Michael Joswig, Marc E. Pfetsch
Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.
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In this paper, we determine the homotopy type of the Morse complex and matching complex of multiple families of complexes by utilizing star cluster collapses and the Cluster Lemma. We compute the homotopy type of the Morse complex of an extended notion of a star graph, as well as the homotopy type of the matching complex of a Dutch windmill graph. Additionally, we provide alternate computations of the homotopy type of the Morse complex of paths, the homotopy type of the matching complex of paths, and the homotopy type of the matching complex of cycles. We then use this same method of computing…
We introduce a Morse theory for posets of Bestvina-Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.
We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes.
We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these matchings to classify every subcomplex whose reduced homology groups are concentrated in a single degree and describe a homology basis for each of them.
Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant of an input simplicial complex to computing the same topological invariant of a (significantly smaller) collapsed cell or chain complex. Consequently, devising methods for obtaining gradient vector fields on complexes to reduce the size of the problem instance has become an emerging theme over the last decade. While computing the optimal gradient vector fie…
Corrigendum to "Topology of matching complexes of complete graphs via discrete Morse theory'' [arXiv:2305.02973, Discrete Math. Theor. Comput. Sci. 26:3#13 (2024)]
本文是对完全图匹配复形拓扑研究的勘误，指出Conjecture 5.1动机有误，并证明了更强的定理反驳该猜想，同时在阶为7的完全图匹配复形上获得了最优梯度向量场。
The complex of discrete Morse matchings of the $n$-simplex: homotopy types and structural results
本文研究了离散Morse配对复形的同伦类型和结构，计算了Δ³和∂Δ³的同伦类型，并证明了对于一般n，f(n)与n-2维骨架的配对数关系，同时提出两个猜想扩展M_P和GM等价性。
We develop Morse-Bott theory on posets, generalizing both discrete Morse-Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik-Schnirelmann theorem for general matchings on posets, in particular, for Morse-Bott functions.
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations of $C$ can be efficiently computed, in particular its homology or Morse-Smale decomposition. Given a function $f$ sampled on $C$, it is possible to derive a discrete gradient that mimics the dynamics of $f$. Many such constructions are based on some variant of a greedy pairing of adjacent cells, given an appropriate weighting. However, proving that the dyna…
We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on $S^2$, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, …
Morse matchings and Khovanov homology of 4-strand torus links
本文提出一种推迟消去的Morse理论简化算法用于计算Khovanov同调，并借此完整计算了4股环面链环的整系数Khovanov同调群，发现大量4-挠元，验证了极限情形下的相关猜想并给出有理Gordian距离下界。
Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word.…
Complexes of strong discrete Morse matchings
本文基于强离散 Morse 理论定义强 Morse 匹配复形，并计算了路径、圈、星、单纯形及其边界等复形上这些复形的同伦类型及圈情形的同调。
The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(Δ_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(Δ_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(Δ_n^m)$ and prove that several homology groups of $\mathrm{Ind}(Δ_n^m)$ are non-zero.
In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular realization by a complete matching on the Hasse diagram of a triangulation of the manifold.
In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse functions based on the induced merge tree. We then relate the matching number of a tree to a certain invariant of the induced merge tree. Finally, we count the number of merge trees that can be induced on a star graph and characterize the induced merge tree.
We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's discrete Morse theory for CW-complexes and generalizes Forman and Chari's results on the face posets of regular CW-complexes. We also introduce a homological variant of the theory that can be used to study the topology of triangulable homology manifolds by means of their order triangulations.
Discrete Morse theory and the topology of matching complexes of complete graphs
本文利用离散莫尔斯理论技术研究完全图匹配复形 $M_n$ 的拓扑，证明其具有比同伦连通性更强的几何 $(ν_n-1)$-连通性，并确定 $M_8$ 同伦等价于 132 个 2 维球面的楔和。
The Dowker theorem via discrete Morse theory
本文通过构造离散 Morse 理论中的显式无环匹配，重新证明了 Dowker 定理的 Barmak 加强版本，即二元关系定义的两个复形是简单同伦等价的。
We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them "precise matchings"). The existence of precise matchings implies that the homology has a square-free torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.
