Related papers: On Vietoris--Rips complexes of hypercube graphs
Computing Persistent Homology for large point clouds remains a bottleneck for the wider adoption of persistent homology by the scientific community. We present an algorithm which can compute the degree-1 Vietoris-Rips Persistent Homology of…
The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been…
We determine the homotopy types of the independence complexes of the $(n \times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.
For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges corresponding to elementary transfers of one cell between two parts, followed by reordering. Let $K_n := \mathrm{Cl}(G_n)$ be the…
This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to an inclusion of data sets. Consequences of these assumptions are investigated on the…
We determine the homotopy types of the independence complexes of $(n \times 4)$ and $(n \times 5)$-square grid graphs. In fact, they are homotopy equivalent to wedges of spheres.
The matching complex $\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appear in various places and found applications in many areas of mathematics including computational geometry,…
The Vietoris-Rips filtration for an $n$-point metric space is a sequence of large simplicial complexes adding a topological structure to the otherwise disconnected space. The persistent homology is a key tool in topological data analysis…
We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is…
A challenge in computational topology is to deal with large filtered geometric complexes built from point cloud data such as Vietoris-Rips filtrations. This has led to the development of schemes for parallel computation and compression…
Schrijver identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$. Bj\"{o}rner and de Longueville proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is…
If V is the vertex sequence of a symmetric 2t-cycle in the hypercube graph with the vertices {1,-1}^t, then for any vertex T of the graph there exists a unique inclusion-minimal subset of V such that T is the sum of its elements. We present…
For $r\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at…
We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we…
In connection with commutative algebra, Bayer et al. introduced cut complexes in [Topology of cut complexes of graphs, SIAM J.\ Discrete Math., 38(2):1630-1675, 2024]. For a positive integer $k$, the $k$-cut complex of a graph $G$, denoted…
For a positive integer $k$, the \emph{ total $k$-cut complex} of a graph $G$, denoted as $\Delta_k^t(G)$, is the simplicial complex whose facets are $\sigma \subseteq V(G)$ such that $|\sigma| = |V(G)|-k$ and the induced subgraph $G[V(G)…
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.
We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its…
Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{Hom}(G,H)$…
We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has…