A Splitter Theorem for Internally 4‐Connected Binary Matroids

Abstract

We prove that if N is an internally 4‐connected minor of an internally 4‐connected binary matroid M with $E(N) \geq 4$, then there exist matroids $M_0, M_1, \ldots, M_n$ such that $M_0 \cong N$, $M_n = M$, and, for each $i\in\{1,\ldots,i\}$, $M_{i-1}$ is a minor of $M_{i}$, $|E(M_{i-1})|\ge |E(M_i)|-2$, and $M_i$ is 4‐connected up to separators of size 5.