Dual matroid

In matroid theory, the dual of a matroid is another matroid that has the same elements as , and in which a set is independent if and only if has a basis set disjoint from it.[1][2][3]

Matroid duals go back to the original paper by Hassler Whitney defining matroids.[4] They generalize to matroids the notions of planar graph duality and of dual spaces in linear algebra.

Basic properties

Duality is an involution: for all , .[1][3][4]

An alternative definition of the dual matroid is that its basis sets are the complements of the basis sets of . The basis exchange axiom, used to define matroids from their bases, is self-complementary, so the dual of a matroid is necessarily a matroid.[3]

The flats of are complementary to the cycles (unions of circuits) of , and vice versa.[3]

If is the rank function of a matroid on ground set , then the rank function of the dual matroid is .[1][3][4]

Minors

A matroid minor is formed from a larger matroid by two operations: the restriction deletes element from without changing the independence or rank of the remaining sets, and the contraction deletes from after subtracting one from the rank of every set it belongs to. These two operations are dual: and . Thus, a minor of a dual is the same thing as a dual of a minor.[5]

Self-dual matroids

An individual matroid is self-dual (generalizing e.g. the 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 independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.[6]

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:

It is an open problem whether the family of algebraic matroids is self-dual.

References

  1. 1 2 3 Schrijver, Alexander (2003), Combinatorial Optimization: Polyhedra and Efficiency. Vol. B: Matroids, Trees, Stable Sets, Algorithms and Combinatorics, 24, Berlin: Springer-Verlag, p. 652, ISBN 3-540-44389-4, MR 1956925.
  2. Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 34, ISBN 9780486474397.
  3. 1 2 3 4 5 Oxley, James G. (2006), Matroid Theory, Oxford Graduate Texts in Mathematics, 3, Oxford University Press, pp. 69–70, ISBN 9780199202508.
  4. 1 2 3 Whitney, Hassler (1935), "On the abstract properties of linear dependence", American Journal of Mathematics, The Johns Hopkins University Press, 57 (3): 509–533, doi:10.2307/2371182, JSTOR 2371182, MR 1507091. Reprinted in Kung (1986), pp. 55–79. See in particular section 11, "Dual matroids", pp. 521–524.
  5. Schrijver (2003), p. 653.
  6. Jensen, Per M.; Korte, Bernhard (1982), "Complexity of matroid property algorithms", SIAM Journal on Computing, 11 (1): 184–190, doi:10.1137/0211014, MR 646772.
  7. Whitney (1935), Section 13, "Orthogonal hyperplanes and dual matroids".
  8. Schrijver (2003), pp. 659–661; Welsh (2010), pp. 222–223.
  9. Oxley (2006), pp. 77 & 111.
  10. Tutte, W. T. (1965), "Lectures on matroids", Journal of Research of the National Bureau of Standards, 69B: 1–47, doi:10.6028/jres.069b.001, MR 0179781.
  11. Welsh, D. J. A. (1969), "Euler and bipartite matroids", Journal of Combinatorial Theory, 6: 375–377, doi:10.1016/s0021-9800(69)80033-5, MR 0237368.
  12. Harary, Frank; Welsh, Dominic (1969), "Matroids versus graphs", The Many Facets of Graph Theory (Proc. Conf., Western Mich. Univ., Kalamazoo, Mich., 1968), Lecture Notes in Mathematics, 110, Berlin: Springer, pp. 155–170, doi:10.1007/BFb0060114, MR 0263666.
This article is issued from Wikipedia - version of the 11/8/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.