Binary matroid

In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2).

A matroid ${\displaystyle M}$ is binary if and only if

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 ${\displaystyle K_{5}}$ nor of ${\displaystyle K_{3,3}}$. 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.

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