Maintaining 3-connectivity relative to a fixed basis

A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of E(M)−B. If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M.