# The abstract Matroid class¶

Matroids are combinatorial structures that capture the abstract properties of (linear/algebraic/…) dependence. See the Wikipedia article on matroids for theory and examples. In Sage, various types of matroids are supported: BasisMatroid, CircuitClosuresMatroid, LinearMatroid (and some specialized subclasses), RankMatroid. To construct them, use the function Matroid().

All these classes share a common interface, which includes the following methods (organized by category). Note that most subclasses (notably LinearMatroids) will implement additional functionality (e.g. linear extensions).

In addition to these, all methods provided by SageObject are available, notably save() and rename().

Many methods (such as M.rank()) have a companion method whose name starts with an underscore (such as M._rank()). The method with the underscore does not do any checks on its input. For instance, it may assume of its input that

• It is a subset of the groundset. The interface is compatible with Python’s frozenset type.
• It is a list of things, supports iteration, and recursively these rules apply to its members.

Using the underscored version could improve the speed of code a little, but will generate more cryptic error messages when presented with wrong input. In some instances, no error might occur and a nonsensical answer returned.

A subclass should always override the underscored method, if available, and as a rule leave the regular method alone.

These underscored methods are not documented in the reference manual. To see them, within Sage you can create a matroid M and type M._<tab>. Then M._rank? followed by <tab> will bring up the documentation string of the _rank() method.

## Creating new Matroid subclasses¶

Many mathematical objects give rise to matroids, and not all are available through the provided code. For incidental use, the RankMatroid subclass may suffice. If you regularly use matroids based on a new data type, you can write a subclass of Matroid. You only need to override the __init__, _rank() and groundset() methods to get a fully working class.

EXAMPLES:

In a partition matroid, a subset is independent if it has at most one element from each partition. The following is a very basic implementation, in which the partition is specified as a list of lists:

sage: import sage.matroids.matroid
sage: class PartitionMatroid(sage.matroids.matroid.Matroid):
....:     def __init__(self, partition):
....:         self.partition = partition
....:         E = set()
....:         for P in partition:
....:             E.update(P)
....:         self.E = frozenset(E)
....:     def groundset(self):
....:         return self.E
....:     def _rank(self, X):
....:         X2 = set(X)
....:         used_indices = set()
....:         rk = 0
....:         while len(X2) > 0:
....:             e = X2.pop()
....:             for i in range(len(self.partition)):
....:                 if e in self.partition[i]:
....:                     if i not in used_indices:
....:                         rk = rk + 1
....:                     break
....:         return rk
....:
sage: M = PartitionMatroid([[1, 2], [3, 4, 5], [6, 7]])
sage: M.full_rank()
3
sage: M.tutte_polynomial(var('x'), var('y'))
x^2*y^2 + 2*x*y^3 + y^4 + x^3 + 3*x^2*y + 3*x*y^2 + y^3


Note

The abstract base class has no idea about the data used to represent the matroid. Hence some methods need to be customized to function properly.

Necessary:

• def __init__(self, ...)
• def groundset(self)
• def _rank(self, X)

Representation:

• def _repr_(self)

Comparison:

• def __hash__(self)
• def __eq__(self, other)
• def __ne__(self, other)

In Cythonized classes, use __richcmp__() instead of __eq__(), __ne__().

• def __copy__(self)
• def __deepcopy__(self, memo={})
• def __reduce__(self)

See, for instance, rank_matroid or circuit_closures_matroid for sample implementations of these.

Note

The example provided does not check its input at all. You may want to make sure the input data are not corrupt.

## Some examples¶

EXAMPLES:

Construction:

sage: M = Matroid(Matrix(QQ, [[1, 0, 0, 0, 1, 1, 1],
....:                         [0, 1, 0, 1, 0, 1, 1],
....:                         [0, 0, 1, 1, 1, 0, 1]]))
sage: sorted(M.groundset())
[0, 1, 2, 3, 4, 5, 6]
sage: M.rank([0, 1, 2])
3
sage: M.rank([0, 1, 5])
2


Minors:

sage: M = Matroid(Matrix(QQ, [[1, 0, 0, 0, 1, 1, 1],
....:                         [0, 1, 0, 1, 0, 1, 1],
....:                         [0, 0, 1, 1, 1, 0, 1]]))
sage: N = M / [2] \ [3, 4]
sage: sorted(N.groundset())
[0, 1, 5, 6]
sage: N.full_rank()
2


