Bicircular matroid

In the mathematical subject of matroid theory, the bicircular matroid of a graph G is the matroid B(G) whose points are the edges of G and whose independent sets are the edge sets of pseudoforests of G, that is, the edge sets in which each connected component contains at most one cycle. The bicircular matroid was introduced by and explored further by and others. It is a special case of the frame matroid of a biased graph. The circuits, or minimal dependent sets, of this matroid are the bicircular graphs (or bicycles, but that term has other meanings in graph theory); these are connected graphs whose circuit rank is exactly two. There are three distinct types of bicircular graph: The theta graph consists of three paths joining the same two vertices but not intersecting each other. The figure eight graph (or tight handcuff) consists of two cycles having just one common vertex. The loose handcuff (or barbell) consists of two disjoint cycles and a minimal connecting path. All these definitions apply to multigraphs, i.e., they permit multiple edges (edges sharing the same endpoints) and loops (edges whose two endpoints are the same vertex). 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. Since the flats of a matroid form a geometric lattice when partially ordered by set inclusion, these forests of G also form a geometric lattice. In the partial ordering for this lattice, that F1 ≤ F2 if each component tree of F1 is either contained in or vertex-disjoint from every tree of F2, and each vertex of F2 is a vertex of F1. For the most interesting example, let Go be G with a loop added to every vertex. Then the flats of B(Go) are all the forests of G, spanning or nonspanning. Thus, all forests of a graph G form a geometric lattice, the forest lattice of G . Bicircular matroids can be characterized as the transversal matroids that arise from a family of sets in which each set element belongs to at most two sets.