Matroidal families are defined as families of connected graphs such that, given any graph G, the subgraphs of G isomorphic to a member of the family are the circuits of a matroid on the edge set of G. It will be proved that there are four matroidal families with all members having less than three independent cycles (Theorems 1 and 2) and that all members of any other matroidal family have at least three independent cycles (Theorem 3). The members of these four families are the complete graph on two points, the cycles, the connected graphs with two independent cycles and no pendant edges (which we call the bicircular graphs), and the members of a family formed by the even cycles and the bicircular graphs with no even cycle, respectively. Concerning any other matroidal families, we prove that no graph in such a family has a vertex of degree two (Theorem 5) and consequently that no two graphs in such a family are homeomorphic (Theorem 6).
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