Testing. Note that the abstract base class does not support pickling:

sage: M = sage.matroids.matroid.Matroid()
sage: TestSuite(M).run(skip="_test_pickling")


## REFERENCES¶

AUTHORS:

• Michael Welsh (2013-04-03): Changed flats() to use SetSystem
• Michael Welsh (2013-04-01): Added is_3connected(), using naive algorithm
• Rudi Pendavingh, Stefan van Zwam (2013-04-01): initial version

## Methods¶

class sage.matroids.matroid.Matroid

The abstract matroid class, from which all matroids are derived. Do not use this class directly!

To implement a subclass, the least you should do is implement the __init__(), _rank() and groundset() methods. See the source of rank_matroid.py for a bare-bones example of this.

EXAMPLES:

In a partition matroid, a subset is independent if it has at most one element from each partition. The following is a very basic implementation, in which the partition is specified as a list of lists:

sage: class PartitionMatroid(sage.matroids.matroid.Matroid):
....:     def __init__(self, partition):
....:         self.partition = partition
....:         E = set()
....:         for P in partition:
....:             E.update(P)
....:         self.E = frozenset(E)
....:     def groundset(self):
....:         return self.E
....:     def _rank(self, X):
....:         X2 = set(X)
....:         used_indices = set()
....:         rk = 0
....:         while len(X2) > 0:
....:             e = X2.pop()
....:             for i in range(len(self.partition)):
....:                 if e in self.partition[i]:
....:                     if i not in used_indices:
....:                         rk = rk + 1
....:                     break
....:         return rk
....:
sage: M = PartitionMatroid([[1, 2], [3, 4, 5], [6, 7]])
sage: M.full_rank()
3
sage: M.tutte_polynomial(var('x'), var('y'))
x^2*y^2 + 2*x*y^3 + y^4 + x^3 + 3*x^2*y + 3*x*y^2 + y^3


Note

The abstract base class has no idea about the data used to represent the matroid. Hence some methods need to be customized to function properly.

Necessary:

• def __init__(self, ...)
• def groundset(self)
• def _rank(self, X)

Representation:

• def _repr_(self)

Comparison:

• def __hash__(self)
• def __eq__(self, other)
• def __ne__(self, other)

In Cythonized classes, use __richcmp__() instead of __eq__(), __ne__().

• def __copy__(self)
• def __deepcopy__(self, memo={})
• def __reduce__(self)

See, for instance, rank_matroid.py or circuit_closures_matroid.pyx for sample implementations of these.

Note

Many methods (such as M.rank()) have a companion method whose name starts with an underscore (such as M._rank()). The method with the underscore does not do any checks on its input. For instance, it may assume of its input that

• Any input that should be a subset of the groundset, is one. The interface is compatible with Python’s frozenset type.
• Any input that should be a list of things, supports iteration, and recursively these rules apply to its members.

Using the underscored version could improve the speed of code a little, but will generate more cryptic error messages when presented with wrong input. In some instances, no error might occur and a nonsensical answer returned.

A subclass should always override the underscored method, if available, and as a rule leave the regular method alone.

augment(X, Y=None)

Return a maximal subset $$I$$ of $$Y - X$$ such that $$r(X + I) = r(X) + r(I)$$.

INPUT:

• X – a subset (or any iterable) of the groundset
• Y – (default: the groundset) a subset (or any iterable) of the groundset

OUTPUT:

A subset of $$Y - X$$.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: X = set(['a']); Y = M.groundset()
sage: Z = M.augment(X, Y)
sage: M.is_independent(Z.union(X))
True
sage: W = Z.union(X)
sage: all(M.is_dependent(W.union([y])) for y in Y if y not in W)
True
sage: sorted(M.augment(['x']))
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset
sage: sorted(M.augment(['a'], ['x']))
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset

bases()

Return the list of bases of the matroid.

A basis is a maximal independent set.

OUTPUT:

An iterable containing all bases of the matroid.

EXAMPLES:

sage: M = matroids.Uniform(2, 4)
sage: sorted([sorted(X) for X in M.bases()])
[[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]]


ALGORITHM:

Test all subsets of the groundset of cardinality self.full_rank()

basis()

Return an arbitrary basis of the matroid.

