# Matroids similar to the cycle matroid

Let $G=(V,E)$ be a graph (loops and multiple edges are permitted). Three following systems of dependent sets in $E$ define matroids:

1) Set $A\subset E$ is dependent if $A$ contains cycle. This is a cycle matroid, maybe the most popular matroid.

2) Set $A\subset E$ is dependent if $A$ contains two different cycles in the same connected component. In other words, independent set is characterized by the property that number of edges in any connected component does not exceed number of vertices.

3) Set $A\subset E$ is dependent if $A$ contains either even cycle or two odd cycles in the same connected component. In other words, rank of a connected subgraph $G_1=(V_1,E_1)$ equals $|V_1|-1$ if $G_1$ is bipartite and equals $V_1$ if $G_1$ has an odd cycle.

My question is whether these examples may be included in a more large picture, and may I read about those matroids.

• You should look up the work of Zaslavsky on frame matroids and biased graphs, as these seem at first glance to be what you are describing. The paper "sciencedirect.com/science/article/pii/0095895691900055" is a good starting point. Jul 31 '15 at 9:54

The first two examples fall neatly into the class of $(k,l)$-sparsity matroids (see the references in this Wikipedia article and references contained within). The cycle matroid (1) is the (1,1)-sparsity matroid on $G$ and the second matroid you describe is the (1,0)-sparsity matroid.
• Just to add, if $(k,l)$-sparsity matroids are the generalisation you are interested in, then the natural way to start is with Nash-Williams result: the $(k,k)$-tight count is equivalent to $k$ edge-disjoint spanning trees. So the $(k,k)$-sparse matroid is the union of $k$ cycle matroids. Then, for example, the $(2,3)$-sparse matroid is formed from the $(2,2)$-sparse matroid by a Dilworth truncation. Aug 5 '15 at 12:24