AATRN Watch Party: On the inverse problem between discrete Morse functions on graphs, and merge trees

Discrete Morse theory is a versatile tool from combinatorial algebraic topology. In a nutshell, discrete Morse theory uses certain well-behaved functions, the so-called discrete Morse functions, from the face poset of a regular CW complex to the real numbers. Additional to providing sublevel-filtrations, discrete Morse functions provide numerous tools to investigate topological properties and to find theoretic guarantees for various algorithms.

Merge trees are combinatorial descriptors for the development of path components within different levels of filtered spaces. They have been introduced in the context of visualization, where they are used to approximate and compute contour trees.

In this talk, we consider an instance of an inverse problem: Given a specific merge tree, what are all the different discrete Morse functions on graphs that induce the given merge tree? Moreover, how can one find all of them in structured way? This talk summarizes results from [1] and [2].

[1] J. Brüggemann, On merge trees and discrete Morse functions on paths and trees, Journal of Applied and Computational Topology, 2022, Volume 7, 103 – 138 [2] J. Brüggemann, N. A. Scoville, On cycles and merge trees, Journal of Pure and Applied Algebra, 2025, Volume 229, Issue 7