The closed sets (flats) of the bicircular matroid of a graph G can be described as the forests F of G such that in the induced subgraph of V(G) − V(F), every connected component has a cycle.
In the partial ordering for this lattice, that F1 ≤ F2 if For the most interesting example, let G o be G with a loop added to every vertex.
To contract a loop e at vertex v, delete e and v but not the other edges incident with v; rather, each edge incident with v and another vertex w becomes a loop at w. Any other graph loops at v become matroid loops—to describe this correctly in terms of the graph one needs half-edges and loose edges; see biased graph minors.
However, unlike graphic matroids, they are not regular: they cannot be represented by vectors over an arbitrary finite field.
The question of the fields over which a bicircular matroid has a vector representation leads to the largely unsolved problem of finding the fields over which a graph has multiplicative gains.