No binary matroid has a minor isomorphic to U 4 2, the "four-point line", and Tutte showed that, conversely, every non-binary matroid has a U 4 2 minor. However, more can be said about the element sets of U 4 2 minors and their distribution. Bixby characterized those elements which are in U 4 2 minors; a matroid M has a U 4 2 minor using element x if and only if the connected component of M containing x is non-binary. We give a similar (but more complicated) characterization for pairs of elements. In particular, we prove that for every two elements of a 3-connected non-binary matroid, there is a U 4 2 minor using them both.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics
- AMS subject classification: (1980): 05B35