Minimal counterexamples for contractible graphs and related notions

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be ${mathcal I}$-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of ${mathcal I}$-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ, as well as a miminal counterexample to an erroneous claim of Ivashchenko from the literature. We also relate these classes of graphs to the notion of $k$-dismantlable graphs and $k$-collapsible complexes, and discuss some open questions.

On smooth families of exact forms

For a smooth family of exact forms on a smooth manifold, an algorithm for computing a primitive family smoothly dependent on parameters is given. The algorithm is presented in the context of a diagram chasing argument in the v{C}ech-de Rham complex. In addition, explicit formulas for such primitive family are presented.

A numerical approach for the filtered generalized Cech complex

In this paper, we present an algorithm to compute the filtered generalized v{C}ech complex for a finite collection of disks in the plane, which dont necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We present two applications of our main results. We give an algorithm for computing the 2-dimensional filtered generalized v{C}ech complex of a finite collection of $d$-dimensional disks in $mathbb{R}^d$. In addition, we show how the algorithm yields the minimal enclosing ball for a finite set of points in the plane.

Global product structure for a space of special matrices

The importance of the Hurwitz Metzler matrices and the Hurwitz symmetric matrices can be appreciated in different applications: communication networks, biology and economics are some of them. In this paper, we use an approach of differential topology for studying such matrices. Our results are as follows: the space of the $ntimes n$ Hurwitz symmetric matrices has a product manifold structure given by the space of the $(n-1) times (n-1)$ Hurwitz symmetric matrices and the euclidean space. Additionally we study the space of Hurwitz Metzler matrices and these ideas let us do an analysis of robustness of Hurwitz Metzler matrices. In particular, we study the Insulin Model as application.

Homological and collapsibility properties of clique complexes

The family of contractible graphs, introduced by A. Ivashchenko, consists of the collection $mathfrak{I}$ of graphs constructed recursively from $K_1$ by contractible transformations. In this paper we show that every graph in a subfamily of $mathfrak{I}$ (the strongly contractible ones) is a collapsible graph (in the simplicial sense), by providing a sequence of elementary collapses induced by removing contractible vertices or edges. In addition, we introduce an algorithm to identify the contractible vertices in any graph and show that there is a natural homomorphism, induced by the inclusion map of graphs, between the homology groups of the clique complex of graphs with the contractible vertices removed. Finally, we show an application of this result to the computation of the persistent homology for the Vietoris-Rips filtration.