# Binary matroid

In matroid theory, a **binary matroid** is a matroid that can be represented over the finite field GF(2).^{[1]} 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).

## Alternative characterizations

A matroid is binary if and only if

- It is the matroid defined from a symmetric (0,1)-matrix.
^{[2]} - For every set of circuits of the matroid, the symmetric difference of the circuits in 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 where is a circuit of and is a circuit of the dual matroid of , is an even number.
^{[4]}^{[5]} - For every pair where is a basis of and is a circuit of , is the symmetric difference of the fundamental circuits induced in by the elements of .
^{[4]}^{[5]} - No matroid minor of is the uniform matroid , 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]}

## Related matroids

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 nor of .^{[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]}

## Additional properties

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

Template:Harvtxt 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]}

## References

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- ↑
^{4.0}^{4.1}^{4.2}^{4.3}Template:Harvtxt, Theorem 10.1.3, p. 162. - ↑
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- ↑
^{8.0}^{8.1}Template:Harvtxt, Section 10.2, "An excluded minor criterion for a matroid to be binary", pp. 167–169. - ↑ Template:Harvtxt, Theorem 10.4.1, p. 175.
- ↑ Template:Harvtxt, Theorem 10.5.1, p. 176.
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