A splitter theorem on 3-connected matroids

We establish the following splitter theorem for graphs and its generalization for matroids: Let G and H be 3-connected simple graphs such that G has an H-minor and k≔|V(G)|−|V(H)|≥2. Let n≔k∕2+1. Then there are pairwise disjoint sets X1,…,Xn⊆E(G) such that each G∕Xi is a 3-connected graph with an H-minor, each Xi is a singleton set or the edge set of a triangle of G with 3 degree-3 vertices and X1∪⋯∪Xn contains no edge sets of circuits of G other than the Xi's. This result extends previous ones of Whittle (for k=1,2) and Costalonga (for k=3).