Bridging Separations in Matroids
Jim Geelen, Petr Hliněný, Geoff Whittle · SIAM Journal on Discrete Mathematics · 2004
Let (X 1 ,X 2 ) be an exact k-separation of a matroid N. If M is a matroid that contains N as a minor and the k-separation (X 1 ,X 2 ) does not extend to a k-separation in M, then we say that Mbridges the k-separation (X 1 ,X 2 ) in N. One would hope that a minor minimal bridge for (X 1 ,X 2 ) would not be much larger than N. Unfortunately there are instances in which one can construct arbitrarily large minor-minimal bridges. We restrict our attention to the class of matroids representable over a fixed finite field and show that here minor-minimal bridges are bounded in size.