Edge Contractions and Simplicial Homology

Lecture Notes in Computer Science, 2006

In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an "elementary simplicial perturbation" process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is a special chain homotopy equivalence connecting the simplicial complex with its homology (working with a field as the ground ring).

Advances in Applied Mathematics, 2001

This paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert-Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids.

unirioja.es

In this paper, an algorithm building the eective homology version of the pushout of the simplicial morphisms f : X → Y and g : X → Z, where X, Y and Z are simplicial sets with eective homology is presented.

arXiv: Combinatorics, 2018

In a known papers, A. Ivashchenko shows the family of contractible graphs, constructed from $K(1)$ by contractible transformations, and he proves that such transformations do not change the homology groups of graphs. In this paper, we show that a contractible graph is actually a collapsible graph (in the simplicial sense), from which the invariance of the homology follows. In addition, we extend a result of A. Ivashchenko about graph homology, to a filtration of graphs, and we prove that the persistent homology is preserved with respect to contractible transformations. We apply this property as an algorithm to preprocess a data cloud and reduce the computation of the persistent homology for the filtered Vietoris-Rips complex.

In this article we give characteristic properties of the simplicial homology groups of digital images which are based on the simplicial homology groups of topological spaces in algebraic topology. We then investigate Eilenberg-Steenrod axioms for the simplicial homology groups of digital images. We state universal coefficient theorem for digital images. We conclude that the Künneth formula for simplicial homology doesn't be hold in digital images. Moreover we show that the Hurewicz Theorem need not be hold in digital images. Copyright © AJCTA, all rights reserved.

Proceedings of the Web Conference 2021, 2021

A simplicial complex is a generalization of a graph: a collection of =-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 generalize the graph notion of truss decomposition to complexes, and show that this more powerful representation gives rise to dierent properties compared to the graph-based one. This power, however, comes with important computational challenges derived from the combinatorial explosion caused by the downward closure property of complexes. Drawing upon ideas from itemset mining and similarity search, we design a memory-aware algorithm, dubbed STD, which is able to eciently compute the truss decomposition of a simplicial complex. STD adapts its behavior to the amount of available memory by storing intermediate data in a compact way. We then devise a variant that computes directly the = simplices of maximum trussness. By applying STD to several datasets, we prove its scalability, and provide an analysis of their structure. Finally, we show that the truss decomposition can be seen as a ltration, and as such it can be used to study the persistent homology of a dataset, a method for computing topological features at dierent spatial resolutions, prominent in Topological Data Analysis.

2021

Given an arbitrary hypergraph H, we may glue toH a family of hypergraphs to get a new hypergraph H having H as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph H is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems.

Journal of Algebraic Combinatorics, 2000

We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For ? a simplicial complex of dimension d - 1, and each r = 0, ...,d, we define rth iterated homology groups of ?. When r = 0, this corresponds to ordinary homology. If ? is a cone over ?', then when

The aim of the paper is to study some of the homological groups in general and related these groups with simplicial complexes. Characterization these groups revealed the successful method to study the simplicial complex which has the following two properties: (a) each q-simplex determines (q +1) faces of dimension q-1, (b) the faces of a simplex determine the simplex and a semi-simplicial complex K is a collection of elements {f} called simplexes together with two functions. The main examples of homological groups are r-chain group, r-cycle group and r-boundary group. When we calculating the Euler characteristic of surface, we need to building a multi-surface equivalent to the original surface, therefore in this paper we achieved that the homological groups are a type of improvement for the Euler characteristic. If there is no simplex of order two (2-simplexs) in K, then B1(K) and H1(K) are equal to Z1(K). Also if K is a simplex complex, then r-chain (Cr(K) is a group. We obtained that if three points and three lines such that is triangulation of the rings and there is no simplex of order two (2-simplexs) in K, in this case the boundary homological group equal zero and H1(K)=Z1(K).

SIAM Conference on Geometric and Physical Modeling (GD/SPM 2011), 2011

We propose a new iterative algorithm for computing the homology of arbitrary shapes discretized through simplicial complexes. We demonstrate how the simplicial homology of a shape can be effectively expressed in terms of the homology of its sub-components. The proposed algorithm retrieves the complete homological information of an input shape including the Betti numbers, the torsion coefficients and the representative homology generators. To the best of our knowledge, this is the first algorithm based on the constructive Mayer-Vietoris sequence, which relates the homology of a topological space to the homologies of its sub-spaces, i.e. the sub-components of the input shape and their intersections. We demonstrate the validity of our approach through a specific shape decomposition, based only on topological properties, which minimizes the size of the intersections between the sub-components and increases the efficiency of the algorithm.