Related papers: A notion of minor-based matroid connectivity

Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following…

In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special…

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…

Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets…

For a matroid $M$ of rank $r$ on $n$ elements, let $b(M)$ denote the fraction of bases of $M$ among the subsets of the ground set with cardinality $r$. We show that $$\Omega(1/n)\leq 1-b(M)\leq O(\log(n)^3/n)\text{ as }n\rightarrow \infty$$…

Given an $n$-connected binary matroid, we obtain a necessary and sufficient condition for its single-element coextensions to be $n$-connected.

Generalizing a well known theorem for finite matroids, we prove that for every (infinite) connected matroid M there is a unique tree T such that the nodes of T correspond to minors of M that are either 3-connected or circuits or cocircuits,…

The Splitter Theorem states that, if $N$ is a 3-connected proper minor of a 3-connected matroid $M$ such that, if $N$ is a wheel or whirl then $M$ has no larger wheel or whirl, respectively, then there is a sequence $M_0,..., M_n$ of…

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…

A clutter consists of a finite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for…

We consider the situation that M and N are 3-connected matroids such that |E(N)| > 3 and C* is a cocircuit of M with the property that M/y has an N-minor for some y in C*. We show that either there is an element x in C* such that si(M/x) or…

Let AG(3,2)xU(1,1) denote the binary matroid obtained from the direct sum of AG(3,2) and a coloop by completing the 3-point lines between every element in AG(3,2) and the coloop. We prove that every internally 4-connected binary matroid…

For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the…

Let $M$ be an internally $4$-connected binary matroid with every element in three triangles. Then $M$ has at least four elements $e$ such that si$(M/e)$ is internally 4-connected.

In this paper we establish a variation of the Splitter Theorem. Let $M$ and $N$ be simple 3-connected matroids. We say that $x\in E(M)$ is vertically $N$-contractible if $si(M/x)$ is a 3-connected matroid with an $N$-minor. Whittle (for…

In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related…

For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem…

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 this sequel to "Foundations of matroids - Part 1", we establish several presentations of the foundation of a matroid in terms of small building blocks. For example, we show that the foundation of a matroid M is the colimit of the…

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…