We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us t…
A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex
本文提出一种递归方法，用于构建图的独立复形上的无环匹配，应用离散Morse理论确定了几种图类的同调类型，并为高效同调计算提供预处理框架。
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. (2) The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naïveté, this approach turns out to be very successful even in the case of huge inputs. (3) In our view the existing libraries of examples in …
Forman introduced discrete Morse theory as a tool for studying CW complexes by essentially collapsing them onto smaller, simpler-to-understand complexes of critical cells in [Fo]. Chari reformulated discrete Morse theory for regular cell complexes in terms of acyclic matchings on face posets in [Ch]. This paper addresses two questions: (1) under what conditions may several gradient paths in a discrete Morse function simultaneously be reversed to cancel several pairs of critical cells, to further collapse the complex, and (2) how to use lexicographically first reduced expressions for permutatio…
A $2$-matching complex is a simplicial complex which captures the relationship between $2$-matchings of a graph. In this paper, we will use discrete Morse Theory and the Matching Tree Algorithm to prove homotopical results. We will consider a class of graphs for which the homotopy type of the $2$-matching complex transforms from a sphere to a point with the addition of leaves. We end the paper by defining $k$-matching sequences and looking at the $1$- and $2$-matching complexes of wheel graphs and perfect caterpillar graphs.
We revisit the periodic complexity function $h_{\bf w}(n)$ introduced by Mignosi and Restivo. This function gives the average of the first $n$ local periods of a recurrent infinite word ${\bf w}$. We give a different method than that of Mignosi and Restivo for computing the asymptotics of the periodic complexity function of the Thue-Morse word and show how to apply the method to other automatic sequences, like the Rudin-Shapiro word and the period-doubling word.
A central problem in topological data analysis is that of computing the homology of a given simplicial complex. Said complexes can have arbitrary large number of simplices, as can happen, for example, if the space is the Rips-Vietoris or Cech complex of a large data cloud. Thus, pre-processing the simplicial complex to get a smaller complex with the same homology groups and then applying the homology algorithm to the smaller one, has been an active research topic in the last years. In this survey, we discuss some recent papers that examine the complexity of this simplification via Discrete Mor…
Morse-Smale complexes on convex polyhedra
受地貌学应用驱动，本文将Morse-Smale理论拓展至定义三维凸多面体的径向距离函数，分析了其与两类相关Morse-Smale复形的异同，并设计实现了相应的显式计算算法。
Computing the k-binomial complexity of generalized Thue--Morse words
本文研究广义 Thue--Morse 词的 k-二项式复杂度，通过刻画因子间的 k-二项式等价关系推导出显式公式，正面解决了其复杂度函数最终以 m^k 为周期的猜想，并发展了阿贝尔 Rauzy 图等工具。
Topology of matching complexes of complete graphs via discrete Morse theory
本文利用离散Morse理论构造并改进完全图匹配复形$M_n$（特别是$M_7$）的梯度向量场，以纯组合方式证明其连通性，并高效计算$M_7$的同调群，同时提出一个猜想为最优的梯度向量场。
Morse Theory for the k-NN Distance Function
本文研究了 $\mathbb{R}^d$ 中有限点集的 $k$-最近邻距离函数，提出了分析其次水平集拓扑的 Morse 理论框架，给出了临界点及其指标的组合几何刻画，并计算了齐次 Poisson 过程的临界点期望数量，为持续同调等分析提供了重要见解与工具。
The (combinatorial) diameter of a polytope $P \subseteq \mathbb R^d$ is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of $P$, that is the graph where the nodes are given by the $0$-dimensional faces of $P$, and the edges are given the 1-dimensional faces of $P$. The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard. In this paper, we show that the problem of computing the diame…
On The Morse Ensemble Polynomial Of Simplicial Complexes
本文引入单纯复形的Morse系综多项式，通过Laplacian公式、顶面递归和独立复形多项式等结果，证明了该多项式是比Tutte多项式更精细的同构不变量。
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $μ$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Forman also proved the existence of a CW complex which is homotopy equivalent to $X$ and whose cells are in one-to-one correspondence with the critical cells of $μ$, but the construction is ad hoc and does not have a combin…
Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators
本文研究了离散Morse理论中消除临界对对(co)边界算子的影响，推导出消除后算子的显式公式，通过行变换实现无需重新枚举梯度轨迹。
Complexity of polytope diameters via perfect matchings
本文证明计算多胞体电路直径是强NP-hard问题，通过二部图完美匹配多胞体直径的NP-hard性实现，并给出单调直径的图论描述。
Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of G, then it is called a Konig-Egervary graph. A graph is unicyclic if it has a unique cycle. In 2010, Bartha conjectured that a unique perfect matching, if it exists, can be found in O(m) time, where m is the number of edges. In this paper we validate this conjecture for Konig-Egervary graphs and unicylic graphs. We propose a variation of Karp-Sipser leaf-rem…
Given non-negative weights w_S on the k-subsets S of a km-element set V, we consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1 cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman-Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect mat…
Enumeration of maximum matchings of graphs
本文利用 Gallai-Edmonds 结构定理推导出计算图的最大匹配数的公式，并据此提出枚举算法，将问题转化为计算导出子图的完美匹配数，并应用于最优树，推广了已有结论。
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