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In some of my recent work, I have 'discovered' a binary matroid which I will describe below.

Given a graph $G$, let $H_1(G, \mathbb{Z}/2\mathbb{Z})$ denote the cycle space. This is a vector space over $\mathbb{Z}/2\mathbb{Z}$. Moreover, every cycle of $G$ corresponds to a vector in $H_2(G, \mathbb{Z}/2\mathbb{Z})$. Let $M(G)$ denote the matroid whose elements are the cycles of $G$. Here, the rank of a set of cycles is the rank of the corresponding set of vectors.

Equivalently, given a graph $G$ and a subset $S$ of cycles is independent if there is an edge of the graph which appears in an odd number of the cycles in $S$.

This matroid clearly has been studied before. However, I have been unable to answer the following question: 1) What is the name of this matroid?

Since it is related to the set of cycles of a graph, it is hard to find papers on it. The best I have found come from papers in complexity theory studying minimum weight cycle bases, or from homework exercises. However, I have not found a paper that decides to give this class of matroids a name.

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You can build such a matroid from the circuits of another binary, or representable in general, matroid as well. Some papers make use of such structure, either regarded as a chain group or simply as a subset of a vector space. But to the best of my knowledge I don't think there is a name for such matroids.

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    $\begingroup$ Can you give references for a few of the papers that use this structure? $\endgroup$ – j.c. Jan 23 at 18:53

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