What is a \(4\)-connected matroid?

Nick Brettell¹ ; Susan Jowett¹ ; James Oxley² ; Charles Semple³ ; Geoff Whittle

¹Victoria University of Wellington
²Louisiana State University
³University of Canterbury

The electronic journal of combinatorics, Tome 32 (2025) no. 2

Résumé

The breadth of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. The matroid $M$ is weakly $4$-connected if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\lambda(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}_N$ 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 $4$-connected if $\lambda(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ 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 $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.

Détail

DOI : 10.37236/12467

Classification : 05B35
Mots-clés : breadth of a tangle in a matroid, weakly \(4\)-connected matroid