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.
Maffioli et al. (Wed,) studied this question.