Related papers: Towards a Splitter Theorem for Internally $4$-conn…

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either ${\rm co}(M\delete e)$, the cosimplification of $M\delete e$, or ${\rm si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural…

We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X…

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected…

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…

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…

Let $M$ be a matroid and let $Q$, $R$, $S$ and $T$ be subsets of the ground set such that the smallest separation that separates $Q$ from $R$ has order $k$ and the smallest separation that separates $S$ from $T$ has order $l$. We prove that…

Let $\mathcal M$ be a class of matroids closed under minors and isomorphism. Let $N$ be a matroid in $\mathcal M$ with an exact $k$-separation $(A, B)$. We say $N$ is a $k$-decomposer for $\mathcal M$ having $(A, B)$ as an inducer, if every…

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when…

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…

A result of Seymour implies that any 3-connected matroid with a modular 3-point line is binary. We prove a similar characterization for 3-connected matroids with modular 4-point lines. We show that such a matroid is either representable…

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 isotropic matroid $M[IAS(G)]$ of a graph $G$ is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic…

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$.…

In 1963, Halin and Jung proved that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. Mills and Turner proved an analog of this theorem by showing that every $3$-connected binary matroid in which…

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

For a matroid $M$, an element $e$ such that both $M\backslash e$ and $M/e$ are regular is called a regular element of $M$. We determine completely the structure of non-regular matroids with at least two regular elements. Besides four small…

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by…

Let $M$ be a 3-connected binary matroid and let $Y(M)$ be the set of elements of $M$ avoiding at least $r(M)+1$ non-separating cocircuits of $M$. Lemos proved that $M$ is non-graphic if and only if $Y(M)\neq\emp$. We generalize this result…

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any…

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