A Splitter Theorem Relative to a Fixed Basis

A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M)–B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M, it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N-minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s ₁ and s ₂, such that for each i ∈ {1, 2}, si(M/s _i ) is 3-connected and has an N-minor when s _i ∈ B, and co(M\s _i ) is 3-connected and has an N-minor when s _i ∈ E(M)–B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N-minor, then (P, E(M)–P) is a sequential 3-separation.