Binary matroid explained

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

M is binary if and only if

• It is the matroid defined from a symmetric (0,1)-matrix.^[2]
• For every set

l{S} of circuits of the matroid, the symmetric difference of the circuits in l{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

C,D where C is a circuit of M and D is a circuit of the dual matroid of M , |C\cap D| is an even number.^{[4] [5]}

• For every pair

B,C where B is a basis of M and C is a circuit of M , C is the symmetric difference of the fundamental circuits induced in B by the elements of C\setminus B .^{[4] [5]}

• No matroid minor of

M is the uniform matroid U{} ^{2} _{4} , 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

K _{5} nor of 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]

Additional properties

If

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

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]

Notes and References

1. .
2. .
3. .
4. , Theorem 10.1.3, p. 162.
5. .
6. .
7. .
8. , Section 10.2, "An excluded minor criterion for a matroid to be binary", pp. 167–169.
9. , Theorem 10.4.1, p. 175.
10. , Theorem 10.5.1, p. 176.
11. /
12. .