Matroid 4-Connectivity: A Deletion–Contraction Theorem

A 3-separation (A, B), in a matroid M, is called sequential if the elements of A can be ordered (a1, …, ak) such that, for i=3, …, k, ({a1, …, ai}, {ai+1, …, ak}∪B) is a 3-separation. A matroid M is sequentially 4-connected if M is 3-connected and, for every 3-separation (A, B) of M, either (A, B) or (B, A) is sequential. We prove that, if M is a sequentially 4-connected matroid that is neither a wheel nor a whirl, then there exists an element x of M such that either M\x or M/x is sequentially 4-connected.