What is a 4-connected matroid?

Document Type

Article

Publication Date

1-1-2025

Abstract

The breadth of a tangle T in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of T . A 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 λ(X) ≥ 3. We prove that if T is a tangle of order k ≥ 4 and breadth l in a matroid M, then M has a weakly 4-connected minor N with a tangle TN of order k, breadth l and has the property that T is the tangle in M induced by TN . A set Z of elements of a matroid M is 4-connected if λ(A) ≥ min{|A ∩ Z|, |Z − A|, 3} for all A ⊆ E(M). As a corollary of our theorems on tangles we prove that if M contains an n-element 4-connected set where n ≥ 7, then M has a weakly 4-connected minor that contains an n-element 4-connected set.

Publication Source (Journal or Book title)

Electronic Journal of Combinatorics

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