On cycles and merge trees

TL;DR

This paper introduces generalized merge trees that incorporate cycle information from discrete Morse functions on 1-dimensional complexes, providing a complete inverse correspondence and an algorithm for optimal Morse functions.

Contribution

It extends merge trees to include cycle data, characterizes which generalized merge trees arise from discrete Morse functions, and offers an algorithm for Morse function simplification.

Findings

Complete solution to the inverse problem between Morse functions and generalized merge trees.

Characterization of generalized merge trees induced by simple graphs.

Algorithm for canceling critical simplices to obtain optimal Morse functions.

Abstract

In this paper, we extend the notion of a merge tree to that of a generalized merge tree, a merge tree that includes 1-dimensional cycle birth information. Given a discrete Morse function on a $1$-dimensional regular CW complex, we construct the induced generalized merge tree. We give several notions of equivalence of discrete Morse functions based on the induced generalized merge tree and how these notions relate to one another. As a consequence, we obtain a complete solution to the inverse problem between discrete Morse functions on $1$-dimensional regular CW complexes and generalized merge trees. After characterizing which generalized merge trees can be induced by a discrete Morse function on a simple graph, we give an algorithm based on the induced generalized merge tree of a discrete Morse function $f\colon X \to \mathbb{R}$ that cancels the critical simplices of $f$ and replaces it with an optimal discrete Morse function.