The Internally 4-Connected Binary Matroids with No $M(K_{5}\backslash e)$-Minor
Dillon Mayhew, Gordon Royle · SIAM Journal on Discrete Mathematics · 2012
Let $\mathrm{AG}(3,2)\>\raisebox{-0.5pt}{\rotatebox{90}{$\Bowtie$}}\>U_{1,1}$ denote the binary matroid obtained from $\mathrm{AG}(3,2)\oplus U_{1,1}$ by completing the 3-point lines between every element in $\mathrm{AG}(3,2)$ and the element of $U_{1,1}$. We prove that every internally 4-connected binary matroid that does not have a minor isomorphic to $M(K_{5}\backslash e)$ is isomorphic to a minor of $(\mathrm{AG}(3,2)\>\raisebox{-0.5pt}{\rotatebox{90}{$\Bowtie$}}\>U_{1,1})^{*}$.