# application: matroid

Matroids encode the concept of "(in)dependence" in an abstract way. You can define matroids via vector configurations or graphs, do basic conversions between different descriptions and perform basic operations such as deletions and contractions.

**imports from:**common, graph

**uses:**group, polytope, topaz

**Objects**

A matroid on the set

*{0,...,n-1}*. Here*n*is the same as N_ELEMENTS.#### Properties of Matroid

**BASES**: common::Array<Set<Int>>Subsets of the ground set which form the bases of the matroid. Note that if you want to define a matroid via its bases, you should also specify N_ELEMENTS, because we allow matroids with loops.

**COCIRCUITS**: common::Array<Set<Int>>Cocircuits (or bonds) of the matroid, i.e., minimal sets intersecting every basis.

**LABELS**: common::Array<String>Unique names assigned to the elements of the matroid.

For a matroid build from scratch, you should create this property by yourself. If you build the matroid with a construction client, (e.g. matroid_from_graph) the labels may be assigend for you in a meaningful way.

**NON_BASES**: common::Array<Set<Int>>All subsets of the ground sets with cardinality RANK that are not bases.

**N_ELEMENTS**: common::IntSize of the ground set. The ground set itself always consists of the first integers starting with zero.

**POINTS**: common::Matrix<Rational, NonSymmetric>If the matroid is realizable over the rationals, this property contains (homogeneous) coordinates for some realization. Specifying coordinates is one way to define a matroid.

**POLYTOPE**: polytope::Polytope<Rational>Polytope whose vertices are the characteristic vectors of the bases.

## User Functions

**matroid_test**(bases)Tests whether the given

*bases*do actually form the bases of a matroid.##### Parameters

Array<Set<Int>> bases ##### Options

Bool print if set to true the output tells which condition fails; default value is 0

**contraction**(m, element) → Matroid

**matroid_from_graph**(g) → Matroid**matroid_from_matroid_polytope**(p) → MatroidCreates a matroid from the corresponding matroid polytope

*p*.**uniform_matroid**(r, n) → Matroid