Merge trees in discrete Morse theory

Johnson, Benjamin D.; Scoville, Nicholas A.

doi:10.48550/arxiv.2007.10272

Cited by 1 publication

(16 citation statements)

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“…In discrete Morse theory, there are various applications for merge trees. For instance, in [JS20] an equivalence relation between discrete Morse functions (dMf) on a given tree is introduced. Two dMfs are considered to be equivalent if and only if their induced merge trees are isomorphic.…”

Section: Introductionmentioning

confidence: 99%

“…We respond to an open question asked in [JS20] by showing that every merge tree is represented by a discrete Morse function. In particular, for any given merge tree we construct a dMf on a path as a representative of the isomorphism class defined by said merge tree: Theorem 5.5.…”

Section: Introductionmentioning

confidence: 99%

“…The notion of merge trees we use originates from [JS20, Def 5] and differs from the one used in [Cur19]: A priori, merge trees T in the sense of [JS20] do not carry a function T → R as part of their data. Instead, the two children of each node have a chirality assigned to them as part of the tree's data.…”

Section: Introductionmentioning

confidence: 99%

“…We obtain a similar correspondence between the chirality of merge trees and orientations on paths using the simplex order, Definition 3.11. Apart from these differences, the notion of merge trees in the sense of [JS20] is closely related to the one from [Cur19]. Chirality in the sense of [JS20] is a specific version of the notion of chirality used in [Cur19].…”

Section: Introductionmentioning

confidence: 99%

“…Apart from these differences, the notion of merge trees in the sense of [JS20] is closely related to the one from [Cur19]. Chirality in the sense of [JS20] is a specific version of the notion of chirality used in [Cur19]. In order to see this, we show that the construction of the merge tree M(X, f ) induced by a discrete Morse function f in the sense of [JS20] can be modified, see Proposition 2.19, to obtain a function T → R from f , similarly to [Cur19].…”

Section: Introductionmentioning

confidence: 99%

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On Merge Trees and Discrete Morse Functions on Paths and Trees

Brüggemann¹

2021

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In this work we answer an open question asked by Johnson-Scoville. We show that each merge tree is represented by a discrete Morse function on a path. Furthermore, we present explicit constructions for two different but related kinds of discrete Morse functions on paths that induce any given merge tree. A refinement of the used methods allows us to define notions of equivalence of discrete Morse functions on trees which give rise to a bijection between equivalence classes of discrete Morse functions and isomorphism classes of certain labeled merge trees. We also compare our work to similar results of Curry.Within this chain, we refer by X f c i −ε to the complex that immediately precedes X c i . If it is clear from the context which dMf is being referred to, we drop the dMf from the notation.Remark 2.6. The aforementioned notion of sublevel complex is only that concise because dMfs are already assumed to be weakly increasing in this paper. If you work with a more general notion of dMf, you will need to include all faces of simplices σ with f (σ) ≤ a as well.Lemma 2.7. Let X be a finite simplicial complex and let f : X → R be a dMf. Then f attains its minimum on a critical 0-simplex. Furthermore, the statement also holds for the restriction to any connected component of sublevel complexes.Proof. Because f is weakly increasing, no higher simplex τ can have a value which is strictly smaller than the value of f on any of τ s boundary 0-simplices. Thus, the minimum is attained on a 0-simplex.Let σ be a 0-simplex such that f (σ) is the minimum of f . We prove by contradiction that σ is critical: If σ is not critical, then there is a 1-simplex τ such that σ is a boundary 0-simplex of τ and f (τ ) = f (σ) holds. Let σ ′ be the other boundary 0-simplex of τ . Then f (σ ′ ) < f (τ ) = f (σ) holds because f is by assumption weakly increasing and at most 2 − 1. This is a contradiction to f attaining its minimum on σ, so f attains its minimum on a critical 0-simplex.Since all three properties of Definition 2.1 are inherited by arbitrary subcomplexes of X, the statement is also true for connected components of sublevel complexes.Remark 2.8. The analogous statement for the maximum of a dMf f on arbitrary 1simplices is false, as the following example shows:

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%