Strong Splitter Theorem

Abstract

The Splitter Theorem states that, if N is a 3-connected proper minor of a 3-connected matroid M such that, if N is a wheel or whirl then M has no larger wheel or whirl, respectively, then there is a sequence M 0, . . . , M n of 3-connected matroids with \({M_0 \cong N}\), M n = M and for \({i \in \{1, \ldots , n}\}\), M i is a single-element extension or coextension of M i−1. Observe that there is no condition on how many extensions may occur before a coextension must occur. We give a strengthening of the Splitter Theorem, as a result of which we can obtain, up to isomorphism, M starting with N and at each step doing a 3-connected single-element extension or coextension, such that at most two consecutive single-element extensions occur in the sequence (unless the rank of thematroids involved is r(M)). Moreover, if two consecutive single-element extensions by elements {e, f} are followed by a coextension by element g, then {e, f , g} form a triad in the resulting matroid.