Related papers: What is a 4-connected matroid?

We show that an adjoint of a loopless matroid is connected if and only if it itself is connected. Our first goal is to study the adjoint of modular matroids. We prove that a modular matroid has only one adjoint (up to isomorphism) which can…

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…

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…

Let $M$ be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. Locked subsets characterize nontrivial facets of the…

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; we order these extensions by the weak…

We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley…

If $G$ is a looped graph, then its adjacency matrix represents a binary matroid $M_{A}(G)$ on $V(G)$. $M_{A}(G)$ may be obtained from the delta-matroid represented by the adjacency matrix of $G$, but $M_{A}(G)$ is less sensitive to the…

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded…

We consider the rank reduction problem for matroids: Given a matroid M and an integer k, find a minimum size subset of elements of M whose removal reduces the rank of M by at least k. When M is a graphical matroid this problem is the…

It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for…

A matroid is uniform if and only if it has no minor isomorphic to $U_{1,1}\oplus U_{0,1}$ and is paving if and only if it has no minor isomorphic to $U_{2,2}\oplus U_{0,1}$. This paper considers, more generally, when a matroid $M$ has no…

A result of Mason, as refined by Ingleton, characterizes transversal matroids as the matroids that satisfy a set of inequalities that relate the ranks of intersections and unions of nonempty sets of cyclic flats. We prove counterparts, for…

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each non-negative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$.…

A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a…

We prove that any element in a matroid can be removed, by either deletion or contraction, in such a way that no tangle "splits".

Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related 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…

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth…

We study the realization spaces of matroids and hyperplane arrangements. First, we define the notion of naive dimension for the realization space of matroids and compare it with the expected dimension and the algebraic dimension, exploring…

One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the…