What is a 4-connected matroid?

Abstract

The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\hat{I}»(X)\ge 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\ge 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by ${\mathcal{T}}_N$. A set $Z$ of elements of a matroid $M$ is $4$-{\em connected} if $\hat{I}»(A)\ge \min \{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\ge 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.

37 pages