Related papers: Computing discrete Morse complexes from simplicial…
We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial…
We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions,…
Discrete Morse theory, a cell complex-analog to smooth Morse theory, has been developed over the past few decades since its original formulation by Robin Forman in 1998. In particular, discrete gradient vector fields on simplicial complexes…
Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…
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…
On a smooth manifold, we associate to any closed differential form a mapping cone complex. The cohomology of this mapping cone complex can vary with the de Rham cohomology class of the closed form. We present a novel Morse theoretical…
We present a structural clustering algorithm for large-scale datasets of small labeled graphs, utilizing a frequent subgraph sampling strategy. A set of representatives provides an intuitive description of each cluster, supports the…
We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit…
Bouc (1992) first studied the topological properties of $M_n$, the matching complex of the complete graph of order $n$, in connection with Brown complexes and Quillen complexes. Bj\"{o}rner et al. (1994) showed that $M_n$ is homotopically…
A simplicial complex is a generalization of a graph: a collection of n-ary relationships (instead of binary as the edges of a graph), named simplices. In this paper, we develop a new tool to study the structure of simplicial complexes: we…
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this…
We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse…
We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
Simplicial complexes are extensively studied in the field of algebraic topology. They have gained attention in recent time due to their applications in fields like theoretical distributed computing and simplicial neural networks. Graphs are…
We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests…
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology…