On the excluded minors for matroids of branch-width three (Q698604)

Summary: Knowing the excluded minors for a minor-closed matroid property provides a useful alternative characterization of that property. It has been shown in [\textit{R. Hall, J. Oxley, C. Semple} and \textit{G. Whittle}, On matroids of branch-width three, J. Comb. Theory, Ser. B 86, 148-171 (2002)] 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 [\textit{P. Hliněný}, The MACEK program, \texttt{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.