Proving excluded-minor characterisations using detachable pairs

<p><strong>Let M be an excluded minor for the class of GF(4)-representable matroids. We give an independent proof, using results on detachable pairs that, except for a single edge case, |E(M)| ≤ 14. Moreover, we compute all excluded minors up to size 10. We also define a class M consisting of matroids obtained by extending the cycle matroid M(Kn) via the principal modular cut consisting of two non-adjacent edges, and their minors. We show that members of this class are representable over all fields of size at least 4, that a maximum-sized simple rank-r member of M has size at most (r + 2)(r + 1)/2 - 1, and that the class is not closed under duality or ∆-Y exchange.</strong></p><p>.</p>