Unavoidable connected matroids retaining a specified minor
A sufficiently large connected matroid M contains a big circuit or a big cocircuit. Wu showed that we can ensure that M has a big circuit or a big cocircuit containing any chosen element of M. In this paper, we prove that, for a fixed connected matroid N, if M is a sufficiently large connected matroid having N as a minor, then, up to duality, either M has a big connected minor in which N is a spanning restriction and the deletion of E(N) is a large connected uniform matroid, or M has, as a minor, the 2-sum of a big circuit and a connected single-element extension or coextension of N. In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.
Publication Source (Journal or Book title)
SIAM Journal on Discrete Mathematics
Chun, C., Ding, G., Mayhew, D., & Oxley, J. (2016). Unavoidable connected matroids retaining a specified minor. SIAM Journal on Discrete Mathematics, 30 (3), 1590-1606. https://doi.org/10.1137/14096089X