Related papers: A splitter theorem for elastic elements in $3$-con…

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…

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…

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…

Let $M$ be a $3$-connected binary matroid; $M$ is called internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a…

Let $M$ be a $3$-connected binary matroid; $M$ is internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a…

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…

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…

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M…

We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important…

Let M and N be internally 4-connected binary matroids such that M has a proper N-minor, and |E(N)| is at least seven. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following…

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 $n:=\left\lceil k/2\right\rceil+1$.…

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an…

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.

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…

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…

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 $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. We say that a pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and…

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 $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As…

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an…