A basis is an inclusionwise maximal independent set.

Note

The output of this method can change in between calls.

OUTPUT:

Set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: B = M.basis()
sage: M.is_basis(B)
True
sage: len(B)
3
sage: M.rank(B)
3
sage: M.full_rank()
3

binary_matroid(randomized_tests=1, verify=True)

Return a binary matroid representing self, if such a representation exists.

INPUT:

• randomized_tests – (default: 1) an integer; the number of times a certain necessary condition for being binary is tested, using randomization
• verify – (default: True), a Boolean; if True, any output will be a binary matroid representing self; if False, any output will represent self if and only if the matroid is binary

OUTPUT:

Either a BinaryMatroid, or None

ALGORITHM:

First, compare the binary matroids local to two random bases. If these matroids are not isomorphic, return None. This test is performed randomized_tests times. Next, if verify is True, test if a binary matroid local to some basis is isomorphic to self.

M.local_binary_matroid()

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.binary_matroid()
Fano: Binary matroid of rank 3 on 7 elements, type (3, 0)
sage: N = matroids.named_matroids.NonFano()
sage: N.binary_matroid() is None
True

broken_circuit_complex(ordering=None)

Return the broken circuit complex of self.

The broken circuit complex of a matroid with a total ordering $$<$$ on the ground set is obtained from the NBC sets under subset inclusion.

INPUT:

• ordering – a total ordering of the groundset given as a list

OUTPUT:

A simplicial complex of the NBC sets under inclusion.

EXAMPLES:

sage: M = Matroid(circuits=[[1,2,3], [3,4,5], [1,2,4,5]])
sage: M.broken_circuit_complex()
Simplicial complex with vertex set (1, 2, 3, 4, 5)
and facets {(1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5)}
sage: M.broken_circuit_complex([5,4,3,2,1])
Simplicial complex with vertex set (1, 2, 3, 4, 5)
and facets {(1, 3, 5), (1, 4, 5), (2, 3, 5), (2, 4, 5)}

broken_circuits(ordering=None)

Return the list of broken circuits of self.

Let $$M$$ be a matroid with ground set $$E$$, and let $$<$$ be a total ordering on $$E$$. A broken circuit for $$M$$ means a subset $$B$$ of $$E$$ such that there exists a $$u \in E$$ for which $$B \cup \{ u \}$$ is a circuit of $$M$$ and $$u < b$$ for all $$b \in B$$.

INPUT:

• ordering – a total ordering of the groundset given as a list

EXAMPLES:

sage: M = Matroid(circuits=[[1,2,3], [3,4,5], [1,2,4,5]])
sage: sorted([sorted(X) for X in M.broken_circuits()])
[[2, 3], [2, 4, 5], [4, 5]]
sage: sorted([sorted(X) for X in M.broken_circuits([5,4,3,2,1])])
[[1, 2], [1, 2, 4], [3, 4]]

sage: M = Matroid(circuits=[[1,2,3], [1,4,5], [2,3,4,5]])
sage: sorted([sorted(X) for X in M.broken_circuits([5,4,3,2,1])])
[[1, 2], [1, 4], [2, 3, 4]]

chordality()

Return the minimal $$k$$ such that the matroid M is $$k$$-chordal.

EXAMPLES:

sage: M = matroids.Uniform(2,4)
sage: M.chordality()
4
sage: M = matroids.named_matroids.NonFano()
sage: M.chordality()
5
sage: M = matroids.named_matroids.Fano()
sage: M.chordality()
4

chow_ring(R=None)

Return the Chow ring of self over R.

Let $$M$$ be a matroid and $$R$$ be a commutative ring. The Chow ring of $$M$$ is the quotient ring

$A^*(M)_R := R[x_{F_1}, \ldots, x_{F_k}] / (Q_M + L_M),$

where

• $$F_1, \ldots, F_k$$ are the non-empty proper flats of $$M$$,

• $$Q_M$$ is the ideal generated by all $$x_{F_i} x_{F_j}$$ where $$F_i$$ and $$F_j$$ are incomparable elements in the lattice of flats, and

• $$L_M$$ is the ideal generated by all linear forms

$\sum_{i_1 \in F} x_F - \sum_{i_2 \in F} x_F$

for all $$i_1 \neq i_2 \in E$$.

