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In [[matroid theory]], the '''dual''' of a matroid <math>M</math> is another matroid <math>M^\ast</math> that has the same elements as <math>M</math>, and in which a set is independent if and only if <math>M</math> has a basis set disjoint from it.<ref name="s03">{{citation
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  | last = Schrijver | first = Alexander | author-link = Alexander Schrijver
| isbn = 3-540-44389-4
| location = Berlin
| mr = 1956925
| pages = 652
| publisher = Springer-Verlag
| series = Algorithms and Combinatorics
| title = Combinatorial Optimization: Polyhedra and Efficiency. Vol. B: Matroids, Trees, Stable Sets
| url = http://books.google.com/books?id=mqGeSQ6dJycC&pg=RA1-PA652
| volume = 24
| year = 2003}}.</ref><ref>{{citation
| last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
| isbn = 9780486474397
| page = 34
| publisher = Courier Dover Publications
| title = Matroid Theory
| url = http://books.google.com/books?id=QL2iYMBLpFwC&pg=PA222
| year = 2010}}.</ref><ref name="oxley">{{citation
| last = Oxley | first = James G. | authorlink = James Oxley
| isbn = 9780199202508
| pages = 69–70
| publisher = Oxford University Press
| series = Oxford Graduate Texts in Mathematics
| title = Matroid Theory
| url = http://books.google.com/books?id=puKta1Hdz-8C&pg=PA69
| volume = 3
| year = 2006}}.</ref>
 
Matroid duals go back to the original paper by [[Hassler Whitney]] defining matroids.<ref name="w35">{{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.&nbsp;55–79. See in particular section 11, "Dual matroids", pp. 521–524.</ref> They generalize to matroids the notions of [[dual graph|planar graph duality]] and of [[dual space]]s in [[linear algebra]].
 
==Basic properties==
Duality is an [[Involution (mathematics)|involution]]: for all <math>M</math>, <math>(M^\ast)^\ast=M</math>.<ref name="s03"/><ref name="oxley"/><ref name="w35"/>
 
An alternative definition of the dual matroid is that its basis sets are the [[complement (set theory)|complements]] of the basis sets of <math>M</math>. The basis exchange axiom, used to define matroids from their bases, is self-complementary, so the dual of a matroid is necessarily a matroid.<ref name="oxley"/>
 
The [[Matroid_theory#Flats|flats]] of <math>M</math> are complementary to the circuits of <math>M^\ast</math>, and vice versa.<ref name="oxley"/>
 
If <math>r</math> is the [[matroid rank|rank function]] of a matroid <math>M</math> on ground set <math>E</math>, then the rank function of the dual matroid is <math>r^\ast(S)=r(E \setminus S)+|S|-r(E)</math>.<ref name="s03"/><ref name="oxley"/><ref name="w35"/>
 
==Minors==
A [[matroid minor]] is formed from a larger matroid <math>M</math> by two operations: the restriction <math>M\setminus x</math> deletes element <math>x</math> from <math>M</math> without changing the independence or rank of the remaining sets, and the contraction <math>M/x</math> deletes <math>x</math> from <math>M</math> after subtracting one from the rank of every set it belongs to. These two operations are dual: <math>M\setminus X=(M^\ast/x)^\ast</math> and <math>M/X=(M^\ast\setminus x)^\ast</math>. Thus, a minor of a dual is the same thing as a dual of a minor.<ref>{{harvtxt|Schrijver|2003}}, p. 653.</ref>
 
==Self-dual matroids==
An individual matroid is self-dual (generalizing e.g. the [[Dual polyhedron#Self-dual polyhedra|self-dual polyhedra]] for graphic matroids) if it is isomorphic to its own dual. The isomorphism may, but is not required to, leave the elements of the matroid fixed. Any algorithm that tests whether a given matroid is self-dual, 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
| last1 = Jensen | first1 = Per M.
| last2 = Korte | first2 = Bernhard
| doi = 10.1137/0211014
| issue = 1
| journal = [[SIAM Journal on Computing]]
| mr = 646772
| pages = 184–190
| title = Complexity of matroid property algorithms
| volume = 11
| year = 1982}}.</ref>
 
==Matroid families==
Many important matroid families are self-dual, meaning that a matroid belongs to the family if and only if its dual does. Many other matroid families come in dual pairs. Examples of this phenomenon include:
*The [[binary matroid]]s (matroids representable over [[GF(2)]]), the matroids representable over any other field, and the [[regular matroid]]s, are all self-dual families.<ref>{{harvtxt|Whitney|1935}}, Section 13, "Orthogonal hyperplanes and dual matroids".</ref>
*The [[gammoid]]s are self-dual. The strict gammoids are dual to the [[transversal matroid]]s.<ref>{{harvtxt|Schrijver|2003}}, pp. 659–661; {{harvtxt|Welsh|2010}}, pp. 222–223.</ref>
*The [[uniform matroid]]s and [[partition matroid]]s are self-dual. The dual to a uniform matroid <math>U{}^r_n</math> is the uniform matroid <math>U{}^{n-r}_n</math>.<ref>{{harvtxt|Oxley|2006}}, pp. 77 & 111.</ref>
*The dual of a [[graphic matroid]] is itself graphic if and only if the underlying graph is planar; the matroid of the dual of a planar graph is the same as the dual of the matroid of the graph. Thus, the graphic matroids of planar graphs are self-dual.<ref>{{citation
| last = Tutte | first = W. T.
| journal = Journal of Research of the National Bureau of Standards
| mr = 0179781
| pages = 1–47
| title = Lectures on matroids
| url = http://cdm16009.contentdm.oclc.org/cdm/ref/collection/p13011coll6/id/66650
| volume = 69B
| year = 1965}}.</ref>
*Among the graphic matroids, and more generally among the binary matroids, the [[bipartite matroid]]s (matroids in which every circuit is even) are dual to the [[Eulerian matroid]]s (matroids that can be partitioned into disjoint circuits).<ref>{{citation
| last = Welsh | first = D. J. A. | authorlink = Dominic Welsh
| journal = [[Journal of Combinatorial Theory]]
| mr = 0237368
| pages = 375–377
| title = Euler and bipartite matroids
| volume = 6
| year = 1969}}.</ref><ref>{{citation
| last1 = Harary | first1 = Frank | author1-link = Frank Harary
| last2 = Welsh | first2 = Dominic | author2-link = Dominic Welsh
| contribution = Matroids versus graphs
| doi = 10.1007/BFb0060114
| location = Berlin
| mr = 0263666
| pages = 155–170
| publisher = Springer
| series = Lecture Notes in Mathematics
| title = The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968)
| volume = 110
| year = 1969}}.</ref>
 
==References==
{{reflist}}
 
[[Category:Matroid theory]]
[[Category:Duality theories|Matroid]]

Latest revision as of 15:51, 29 November 2014

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