When does a biased graph come from a group labelling?

In the second part of this article, we use this theorem to construct some exceptional biased graphs. Notably, we prove that for every m≥3 and ℓ there exists a minor-minimal not group-labellable biased graph on m vertices where every pair of vertices is joined by at least ℓ edges. In particular, this shows that biased graphs are not well-quasi-ordered under minors. Finally, we show that these results extend to give infinite sets of excluded minors for certain natural families of frame and lift matroids, and to show that neither are these families well-quasi-ordered under minors.