# Categorified Reeb Graphs

arXiv: https://arxiv.org/abs/1501.04147 · HTML: https://mlsanity.com/paper/1501.04147/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jan 2015.

## Built on by (most cited first, 2 of 2)

- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)
- [Algebraic and Geometric Models for Space Networking](https://mlsanity.com/paper/2304.01150/) (2023)

## Abstract (arXiv)

The Reeb graph is a construction which originated in Morse theory to study a real valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an `interleaving' distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.