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| In [[Matroid|matroid theory]], a '''binary matroid''' is a matroid that can be [[Matroid representation|represented]] over the [[finite field]] [[GF(2)]].<ref name="w76">{{citation
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| | last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
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| | contribution = 10. Binary Matroids
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| | isbn = 9780486474397
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| | pages = 161–182
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| | publisher = Courier Dover Publications
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| | title = Matroid Theory
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| | year = 2010 | origyear=1976}}.</ref> That is, up to isomorphism, they are the matroids whose elements are the columns of a [[Logical matrix|(0,1)-matrix]] and whose sets of elements are independent if and only if the corresponding columns are [[linearly independent]] in GF(2).
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| ==Alternative characterizations==
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| A matroid <math>M</math> is binary if and only if
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| *It is the matroid defined from a [[symmetric matrix|symmetric]] (0,1)-matrix.<ref>{{citation
| |
| | last = Jaeger | first = F.
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| | contribution = Symmetric representations of binary matroids
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| | location = Amsterdam
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| | mr = 841317
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| | pages = 371–376
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| | publisher = North-Holland
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| | series = North-Holland Math. Stud.
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| | title = Combinatorial mathematics (Marseille-Luminy, 1981)
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| | volume = 75
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| | year = 1983}}.</ref>
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| *For every set <math>\mathcal{S}</math> of circuits of the matroid, the [[symmetric difference]] of the circuits in <math>\mathcal{S}</math> can be represented as a [[disjoint union]] of circuits.<ref>{{citation|last=Whitney|first=Hassler|authorlink=Hassler Whitney|year=1935|title=On the abstract properties of linear dependence|journal=American Journal of Mathematics|volume=57|pages=509–533|doi=10.2307/2371182|issue=3|publisher=The Johns Hopkins University Press|mr=1507091|jstor=2371182}}. Reprinted in {{harvtxt|Kung|1986}}, pp. 55–79.</ref><ref name="w-thm3">{{harvtxt|Welsh|2010}}, Theorem 10.1.3, p. 162.</ref>
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| *For every pair of circuits of the matroid, their symmetric difference contains another circuit.<ref name="w-thm3"/>
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| *For every pair <math>C,D</math> where <math>C</math> is a circuit of <math>M</math> and <math>D</math> is a circuit of the [[dual matroid]] of <math>M</math>, <math>|C\cap D|</math> is an even number.<ref name="w-thm3"/><ref name="vs">{{citation
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| | last1 = Harary | first1 = Frank | author1-link = Frank Harary
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| | last2 = Welsh | first2 = Dominic | author2-link = Dominic Welsh
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| | contribution = Matroids versus graphs
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| | doi = 10.1007/BFb0060114
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| | location = Berlin
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| | mr = 0263666
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| | pages = 155–170
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| | publisher = Springer
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| | series = Lecture Notes in Mathematics
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| | title = The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968)
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| | volume = 110
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| | year = 1969}}.</ref>
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| *For every pair <math>B,C</math> where <math>B</math> is a basis of <math>M</math> and <math>C</math> is a circuit of <math>M</math>, <math>C</math> is the symmetric difference of the fundamental circuits induced in <math>B</math> by the elements of <math>C\setminus B</math>.<ref name="w-thm3"/><ref name="vs"/>
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| *No [[matroid minor]] of <math>M</math> is the [[uniform matroid]] <math>U{}^2_4</math>, the four-point line.<ref>{{citation
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| | last = Tutte | first = W. T. | authorlink = W. T. Tutte
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| | journal = [[Transactions of the American Mathematical Society]]
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| | mr = 0101526
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| | pages = 144–174
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| | title = A homotopy theorem for matroids. I, II
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| | volume = 88
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| | year = 1958}}.</ref><ref name="tutte">{{citation
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| | last = Tutte | first = W. T.
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| | journal = Journal of Research of the National Bureau of Standards | |
| | mr = 0179781
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| | pages = 1–47
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| | title = Lectures on matroids
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| | url = http://cdm16009.contentdm.oclc.org/cdm/ref/collection/p13011coll6/id/66650
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| | volume = 69B
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| | year = 1965}}.</ref><ref name="w-10-2">{{harvtxt|Welsh|2010}}, Section 10.2, "An excluded minor criterion for a matroid to be binary", pp. 167–169.</ref>
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| *In the [[geometric lattice]] associated to the matroid, every interval of height two has at most five elements.<ref name="w-10-2"/>
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| ==Related matroids==
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| Every [[regular matroid]], and every [[graphic matroid]], is binary.<ref name="vs"/> 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 [[matroid minor|minor]].<ref>{{harvtxt|Welsh|2010}}, Theorem 10.4.1, p. 175.</ref> A binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of <math>K_5</math> nor of <math>K_{3,3}</math>.<ref>{{harvtxt|Welsh|2010}}, Theorem 10.5.1, p. 176.</ref> 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]].<ref name="vs"/>
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| | |
| ==Additional properties== | |
| If <math>M</math> is a binary matroid, then so is its dual, and so is every [[matroid minor|minor]] of <math>M</math>.<ref name="vs"/> Additionally, the direct sum of binary matroids is binary.
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| {{harvtxt|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 graph]]s and [[Eulerian graph]]s (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.<ref name="vs"/><ref>{{citation
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| | last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
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| | journal = [[Journal of Combinatorial Theory]]
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| | mr = 0237368
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| | pages = 375–377
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| | title = Euler and bipartite matroids
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| | volume = 6
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| | year = 1969}}/</ref>
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| Any algorithm that tests whether a given matroid is binary, given access to the matroid via an [[matroid oracle|independence oracle]], must perform an exponential number of oracle queries, and therefore cannot take polynomial time.<ref>{{citation
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| | last = Seymour | first = P. D. | authorlink = Paul Seymour (mathematician)
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| | doi = 10.1007/BF02579179
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| | issue = 1
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| | journal = [[Combinatorica]]
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| | mr = 602418
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| | pages = 75–78
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| | title = Recognizing graphic matroids
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| | volume = 1
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| | year = 1981}}.</ref>
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| | |
| ==References==
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| {{reflist|colwidth=30em}}
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| [[Category:Matroid theory]]
| |
The hotel is a few miles from the Rogue Valley International Medford Airport as well as Southern Oregon University and the Rogue Valley Mall.
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