706:(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.
624:. 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
135:
107:
466:
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909:
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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
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564:. A binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of
28:
110:
43:
116:
88:
713:, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.
445:
1104:
926:
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501:
709:
Any algorithm that tests whether a given matroid is binary, given access to the matroid via an
260:
144:
For every pair of circuits of the matroid, their symmetric difference contains another circuit.
737:
687:
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702:(not-necessarily-connected graphs in which all vertices have even degree), respectively. For
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The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968)
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42:
and whose sets of elements are independent if and only if the corresponding columns are
38:. That is, up to isomorphism, they are the matroids whose elements are the columns of a
761:, North-Holland Math. Stud., vol. 75, Amsterdam: North-Holland, pp. 371–376,
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39:
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998:, Section 10.2, "An excluded minor criterion for a matroid to be binary", pp. 167–169.
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877:, Lecture Notes in Mathematics, vol. 110, Berlin: Springer, pp. 155–170,
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associated to the matroid, every interval of height two has at most five elements.
904:
556:, is binary. A binary matroid is regular if and only if it does not contain the
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814:
20:
922:
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to be a matroid in which every circuit has even cardinality, and an
757:
Jaeger, F. (1983), "Symmetric representations of binary matroids",
422:
is the symmetric difference of the fundamental circuits induced in
35:
560:(a seven-element non-regular binary matroid) or its dual as a
680:. Additionally, the direct sum of binary matroids is binary.
122:
94:
783:(1935), "On the abstract properties of linear dependence",
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is a binary matroid, then so is its dual, and so is every
958:
Journal of
Research of the National Bureau of Standards
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907:(1958), "A homotopy theorem for matroids. I, II",
791:(3), The Johns Hopkins University Press: 509–533,
759:Combinatorial mathematics (Marseille-Luminy, 1981)
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1077:(1981), "Recognizing graphic matroids",
1035:(1969), "Euler and bipartite matroids",
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16:Abstraction of mod-2 vector independence
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873:(1969), "Matroids versus graphs",
14:
78:It is the matroid defined from a
732:(2010) , "10. Binary Matroids",
109:of circuits of the matroid, the
1038:Journal of Combinatorial Theory
785:American Journal of Mathematics
279:
265:
130:{\displaystyle {\mathcal {S}}}
102:{\displaystyle {\mathcal {S}}}
1:
1052:10.1016/s0021-9800(69)80033-5
50:Alternative characterizations
461:{\displaystyle C\setminus B}
525:{\displaystyle U{}_{4}^{2}}
1144:
1022:, Theorem 10.5.1, p. 176.
1010:, Theorem 10.4.1, p. 175.
845:, Theorem 10.1.3, p. 162.
684:Harary & Welsh (1969)
286:{\displaystyle |C\cap D|}
74:is binary if and only if
27:is a matroid that can be
137:can be represented as a
617:{\displaystyle K_{3,3}}
954:"Lectures on matroids"
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532:, the four-point line.
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952:Tutte, W. T. (1965),
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632:Additional properties
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584:{\displaystyle K_{5}}
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111:symmetric difference
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44:linearly independent
711:independence oracle
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442:by the elements of
315:{\displaystyle B,C}
166:{\displaystyle C,D}
113:of the circuits in
1093:10.1007/BF02579179
883:10.1007/BFb0060114
806:10338.dmlcz/100694
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293:is an even number.
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688:bipartite matroid
673:{\displaystyle M}
649:{\displaystyle M}
537:geometric lattice
488:{\displaystyle M}
435:{\displaystyle B}
415:{\displaystyle C}
395:{\displaystyle M}
375:{\displaystyle C}
355:{\displaystyle M}
335:{\displaystyle B}
250:{\displaystyle M}
226:{\displaystyle D}
206:{\displaystyle M}
186:{\displaystyle C}
67:{\displaystyle M}
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696:bipartite graphs
692:Eulerian matroid
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544:Related matroids
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1033:Welsh, D. J. A.
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797:10.2307/2371182
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730:Welsh, D. J. A.
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700:Eulerian graphs
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554:graphic matroid
550:regular matroid
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296:For every pair
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147:For every pair
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1128:Matroid theory
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1075:Seymour, P. D.
1066:
1045:(4): 375–377,
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985:
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917:(1): 144–174,
896:
871:Welsh, Dominic
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734:Matroid Theory
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342:is a basis of
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139:disjoint union
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85:For every set
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25:binary matroid
21:matroid theory
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3:
2:
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1080:Combinatorica
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867:Harary, Frank
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743:9780486474397
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704:planar graphs
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473:matroid minor
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84:
82:(0,1)-matrix.
81:
77:
76:
75:
61:
49:
47:
45:
41:
37:
34:
30:
26:
22:
1087:(1): 75–78,
1084:
1078:
1069:
1042:
1036:
1027:
1020:Welsh (2010)
1015:
1008:Welsh (2010)
1003:
996:Welsh (2010)
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957:
947:
914:
908:
905:Tutte, W. T.
899:
874:
843:Welsh (2010)
788:
784:
775:
758:
752:
733:
724:
708:
682:
635:
626:cactus graph
552:, and every
547:
235:dual matroid
141:of circuits.
53:
40:(0,1)-matrix
33:finite field
24:
18:
29:represented
717:References
558:Fano plane
54:A matroid
46:in GF(2).
686:define a
453:∖
273:∩
80:symmetric
31:over the
1122:Category
1109:35579707
964:: 1–47,
1101:0602418
1061:0237368
980:0179781
939:0101526
931:1993244
891:0263666
823:1507091
815:2371182
767:0841317
591:nor of
535:In the
495:is the
1107:
1099:
1059:
978:
937:
929:
889:
821:
813:
765:
740:
548:Every
322:where
173:where
1105:S2CID
927:JSTOR
811:JSTOR
658:minor
562:minor
36:GF(2)
738:ISBN
698:and
362:and
213:and
23:, a
1089:doi
1047:doi
966:doi
962:69B
919:doi
879:doi
801:hdl
793:doi
660:of
636:If
475:of
471:No
237:of
19:In
1124::
1103:,
1097:MR
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1057:MR
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668:M
644:M
610:3
607:,
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600:K
577:5
573:K
518:2
513:4
506:U
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468:.
456:B
450:C
430:B
410:C
390:M
370:C
350:M
330:B
310:C
307:,
304:B
280:|
276:D
270:C
266:|
245:M
221:D
201:M
181:C
161:D
158:,
155:C
123:S
95:S
62:M
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