On cycles and merge trees
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910781412474880 |
|---|---|
| author | Brüggemann, Julian Scoville, Nicholas A. |
| author_facet | Brüggemann, Julian Scoville, Nicholas A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01316 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On cycles and merge trees Brüggemann, Julian Scoville, Nicholas A. Algebraic Topology Combinatorics 57Q70 (Primary), 05C90, 55N31 (Secondary) 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. |
| title | On cycles and merge trees |
| topic | Algebraic Topology Combinatorics 57Q70 (Primary), 05C90, 55N31 (Secondary) |
| url | https://arxiv.org/abs/2301.01316 |