## 173 Citations

### A New Construction of the Vietoris-Rips Complex

- MathematicsArXiv
- 2023

. We present a new, inductive construction of the Vietoris-Rips complex, in which we take advantage of a small amount of unexploited combinatorial structure in the k -skeleton of the complex in order…

### Vietoris-Rips Homology Theory for Semi-Uniform Spaces

- Mathematics
- 2020

While the Vietoris-Rips complex is now widely used in both topological data analysis and the theory of hyperbolic groups, many of the fundamental properties of its homology have remained elusive. In…

### Approximating Cfree Space Topology by Constructing Vietoris-Rips Complex

- Mathematics2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
- 2019

A new way of constructing sparse roadmaps using point clouds that approximates and measures the underlying topology of the Cfree space and performs a series of topological collapses to remove vertices from the graph while still preserving its topological properties.

### Topological Data Analysis with ε-net Induced Lazy Witness Complex

- Computer ScienceArXiv
- 2019

It is proved that -net, as a choice of landmarks, is an -approximate representation of the point cloud and the induced lazy witness complex is a 3-approximation of the induced Vietoris-Rips complex and three algorithms to construct -net landmarks are proposed.

### The tidy set: a minimal simplicial set for computing homology of clique complexes

- Computer ScienceSCG
- 2010

The tidy set is introduced, a minimal simplicial set that captures the topology of a simplicial complex that is particularly effective for computing the homology of clique complexes.

### An improved algorithm for Generalized Čech complex construction

- Computer ScienceArXiv
- 2022

An algorithm that computes the generalized ˇCech complex for a set of disks where each may have a diﬀerent radius in 2D space is presented and an eﬃcient veriﬁcation method is proposed to see if a k -simplex can be constructed on the basis of the ( k − 1)-simplices.

### Faster Enumeration of All Maximal Cliques in Unit Disk Graphs Using Geometric Structure

- Mathematics, Computer ScienceIEICE Trans. Inf. Syst.
- 2015

A faster algorithm is proposed based on two well-known algorithms called Bron-Kerbosch and Tomita-Tanaka- Takahashi for enumerating all max- imal cliques in unit disk graphs, which is a plausible setting for applications of similar data groups.

### Constructing simplicial complexes over topological spaces

- Mathematics, Computer Science
- 2012

The utility of the oracle-based framework for constructing simplicial complexes over arbitrary topological spaces is demonstrated by presenting three applications: to multiword search in Google, to the computational analysis of a language and to the study of protein structure.

### Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes

- MathematicsSoCG '11
- 2011

This work shows that Rips complexes can also be used to provide topologically correct reconstruction of shapes and compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach.

### On contractible transformations of graphs: Collapsibility and homological properties

- Mathematics
- 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…

## References

SHOWING 1-10 OF 49 REFERENCES

### The tidy set: a minimal simplicial set for computing homology of clique complexes

- Computer ScienceSCG
- 2010

The tidy set is introduced, a minimal simplicial set that captures the topology of a simplicial complex that is particularly effective for computing the homology of clique complexes.

### Topological estimation using witness complexes

- MathematicsPBG
- 2004

This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.

### Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension

- Computer ScienceSCG '09
- 2009

A new implementation of the well-known incremental algorithm for constructing Delaunay triangulations in any dimension is described and a modification of the algorithm that uses and stores only theDelaunay graph (the edges of the full triangulation) is proposed.

### Topological Persistence and Simplification

- EconomicsProceedings 41st Annual Symposium on Foundations of Computer Science
- 2000

Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.

### A New Algorithm for Generating All the Maximal Independent Sets

- Mathematics, Computer ScienceSIAM J. Comput.
- 1977

This paper presents a new efficient algorithm for generating all the maximal independent sets, for which processing time and memory space are bounded by $O(nm\mu)$ and $O (n+m)$, respectively, where n, m, and $\mu$ are the numbers of vertices, edges, and maximalIndependent sets of a graph.

### Combinatorial Algebraic Topology

- MathematicsAlgorithms and computation in mathematics
- 2008

Concepts of Algebraic Topology, Applications of Spectral Sequences to Hom Complexes, and Structural Theory of Morphism Complexes are presented.

### Computing Persistent Homology

- MathematicsSCG '04
- 2004

The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.

### Reporting maximal cliques: new insights into an old problem

- Computer Science
- 2005

The bottom-line of this study is the investigation of the trade-off between output sensitivity and the overhead associated to the identification of dominated nodes.