Unavoidable connected matroids retaining a specified minor

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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.

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SIAM Journal on Discrete Mathematics

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