Related papers: On merge trees with a given homological sequence

Suppose $M$ is a finite simplicial complex and that for $0=t_0,t_1,...,t_r=1$ we have a discrete Morse function $F_{t_i}:M\to \zr$. In this paper, we study the births and deaths of critical cells for the functions $F_{t_i}$ and present an…

We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3) prove that the icosahedral graph is…

This paper reveals the tree structure as an intermediate result of clustering by fast search and find of density peaks (DPCLUS), and explores the power of using this tree to perform hierarchical clustering. The array used to hold the index…

This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex…

Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…

We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a…

The induction of additional randomness in parallel and sequential ensemble methods has proven to be worthwhile in many aspects. In this manuscript, we propose and examine a novel random tree depth injection approach suitable for sequential…

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex…

In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure. We recommend a simple variant of the standard algorithm, in which clusters are merged by…

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed…

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…

We study the homology of an explicit finite-index subgroup of the automorphism group of a partially commutative group, in the case when its defining graph is a tree. More concretely, we give a lower bound on the first Betti number of this…

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…

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…

In this paper we introduce the horizon visibility graph, a simple extension to the popular horizontal visibility graph representation of a time series, and show that it possesses a rigorous mathematical foundation in computational algebraic…

Two of the most useful tools in topological combinatorics are the nerve lemma and discrete Morse theory. In this note we introduce a theorem that interpolates between them and allows decompositions of complexes into non-contractible pieces…

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric…

A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We…

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…

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize…