相关论文: Topology of matching complexes of complete graphs …
We denote the matching complex of the complete graph with $n$ vertices by $M_n$. Bouc first studied the topological properties of $M_n$ in connection with the Quillen complex. Later Bj\"{o}rner, Lov\'{a}sz, Vre\'{c}ica, and…
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…
The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from…
Robin Forman's highly influential 2002 paper A User's Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
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 $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this…
In this corrigendum to arXiv:2305.02973, we justify that the motivation behind Conjecture 5.1 is based on a mistaken notion. Moreover, we prove a stronger theorem that disproves the conjecture, and obtain an optimal gradient vector field on…
We introduce an algorithm that constructs a discrete gradient field on any simplicial complex. We show that, in all situations, the gradient field is maximal possible and, in a number of cases, optimal. We make a thorough analysis of the…
Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex $M_n$, which is the simplicial complex of matchings in the complete graph $K_n$.…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [11], a hypergraph gives an associated simplicial complex. By [4], the embedded homology of a hypergraph is the homology of the infimum chain…
The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient…
The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not…
Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes.…
We consider the problem of efficiently computing a discrete Morse complex on simplicial complexes of arbitrary dimension and very large size. Based on a common graph-based formalism, we analyze existing data structures for simplicial…
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient…
Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or…
Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the…
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…
Topological abstractions offer a method to summarize the behavior of vector fields but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach,…
In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous…