We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor.
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Academic Press
Journal of Combinatorial Theory - Series B
Matroid Structure for Efficiency
Networks and Optimization

van Zwam, S., & Geelen, J. (2015). Matroid 3-connectivity and branch width. Journal of Combinatorial Theory - Series B, 112(0), 104–123. doi:10.1016/j.jctb.2014.12.003