Bicircular Matroid - Minors

Minors

Unlike transversal matroids in general, bicircular matroids form a minor-closed class; that is, any submatroid or contraction of a bicircular matroid is also a bicircular matroid. (That can be seen from their description in terms of biased graphs (Zaslavsky 1991).) Here is a description of deletion and contraction of an edge in terms of the underlying graph: To delete an edge from the matroid, remove it from the graph. The rule for contraction depends on what kind of edge it is. To contract a link (a non-loop) in the matroid, contract it in the graph in the usual way. 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.