On preserving matroid 3-connectivity relative to a fixed basis

We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M , there exists a set K ⊆ E ( M ) of four distinct elements such that for all k ∈ K , si ( M / k ) is 3-connected whenever k ∈ B , and co ( M ∖ k ) is 3-connected whenever k ∈ E ( M ) − B . Moreover, we show that if no other elements of E ( M ) − K satisfy this property, then M necessarily has path-width 3.