A Splitter Theorem Relative to a Fixed Basis
نویسنده
چکیده
A standard matrix representation of a matroid M represents M relative to a fixed basis B, where contracting elements of B and deleting elements of E(M)−B correspond to removing rows and columns of the matrix, while retaining standard form. If M is 3-connected and N is a 3-connected minor of M , it is often desirable to perform such a removal while maintaining both 3-connectivity and the presence of an N -minor. We prove that, subject to a mild essential restriction, when M has no 4-element fans with a specific labelling relative to B, there are at least two distinct elements, s1 and s2, such that for each i ∈ {1, 2}, si(M/si) is 3-connected and has an N -minor when si ∈ B, and co(M\si) is 3-connected and has an N -minor when si ∈ E(M)−B. We also show that if M has precisely two such elements and P is the set of elements that, when removed in the appropriate way, retain the N -minor, then (P,E(M)− P ) is a sequential 3-separation.
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تاریخ انتشار 2012