On rounded five-element lines of matroids Original Research Article

We show that if X is a five-element line in a 3-connected matroid M which has a six-element line minor, then M has a minor using X which is isomorphic to a six-element line or one of two rank-3 matroids on ten elements. The two rank-3 matroids are constructed by freely adding an element to a four-element line of either the rank-3 Dowling geometry over GF(3) or the ternary Reid geometry. A corresponding result for four-element lines was provided by Oxley, Vertigan, and Whittle.