INPUT:

• R – (default: $$\ZZ$$) the base ring

EXAMPLES:

sage: M = matroids.Wheel(2)
sage: A = M.chow_ring()
sage: A
Chow ring of Wheel(2): Regular matroid of rank 2 on 4 elements
with 5 bases over Integer Ring
sage: A.gens()
(A23, A23, A23)
sage: A23 = A.gen(0)
sage: A23*A23
0


We construct a more interesting example using the Fano matroid:

sage: M = matroids.named_matroids.Fano()
sage: A = M.chow_ring(QQ)
sage: A
Chow ring of Fano: Binary matroid of rank 3 on 7 elements,
type (3, 0) over Rational Field


Next we get the non-trivial generators and do some computations:

sage: G = A.gens()[6:]
sage: Ag * Ag
sage: Ag * Abeg
sage: matrix([[x * y for x in G] for y in G])
[       0   Adef^2        0        0        0        0        0        0]
[       0        0   Adef^2        0        0        0        0        0]
[       0        0        0        0   Adef^2        0        0        0]
[       0        0        0        0        0        0        0   Adef^2]


REFERENCES:

circuit(X=None)

Return a circuit.

A circuit of a matroid is an inclusionwise minimal dependent subset.

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset

OUTPUT:

Set of elements.

• If X is not None, the output is a circuit contained in X if such a circuit exists. Otherwise an error is raised.
• If X is None, the output is a circuit contained in self.groundset() if such a circuit exists. Otherwise an error is raised.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.circuit(['a', 'c', 'd', 'e', 'f']))
['c', 'd', 'e', 'f']
sage: sorted(M.circuit(['a', 'c', 'd']))
Traceback (most recent call last):
...
ValueError: no circuit in independent set
sage: M.circuit(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset
sage: C = M.circuit()
sage: sorted(C) # py2
['a', 'b', 'c', 'd']
sage: sorted(C) # py3 random
['a', 'b', 'c', 'd']
sage: M.is_circuit(C)
True

circuit_closures()

Return the list of closures of circuits of the matroid.

A circuit closure is a closed set containing a circuit.

OUTPUT:

A dictionary containing the circuit closures of the matroid, indexed by their ranks.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: CC = M.circuit_closures()
sage: len(CC[2])
7
sage: len(CC[3])
1
sage: len(CC[1])
Traceback (most recent call last):
...
KeyError: 1
sage: [sorted(X) for X in CC[3]]
[['a', 'b', 'c', 'd', 'e', 'f', 'g']]

circuits()

Return the list of circuits of the matroid.

OUTPUT:

An iterable containing all circuits.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted([sorted(C) for C in M.circuits()])
[['a', 'b', 'c', 'g'], ['a', 'b', 'd', 'e'], ['a', 'b', 'f'],
['a', 'c', 'd', 'f'], ['a', 'c', 'e'], ['a', 'd', 'g'],
['a', 'e', 'f', 'g'], ['b', 'c', 'd'], ['b', 'c', 'e', 'f'],
['b', 'd', 'f', 'g'], ['b', 'e', 'g'], ['c', 'd', 'e', 'g'],
['c', 'f', 'g'], ['d', 'e', 'f']]

closure(X)

Return the closure of a set X.

A set is closed if adding any extra element to it will increase the rank of the set. The closure of a set is the smallest closed set containing it.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Set of elements containing X.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.closure(set(['a', 'b', 'c'])))
['a', 'b', 'c', 'd']
sage: M.closure(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset

cobasis()

Return an arbitrary cobasis of the matroid.

A cobasis is the complement of a basis. A cobasis is a basis of the dual matroid.

Note

Output can change between calls.

OUTPUT:

A set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: B = M.cobasis()
sage: M.is_cobasis(B)
True
sage: len(B)
6
sage: M.corank(B)
6
sage: M.full_corank()
6

cocircuit(X=None)

Return a cocircuit.

A cocircuit is an inclusionwise minimal subset that is dependent in the dual matroid.

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset

OUTPUT:

A set of elements.

