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We give "if and only if" characterization of graphs with the following property: given n ≥ 3, edges of such graphs form matroids with circuits from the collection of all graphs with n fundamental cycles. In this way we refer to the notion of matroidal family defined by Simões-Pereira .doi:10.7151/dmgt.1505 fatcat:6xzc2ntq6zcmzaibcdfzyokhrq