Towards a splitter theorem for internally 4-connected binary matroids II

Let M and N be internally 4-connected binary matroids such that M has a proper N -minor, and | E ( N ) | ≥ 7 . As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M ∖ e has no N -minor whenever e is in a triangle of M , and M / e has no N -minor whenever e is in a triad of M , then M has a minor, M ′ , such that M ′ is internally 4-connected with an N - minor, and 1 ≤ | E ( M ) | − | E ( M ′ ) | ≤ 2 .