A splitter theorem on 3-connected matroids

Abstract

ABSTRACT. 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 X 1 , . . . , X n ⊆ E (G) such that each G/X i is a 3-connected graph with an H -minor, each X i is a singleton set or the edge set of a triangle of G with 3 degree-3 vertices and X 1 ∪ · · · ∪ X n contains no edge sets of circuits of G other than the X i 's. This result extends previous ones of Whittle (for k = 1, 2) and Costalonga (for k = 3).