An obstacle to a decomposition theorem for near-regular matroids

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Mayhew, Dillon
Whittle, Geoff
van Zwam, Stefan H. M.
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Abstract
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Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographic matroids. We show that in that case 3-sums will not suffice.
Comment: 11 pages, 1 figure
Keywords
Mathematics - Combinatorics, 05B35
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