# Name of a binary matroid coming from the cycle space of a graph

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.

## 1 Answer

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.

• Can you give references for a few of the papers that use this structure?
– j.c.
Jan 23 '19 at 18:53