Collapsibility and homological properties of $\mathfrak{I}$-contractible transformations

TL;DR

This paper investigates the collapsibility and homological features of clique complexes derived from $rak{I}$-contractible graphs, introduces algorithms for identifying strong $rak{I}$-contractibility, and applies these methods to compute persistent homology in topological data analysis.

Contribution

It provides new insights into the collapsibility of clique complexes of $rak{I}$-contractible graphs and introduces algorithms for their recognition and simplification.

Findings

Clique complexes of strong $rak{I}$-contractible graphs are collapsible.

Algorithms for verifying strong $rak{I}$-contractibility and vertex deletion are developed.

Application of these algorithms to compute persistent homology in topological data analysis.

Abstract

The family of $\mathfrak{I}$-contractible graphs and contractible transformations was defined by A. Ivashchenko in the mid-90's. In this paper we study the collapsibility and homological properties of the clique complex associated to $\mathfrak{I}$-contractible graphs. We show that for any graph in a special subfamily of the $\mathfrak{I}$-contractible graphs (the strong $\mathfrak{I}$-contractible ones) its clique complex is collapsible. Moreover, we present an algorithm that allows us to verify if any graph is strong $\mathfrak{I}$-contractible, as well as an algorithm to delete those vertices whose open neighborhood is also strong $\mathfrak{I}$-contractible. Finally, we show how to use these algorithms to compute the persistent homology of an arbitrary Vietoris-Rips complex for applications in topological data analysis.