Strong Splitter Theorem |
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Authors: | S R Kingan Manoel Lemos |
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Institution: | 1. Department of Mathematics, Brooklyn College, City University of New York, Brooklyn, NY, 11210, USA 2. Departamento de Matematica, Universidade Federal de Pernambuco, Recife, Pernambuco, 50740-540, Brazil
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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. |
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