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Can anyone explain to me why a cycle basis hones the properties of a matroid? Especially points 2 & 3.
How can a subset of I also be a member of I? Isn't a cycle basis supposed to be consisted of cycles and cycles only? If the cycle basis consisted a triangle, wouldn't a subset of the triangle become an edge? And by point 2 am I supposed to say that the edge is also a member of I?
I would really appreciate it if anyone can use a square or pentagon graph to explain, cuz my mind is really messed up...

Thanks in advance.

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Are you referring to points 2 & 3 in the definition of the matroid in terms of independent sets? That's obviously not satisfied by the set of cycles in a graph. The independent sets in the "cycle matroid" of a graph $G$ are the forests in $G$, and the bases are the spanning trees. The cycles themselves are the circuits in that matroid, i.e., the minimal dependent sets. (If I've understood things correctly; I'm not an expert.) –  Hans Lundmark Mar 28 '11 at 14:56
youre confusing the various definitions i think... here is a survey paper –  yoyo Mar 28 '11 at 15:27

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