Theory of linear and integer programming
Theory of linear and integer programming
The incidence structure of subspaces with well-scaled frames
Journal of Combinatorial Theory Series B
A characterization of the matroids representable over GF(3) and the rationals
Journal of Combinatorial Theory Series B
Regular Article: Partial Fields and Matroid Representation
Advances in Applied Mathematics
Lifts of matroid representations over partial fields
Journal of Combinatorial Theory Series B
Confinement of matroid representations to subsets of partial fields
Journal of Combinatorial Theory Series B
The excluded minors for near-regular matroids
European Journal of Combinatorics
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Seymour's decomposition theorem [J. Combin. Theory Ser. B, 28 (1980), pp. 305-359] for regular matroids states that any matroid representable over both $\mathrm{GF}(2)$ and $\mathrm{GF}(3)$ can be obtained from matroids that are graphic, cographic, or isomorphic to $R_{10}$ 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.