Related papers: Computing excluded minors for classes of matroids …
The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of…
If $\mathcal{C}$ is a minor-closed class of matroids, the class $\mathcal{C}'$ of integer polymatroids whose natural matroids are in $\mathcal{C}$ is also minor closed, as is the class $\mathcal{C}'_k$ of $k$-polymatroids in $\mathcal{C}'$.…
Mayhew and Royle (2008) showed that there are 564 excluded minors for the class of GF(5)-representable matroids having at most 9 elements. We enumerate the excluded minors for GF(5)-representable matroids having 10 elements: there are…
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…
We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show…
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
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…
We find the excluded minors for the minor-closed class of lattice path polymatroids as a subclass of the minor-closed class of Boolean polymatroids. Like lattice path matroids and Boolean polymatroids, there are infinitely many excluded…
If $\mathcal{C}$ is a minor-closed class of matroids, then the class $\widetilde{\mathcal{C}}'_k$ of $k$-polymatroids whose $k$-natural matroids are in $\mathcal{C}$ is also minor-closed. We investigate the following question: When…
For all positive integers $\ell$ and $r$, we determine the maximum number of elements of a simple rank-$r$ positroid without the rank-$2$ uniform matroid $U_{2,\ell+2}$ as a minor, and characterize the matroids with the maximum number of…
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic…
The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minors for these classes can be drastically reduced if we…
In earlier work, we characterized the class of matroids with no $M(C_4)$ as an induced minor and the class of matroids with no member of $\{M(C_4),M(K_4)\}$ as an induced minor. In this paper, for every two matroids in…
The classes of bicircular matroids and lattice path matroids are closed under minors. The complete list of excluded minors for the class of lattice path matroids is known, and it has been recently shown that the analogous list for the class…
We give an excluded-minor characterization for the class of matroids M in which M\e or M/e is binary for all e in E(M). This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary…
A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of $2$-polymatroids, up to isomorphism. When we implemented this algorithm on a…
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular…
Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A…
A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number…
Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list…