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On cycles and merge trees

Julian Brüggemann, Nicholas A. Scoville

TL;DR

The paper develops a generalized merge-tree framework for $1$-dimensional CW complexes by introducing generalized merge trees with cycle birth information and Morse labeling. It establishes a complete correspondence between CM-equivalence classes of critical discrete Morse functions on graphs and isomorphism classes of gML trees, and provides realizability criteria for simple graphs along with a constructive method. A cancellation algorithm on the induced gML tree enables reduction of critical cells while preserving targeted properties, linking to symmetry and sublevel automorphisms. The approach clarifies the inverse problem for 1D complexes, offers practical tools for simplifying Morse filtrations, and lays groundwork for higher-dimensional extensions and geometric group-theoretic perspectives on sublevel symmetries.

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.

On cycles and merge trees

TL;DR

The paper develops a generalized merge-tree framework for -dimensional CW complexes by introducing generalized merge trees with cycle birth information and Morse labeling. It establishes a complete correspondence between CM-equivalence classes of critical discrete Morse functions on graphs and isomorphism classes of gML trees, and provides realizability criteria for simple graphs along with a constructive method. A cancellation algorithm on the induced gML tree enables reduction of critical cells while preserving targeted properties, linking to symmetry and sublevel automorphisms. The approach clarifies the inverse problem for 1D complexes, offers practical tools for simplifying Morse filtrations, and lays groundwork for higher-dimensional extensions and geometric group-theoretic perspectives on sublevel symmetries.

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 -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 -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 that cancels the critical simplices of and replaces it with an optimal discrete Morse function.
Paper Structure (8 sections, 12 theorems, 15 equations, 1 figure)

This paper contains 8 sections, 12 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on DMfs and Merge Trees
  3. Inverse Problem for Multigraphs
  4. Realization Problem with Simple Graphs
  5. How to Find Cancellations with Merge Trees
  6. Future Directions
  7. Conflict of Interest Statement
  8. Data Availability Statement

Key Result

Lemma 2.1

Let $(X,f)$ be a dMf on a graph and let $M(X,f)$ be the induced gML tree. For any critical cell $s\in X$, the generalized merge subtreeRecall iMLT for the definition of $M(s)$ and def: generalized merge tree for the definition of generalized merge subtrees $T(\_)$.$T(M(s))$ of $M(X,f)$ is induced by

Figures (1)

  • Figure 1: Relationships between dMfs and merge trees

Theorems & Definitions (64)

  • Definition 2.1: JohnsonScoville2022/Bene
  • Example 2.1
  • Remark 2.1
  • Definition 2.2: JohnsonScoville2022
  • Remark 2.2
  • Definition 2.3
  • Remark 2.3
  • Example 2.2
  • Definition 2.4: brueggemann2021merge
  • Remark 2.4
  • ...and 54 more