The Structure of Connectivity Functions

Graphs, matroids and polymatroids all have associated connectivity functions, and many properties of these structures follow from properties of their connectivity functions. This motivates the study of connectivity functions in general. It turns out that connectivity functions are surprisingly highly structured. We prove some interesting results about connectivity functions. In particular we show that every connectivity function is a connectivity function of a half-integral polymatroid

Generic Rigidity Matroids with Dilworth Truncations

We prove that the linear matroid that defines generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$ graphic matroids by applying variants of Dilworth truncation $n_r$ times, where $n_r$ denotes the number of rods. This leads to an alternative proof of Tay's combinatorial characterizations of generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks

Branch-Width and Rota's Conjecture

AbstractFor a fixed finite field F and an integer k there are a finite number of matroids of branch-width k that are excluded minors for F-representability

A note on the connectivity of 2-polymatroid minors

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but somewhat weaker result for $2$-polymatroids. Specifically, if $M$ is a connected $2$-polymatroid with a proper connected minor $N$, then there is an element $e$ of $E(M) - E(N)$ such that $M \backslash e$ or $M / e$ is connected having $N$ as a minor. We also consider what can be said about the uniqueness of the way in which the elements of $E(M) - E(N)$ can be removed so that connectedness is always maintained.Comment: 9 page

On Matroid and Polymatroid Connectivity

Matroids were introduced in 1935 by Hassler Whitney to provide a way to abstractly capture the dependence properties common to graphs and matrices. One important class of matroids arises by taking as objects some finite collection of one-dimensional subspaces of a vector space. If, instead, one takes as objects some finite collection of subspaces of dimensions at most k in a vector space, one gets an example of a k-polymatroid. Connectivity is a pivotal topic of study in the endeavor to understand the structure of matroids and polymatroids. In this dissertation, we study the notion of connectivity from several angles. It is a well-known result of Tutte that, for every element x of a connected matroid M, at least one of the deletion and contraction of x from M is connected. Our first result shows that, in a connected k-polymatroid, only two such elements are guaranteed. We show that this bound is sharp and characterize those 2-polymatroids that achieve this minimum. It is well known that, for any integer n greater than one, there is a number r such that every 2-connected simple graph with at least r edges has a minor isomorphic to an n-edge cycle or K2,n. This result was extended to matroids by Lovász, Schrijver, and Seymour who proved that every sufficiently large connected matroid has an n-element circuit or an n-element cocircuit as a minor. As our second result, we generalize these theorems by providing an analogous result for connected 2-polymatroids. Significant progress on the corresponding problem for k-polymatroids is also described. Finally, we look at tangles, a tool that has been used extensively in recent results in matroid structure theory. We prove that a matroid with at least two elements is a tangle matroid if and only if it cannot be covered by three hyperplanes. Some consequences of this theorem are also noted. In particular, no binary matroid of rank at least two is a tangle matroid