On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.