Frame matroids with a distinguished frame element

A matroid is frame if it can be extended such that it possesses a basis $B$ (a frame) such that every element is spanned by at most two elements of $B$. Frame matroids extend the class of graphic matroids and also have natural graphical representations. We characterise the inequivalent graphical representations of 3-connected frame matroids that have a fixed element $\ell$ in their frame $B$. One consequence is a polynomial time recognition algorithm for frame matroids with a distinguished frame element.

Joint work with Jim Geelen and Cynthia Rodríquez.