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 , s i ( 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.