold data from the Reeb graph structure. We demonstrate the running time of our algorithm on standard datasets. As an application, we show how our algorithm can be utilized in mesh segmentation algorithms.

Deformations of smooth functions on 2-torus

Let $f$ be a Morse function on a smooth compact surface $M$ and $\mathcal{S}'(f)$ be the group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the Kronrod-Reeb graph of $f$ induced by elements from $\mathcal{S}'(f)$, and $\Delta'$ be the subgroup of $\mathcal{S}'(f)$ consisting of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $\pi_0\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. The groups $\pi_0\Delta'(f)$ and $G(f)$ encode ``combinatorially trivial'' and ``combinatorially nontrivial'' counterparts of $\pi_0\mathcal{S}'(f)$ respectively. In the paper we compute groups $\pi_0\mathcal{S}'(f)$, $G(f)$, and $\pi_0\Delta'(f)$ for Morse functions on $2$-torus $T^2$.

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere. The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

Approximation of metric spaces by Reeb graphs: Cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifolds

For a connected locally path-connected topological space X and a continuous function f on it such that its Reeb graph Rf is a finite topological graph, we show that the cycle rank of Rf, i.e., the first Betti number b1(Rf), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group ?1(X), the condition of local path-connectedness being important since generally b1(Rf) can even exceed b1(X). We give some practical methods for calculating the co-rank of ?1(X) and a closely related value, the isotropy index. We apply our bound to improve upper bounds on the distortion of the Reeb quotient map, and thus on the Gromov-Hausdorff approximation of the space by Reeb graphs, for the distance function on a compact geodesic space and for a simple Morse function on a closed Riemannian manifold. This distortion is bounded from below by what we call the Reeb width b(M) of a metric space M, which guarantees that any real-valued continuous function on M has large enough contour (connected component of a level set). We show that for a Riemannian manifold, b(M) is non-zero and give a lower bound on it in terms of characteristics of the manifold. In particular, we show that any real-valued continuous function on a closed Euclidean unit ball E of dimension at least two has a contour C with diam(C??E)??3.