every regular matroid, and every graphic matroid, is binary. a binary matroid is regular if and only if it does not contain the fano plane (a seven-element non-regular binary matroid) or its dual as a minor. a binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of formula_20 nor of formula_21. if every circuit of a binary matroid has odd cardinality, then its circuits must all be disjoint from each other; in this case, it may be represented as the graphic matroid of a cactus graph.