Uniform matroid

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In mathematics, a uniform matroid is a matroid in which every permutation of the elements is a symmetry.

Definition

The uniform matroid is defined over a set of elements. A subset of the elements is independent if and only if it contains at most elements. A subset is a basis if it has exactly elements, and it is a circuit if it has exactly elements. The rank of a subset is and the rank of the matroid is .[1][2]

A matroid of rank is uniform if and only if all of its circuits have exactly elements.[3]

The matroid is called the -point line.

Duality and minors

The dual matroid of the uniform matroid is another uniform matroid . A uniform matroid is self-dual if and only if .[4]

Every minor of a uniform matroid is uniform. Restricting a uniform matroid by one element (as long as ) produces the matroid and contracting it by one element (as long as ) produces the matroid .[5]

Realization

The uniform matroid may be represented as the matroid of affinely independent subsets of points in general position in -dimensional Euclidean space, or as the matroid of linearly independent subsets of vectors in general position in an -dimensional real vector space.

Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields.[6] However, the field must be large enough to include enough independent vectors. For instance, the -point line can be realized only over finite fields of or more elements (because otherwise the projective line over that field would have fewer than points): is not a binary matroid, is not a ternary matroid, etc. For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields.[7]

Algorithms

The problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem. It may be solved in linear time.[8]

Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.[9]

Related matroids

Unless , a uniform matroid is connected: it is not the direct sum of two smaller matroids.[10] The direct sum of a family of uniform matroids (not necessarily all with the same parameters) is called a partition matroid.

Every uniform matroid is a paving matroid,[11] a transversal matroid[12] and a strict gammoid.[6]

Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphic matroid, . The uniform matroid is the graphic matroid of an -edge dipole graph, and the dual uniform matroid is the graphic matroid of its dual graph, the -edge cycle graph. is the graphic matroid of a graph with self-loops, and is the graphic matroid of an -edge forest. Other than these examples, every uniform matroid with contains as a minor and therefore is not graphic.[13]

The -point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.[14]

See also

References

  1. {{#invoke:citation/CS1|citation |CitationClass=citation }}. For the rank function, see p. 26.
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  3. Template:Harvtxt, p. 27.
  4. Template:Harvtxt, pp. 77 & 111.
  5. Template:Harvtxt, pp. 106–107 & 111.
  6. 6.0 6.1 Template:Harvtxt, p. 100.
  7. Template:Harvtxt, pp. 202–206.
  8. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  9. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  10. Template:Harvtxt, p. 126.
  11. Template:Harvtxt.
  12. Template:Harvtxt, pp. 48–49.
  13. Template:Harvtxt, p. 30.
  14. Template:Harvtxt, p. 297.