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Questions related to the field of Combinatorics called Matroid Theory. Relevant topics include matroids in Combinatorial Optimization, Lattice Theory, Algebraic Geometry, Polyhedral Theory, Rigidity, and Algorithms. For questions about Oriented Matroids, the oriented-matroids tag may be used.
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Complexity of determining if two graphs have same cycle matroid?
The following paper seems to show that this problem is polynomial equivalent to graph isomorphism (see section 5):
http://arxiv.org/abs/0811.3859