A splitter theorem on 3-connected binary matroids and inner fans

Abstract

We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids $N<M$, there is a disjoint family $\{X_1,\dots ,X_n\}\subseteq 2^{E\left(M\right)}$ such that $r\left(X_1\right)+\cdots +r\left(X_n\right)=r(X_1\cup \cdots \cup X_n)\ge r\left(M\right)-r\left(N\right)$, each $\mathrm{si}(M/X_i)$ is 3-connected with an N-minor, and either $\left|X_i\right|=1$ or X is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.