Union of matroids

class sage.matroids.union_matroid.MatroidSum

Bases: Matroid

Matroid Sum.

The matroid sum of a list of matroids $(E_1,I_1),\dots ,(E_n,I_n)$ is a matroid $(E,I)$ where $E=\underset{i=1}{\overset{n}{⨆}}{E}_i$ and $I=\underset{i=1}{\overset{n}{⨆}}{I}_i$.

INPUT:

• matroids – iterator of matroids

OUTPUT:

A MatroidSum instance, it’s a matroid sum of all matroids in matroids.

groundset()

Return the groundset of the matroid.

The groundset is the set of elements that comprise the matroid.

OUTPUT: set

EXAMPLES:

sage: from sage.matroids.union_matroid import *
sage: M = MatroidSum([matroids.Uniform(2,4),matroids.Uniform(2,4)])
sage: sorted(M.groundset())
[(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3)]

class sage.matroids.union_matroid.MatroidUnion

Bases: Matroid

Matroid Union.

The matroid union of a set of matroids $\{(E_1,I_1),\dots ,(E_n,I_n)\}$ is a matroid $(E,I)$ where $E=\bigcup _{i=1}^n{E}_i$ and

$I=\{\bigcup _{i=1}^n{J}_i|J_i\in I_i\}$.

EXAMPLES:

sage: M1 = matroids.Uniform(3,3)
sage: M2 = Matroid(bases = [frozenset({3}), frozenset({4})])
sage: M = M1.union(M2); M
Matroid of rank 4 on 5 elements as matroid union of
Matroid of rank 3 on 3 elements with circuit-closures
{}
Matroid of rank 1 on 2 elements with 2 bases
sage: M.bases()
SetSystem of 2 sets over 5 elements
sage: list(M.circuits())
[frozenset({3, 4})]

INPUT:

• matroids – iterator

OUTPUT: a MatroidUnion instance; a matroid union of all matroids in matroids

groundset()

Return the groundset of the matroid.

The groundset is the set of elements that comprise the matroid.

OUTPUT: set

EXAMPLES:

sage: from sage.matroids.union_matroid import *
sage: M = MatroidUnion([matroids.Uniform(2,4),matroids.Uniform(5,8)])
sage: sorted(M.groundset())
[0, 1, 2, 3, 4, 5, 6, 7]

class sage.matroids.union_matroid.PartitionMatroid

Bases: Matroid

Partition Matroid.

Given a set of disjoint sets $S=\{S_1,\dots ,S_n\}$, the partition matroid on $S$ is $(E,I)$ where $E=\bigcup _{i=1}^n{S}_i$ and

$I=\{X||X\cap S_i|\le 1,X\subset E\}$.

INPUT:

• partition – iterator of disjoint sets

OUTPUT:

A PartitionMatroid instance, it’s partition matroid of the partition.

groundset()

Return the groundset of the matroid.

The groundset is the set of elements that comprise the matroid.

OUTPUT: set

EXAMPLES:

sage: from sage.matroids.union_matroid import *
sage: M = PartitionMatroid([[1,2,3],[4,5,6]])
sage: sorted(M.groundset())
[1, 2, 3, 4, 5, 6]