A characterization of the base-matroids of a graphic matroid

Let M = (E,F) be a matroid on a set E, and B one of its bases. A closed set θ ⊆ E is saturated with respect to B when |θ ∩B| = r(θ), where r(θ) is the rank of θ. The collection of subsets I of E such that |I ∩ θ| ≤ r(θ) for every closed saturated set θ turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by MB . In this paper we prove that a graphic matroid M , isomorphic to a cycle matroid M(G), is isomorphic to MB , for every base B of M , if and only if M is direct sum of uniform graphic matroids or, in equivalent way, if and only if G is disjoint union of cacti. Moreover we characterize simple binary matroids M isomorphic to MB , with respect to an assigned base B.