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

Whittle proved, for k=1,2k=1,2, that if NN is a 33-connected minor of a 33-connected matroid MM, satisfying r(M)−r(N)≥kr(M)−r(N)≥k, then there is a kk-independent set II of MM such that, for every x∈Ix∈I, si(M/x)si(M/x) is a 33-connected matroid with an NN-minor. In this paper, we establish this result for k=3k=3. It is already known that it cannot be extended to greater values of kk. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6r(M)−r(N)≥6, we can guarantee the existence of a 44-independent set of MM with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5r(M)−r(N)≥5, then MM has such a 44-independent set or MM has a triangle TT meeting 33 triads and such that M/TM/T is a 33-connected matroid with an NN-minor.

► We study possibilities for some chains of 3-connected matroids. ► e is vertically NN-contractible if si(M/e) is 3-connected with an NN-minor. ► Some results about sets of vertically NN-contractible elements are set. ► The existence of a 3-independent set of vertically NN-contractible elements. ► The existence of such a 4-independent set in binary matroids and graphs.