Binary matroid

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

• It is the matroid defined from a symmetric (0,1)-matrix.[2]
• For every set ${\displaystyle {\mathcal {S}}}$ of circuits of the matroid, the symmetric difference of the circuits in ${\displaystyle {\mathcal {S}}}$ can be represented as a disjoint union of circuits.[3][4]
• For every pair of circuits of the matroid, their symmetric difference contains another circuit.[4]
• For every pair ${\displaystyle C,D}$ where ${\displaystyle C}$ is a circuit of ${\displaystyle M}$ and ${\displaystyle D}$ is a circuit of the dual matroid of ${\displaystyle M}$, ${\displaystyle |C\cap D|}$ is an even number.[4][5]
• For every pair ${\displaystyle B,C}$ where ${\displaystyle B}$ is a basis of ${\displaystyle M}$ and ${\displaystyle C}$ is a circuit of ${\displaystyle M}$, ${\displaystyle C}$ is the symmetric difference of the fundamental circuits induced in ${\displaystyle B}$ by the elements of ${\displaystyle C\setminus B}$.[4][5]
• No matroid minor of ${\displaystyle M}$ is the uniform matroid ${\displaystyle U{}_{4}^{2}}$, the four-point line.[6][7][8]
• In the geometric lattice associated to the matroid, every interval of height two has at most five elements.[8]

Every regular matroid, and every graphic matroid, is binary.[5] 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.[9] 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 ${\displaystyle K_{3,3}}$.[10] 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.[5]

If ${\displaystyle M}$ is a binary matroid, then so is its dual, and so is every minor of ${\displaystyle M}$.[5] Additionally, the direct sum of binary matroids is binary.

Harary & Welsh (1969) define a bipartite matroid to be a matroid in which every circuit has even cardinality, and an Eulerian matroid to be a matroid in which the elements can be partitioned into disjoint circuits. Within the class of graphic matroids, these two properties describe the matroids of bipartite graphs and Eulerian graphs (not-necessarily-connected graphs in which all vertices have even degree), respectively. For planar graphs (and therefore also for the graphic matroids of planar graphs) these two properties are dual: a planar graph or its matroid is bipartite if and only if its dual is Eulerian. The same is true for binary matroids. However, there exist non-binary matroids for which this duality breaks down.[5][11]

Any algorithm that tests whether a given matroid is binary, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.[12]