• If X is not None, the output is a cocircuit contained in X if such a cocircuit exists. Otherwise an error is raised.
• If X is None, the output is a cocircuit contained in self.groundset() if such a cocircuit exists. Otherwise an error is raised.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.cocircuit(['a', 'c', 'd', 'e', 'f']))
['c', 'd', 'e', 'f']
sage: sorted(M.cocircuit(['a', 'c', 'd']))
Traceback (most recent call last):
...
ValueError: no cocircuit in coindependent set.
sage: M.cocircuit(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset
sage: C = M.cocircuit()
sage: sorted(C) # py2
['e', 'f', 'g', 'h']
sage: sorted(C) # py3 random
['e', 'f', 'g', 'h']
sage: M.is_cocircuit(C)
True

cocircuits()

Return the list of cocircuits of the matroid.

OUTPUT:

An iterable containing all cocircuits.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted([sorted(C) for C in M.cocircuits()])
[['a', 'b', 'c', 'g'], ['a', 'b', 'd', 'e'], ['a', 'c', 'd', 'f'],
['a', 'e', 'f', 'g'], ['b', 'c', 'e', 'f'], ['b', 'd', 'f', 'g'],
['c', 'd', 'e', 'g']]

coclosure(X)

Return the coclosure of a set X.

A set is coclosed if it is closed in the dual matroid. The coclosure of $$X$$ is the smallest coclosed set containing $$X$$.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

A set of elements containing X.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.coclosure(set(['a', 'b', 'c'])))
['a', 'b', 'c', 'd']
sage: M.coclosure(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset

coextension(element=None, subsets=None)

Return a coextension of the matroid.

A coextension of $$M$$ by an element $$e$$ is a matroid $$M'$$ such that $$M' / e = M$$. The element element is placed such that it lies in the coclosure of each set in subsets, and otherwise as freely as possible.

This is the dual method of M.extension(). See the documentation there for more details.

INPUT:

• element – (default: None) the label of the new element. If not specified, a new label will be generated automatically.
• subsets – (default: None) a set of subsets of the matroid. The coextension should be such that the new element is in the cospan of each of these. If not specified, the element is assumed to be in the cospan of the full groundset.

OUTPUT:

A matroid.

EXAMPLES:

Add an element in general position:

sage: M = matroids.Uniform(3, 6)
sage: N = M.coextension(6)
sage: N.is_isomorphic(matroids.Uniform(4, 7))
True


Add one inside the span of a specified hyperplane:

sage: M = matroids.Uniform(3, 6)
sage: H = [frozenset([0, 1])]
sage: N = M.coextension(6, H)
sage: N
Matroid of rank 4 on 7 elements with 34 bases
sage: [sorted(C) for C in N.cocircuits() if len(C) == 3]
[[0, 1, 6]]


Put an element in series with another:

sage: M = matroids.named_matroids.Fano()
sage: N = M.coextension('z', ['c'])
sage: N.corank('cz')
1

coextensions(element=None, coline_length=None, subsets=None)

Return an iterable set of single-element coextensions of the matroid.

A coextension of a matroid $$M$$ by element $$e$$ is a matroid $$M'$$ such that $$M' / e = M$$. By default, this method returns an iterable containing all coextensions, but it can be restricted in two ways. If coline_length is specified, the output is restricted to those matroids not containing a coline minor of length $$k$$ greater than coline_length. If subsets is specified, then the output is restricted to those matroids for which the new element lies in the coclosure of each member of subsets.

This method is dual to M.extensions().

INPUT:

• element – (optional) the name of the newly added element in each coextension.
• coline_length – (optional) a natural number. If given, restricts the output to coextensions that do not contain a $$U_{k - 2, k}$$ minor where k > coline_length.
• subsets – (optional) a collection of subsets of the ground set. If given, restricts the output to extensions where the new element is contained in all cohyperplanes that contain an element of subsets.

OUTPUT:

An iterable containing matroids.

Note

The coextension by a coloop will always occur. The extension by a loop will never occur.

EXAMPLES:

sage: M = matroids.named_matroids.P8()
sage: len(list(M.coextensions()))
1705
sage: len(list(M.coextensions(coline_length=4)))
41
sage: sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h']
sage: len(list(M.coextensions(subsets=[{'a', 'b'}], coline_length=4)))
5

coflats(r)

Return the collection of coflats of the matroid of specified corank.

A coflat is a coclosed set.

INPUT:

• r – a nonnegative integer.

OUTPUT:

An iterable containing all coflats of corank r.

