N-Critical Matroids

Let M and N be 3-connected matroids; we say that M is N-critical if M has an N-minor, but for each $$x\in E(M)$$ , $$M\backslash x$$ is not 3-connected or $$M\backslash x$$ has no N-minor. We establish that if M is an N-critical matroid with $$r^*(M)>\max \{3,r^*(N)\}$$ , then M has an element x such that either $$\mathrm{co}(M\backslash x)$$ is N-critical or M has a coline $$L^*$$ with $$|L^*|\ge 3$$ such that $$M\backslash L^*$$ is N-critical. As a corollary we get a chain theorem for the class of minimally 3-connected matroids. This chain theorem generalizes a previous one of Anderson and Wu for binary matroids.