Abstract
A new matroid decomposition with several attractive properties leads to a new theorem of alternatives for matroids. A strengthened version of this theorem for binary matroids says roughly that to any binary matroid at least one of the following statements must apply: (1) the matroid is decomposable, (2) several elements can be removed (in any order) without destroying 3-connectivity, (3) the matroid belongs to one of 2 well-specified classes or has 10 elements or less. The latter theorem is easily specialized to graphic matroids. These theorems seem particularly useful for the determination of minimal violation matroids, a subject discussed in part II.
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