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. Hlinen

• Type: Journal
• Year: 2002
• Where: COMBINATORICS
• Authors: Petr Hlinený
• Keywords: COMBINATORICS 2002, Matroids, Minor-closed Matroid Property, Useful Alternative Characterization