On the Excluded Minors for Matroids of Branch-Width Three
Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [R. Hall, J. Oxley, C. Semple, G. Whittle, On Matroids of Branch-Width Three, submitted 2001] that if $M$ is an excluded minor for matroids of branch-width $3$, then $ M$ has at most $14$ elements. We show that there are exactly $10$ such binary matroids $M$ (7 of which are regular), proving a conjecture formulated by Dharmatilake in 1994. We also construct numbers of such ternary and quaternary matroids $ M$, and provide a simple practical algorithm for finding a width-$3$ branch-decomposition of a matroid. The arguments in our paper are computer-assisted — we use a program $MACEK$ [P. Hliněný, The MACEK Program, http://www.mcs.vuw.ac.nz/research/macek, 2002] for structural computations with represented matroids. Unfortunately, it seems to be infeasible to search through all matroids on at most $14$ elements.