Generating Internally Four-Connected Graphs

A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. A graph G is internally 4-connected if it is simple, 3-connected, has at least five vertices, and if for every partition (A, B) of the edge-set of G, either |A| ≤ 3, or |B| ≤ 3, or at least four vertices of G are incident with an edge in A and an edge in B. We prove that if H and G are internally 4-connected graphs such that they are not isomorphic, H is a minor of G and they do not belong to a family of exceptional graphs, then there exists a graph H ′ such that H ′ is isomorphic to a minor of G and either H ′ is obtained from H by splitting a vertex, or H ′ is an internally 4-connected graph obtained from H by means of one of four possible constructions. This is a first step toward a more comprehensive theorem along the same lines. 6 March 1997, Revised 21 August 2001. Published in J. Combin. Theory Ser. B 85, 21–58 (2002). 1 Supported by NSF under the program “Research Experiences for Undergraduates,” Grant No. DMS9623031. 2 Partially supported by NSF under Grant No. DMS-9623031 and by ONR under Grant No. N0001493-1-0325.