Related papers: Counting matroids in minor-closed classes

A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $\mathbb{R}$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and…

We are interested in expanding our understanding of symplectic matroids by exploring the properties of a class of symplectic matroids with a "lattice of flats". Taking a well-behaved family of subdivisions of the cross polytope we obtain a…

Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle,…

If S is a set of matroids, then the matroid M is S-fragile if, for every element e in E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on…

In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system…

Given two finite matroids on the same ground set, a celebrated result of Edmonds says that the ground set can be partitioned into two disjoint subsets in a manner that there is a common independent set in both matroids whose intersection…

Polymatroids can be considered as "fractional matroid" where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a…

The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a…

The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is…

The notion of branch-depth for matroids was introduced by DeVos, Kwon and Oum as the matroid analogue of the tree-depth of graphs. The contraction-deletion-depth, another tree-depth like parameter of matroids, is the number of recursive…

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q,…

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden…

The $\mathit{growth\ rate\ function}$ for a nonempty minor-closed class of matroids $\mathcal{M}$ is the function $h_{\mathcal{M}}(n)$ whose value at an integer $n \ge 0$ is defined to be the maximum number of elements in a simple matroid…

We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid is bounded by…

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a…

Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly $\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for all $e \in E(M)$, at least one of $M\backslash e$ and $M/e$ has no minor in $\mathcal{N}$. In…

The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be…

We classify all matroids with at most 8 elements that have the half-plane property, and we provide a list of some matroids on 9 elements that have, and that do not have the half-plane property. Furthermore, we prove that several classes of…

A matroid is a combinatorial structure that captures and generalizes the algebraic concept of linear independence under a broader and more abstract framework. Matroids are closely related with many other topics in discrete mathematics, such…

A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or,…