EXAMPLES:

sage: M = matroids.named_matroids.Q6()
sage: sorted([sorted(F) for F in M.coflats(2)])
[['a', 'b'], ['a', 'c'], ['a', 'd', 'f'], ['a', 'e'], ['b', 'c'],
['b', 'd'], ['b', 'e'], ['b', 'f'], ['c', 'd'], ['c', 'e', 'f'],
['d', 'e']]

coloops()

Return the set of coloops of the matroid.

A coloop is an element $$u$$ of the groundset such that the one-element set $$\{ u \}$$ is a cocircuit. In other words, a coloop is a loop of the dual of the matroid.

OUTPUT:

A set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Fano().dual()
sage: M.coloops()
frozenset()
sage: (M \ ['a', 'b']).coloops()
frozenset({'f'})

components()

Return a list of the components of the matroid.

A component is an inclusionwise maximal connected subset of the matroid. A subset is connected if the matroid resulting from deleting the complement of that subset is connected.

OUTPUT:

A list of subsets.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = Matroid(ring=QQ, matrix=[[1, 0, 0, 1, 1, 0],
....:                              [0, 1, 0, 1, 2, 0],
....:                              [0, 0, 1, 0, 0, 1]])
sage: setprint(M.components())
[{0, 1, 3, 4}, {2, 5}]

connectivity(S, T=None)

Evaluate the connectivity function of the matroid.

If the input is a single subset $$S$$ of the groundset $$E$$, then the output is $$r(S) + r(E\S) - r(E)$$.

If the input are disjoint subsets $$S, T$$ of the groundset, then the output is

$\min \{ r(X) + r(Y) - r(E) \mid X \subseteq S, Y \subseteq T, {X,Y} \text{a partition of} E \}.$

INPUT:

• S – a subset (or any iterable) of the groundset
• T – (optional) a subset (or any iterable) of the groundset disjoint from S

OUTPUT:

An integer.

EXAMPLES:

sage: M = matroids.named_matroids.BetsyRoss()
sage: M.connectivity('ab')
2
sage: M.connectivity('ab', 'cd')
2

contract(X)

Contract elements.

If $$e$$ is a non-loop element, then the matroid $$M / e$$ is a matroid on groundset $$E(M) - e$$. A set $$X$$ is independent in $$M / e$$ if and only if $$X \cup e$$ is independent in $$M$$. If $$e$$ is a loop then contracting $$e$$ is the same as deleting $$e$$. We say that $$M / e$$ is the matroid obtained from $$M$$ by contracting $$e$$. Contracting an element in $$M$$ is the same as deleting an element in the dual of $$M$$.

When contracting a set, the elements of that set are contracted one by one. It can be shown that the resulting matroid does not depend on the order of the contractions.

Sage supports the shortcut notation M / X for M.contract(X).

INPUT:

• X – Either a single element of the groundset, or a collection of elements.

OUTPUT:

The matroid obtained by contracting the element(s) in X.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g']
sage: M.contract(['a', 'c'])
Binary matroid of rank 1 on 5 elements, type (1, 0)
sage: M.contract(['a']) == M / ['a']
True


One can use a single element, rather than a set:

sage: M = matroids.CompleteGraphic(4)
sage: M.contract(1) == M.contract([1])
True
sage: M / 1
Graphic matroid of rank 2 on 5 elements


Note that one can iterate over strings:

sage: M = matroids.named_matroids.Fano()
sage: M / 'abc'
Binary matroid of rank 0 on 4 elements, type (0, 0)


The following is therefore ambiguous. Sage will contract the single element:

sage: M = Matroid(groundset=['a', 'b', 'c', 'abc'],
....:             bases=[['a', 'b', 'c'], ['a', 'b', 'abc']])
sage: sorted((M / 'abc').groundset())
['a', 'b', 'c']

corank(X=None)

Return the corank of X, or the corank of the groundset if X is None.

The corank of a set $$X$$ is the rank of $$X$$ in the dual matroid.

If X is None, the corank of the groundset is returned.

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset

OUTPUT:

Integer.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.corank()
4
sage: M.corank('cdeg')
3
sage: M.rank(['a', 'b', 'x'])
Traceback (most recent call last):
...
ValueError: ['a', 'b', 'x'] is not a subset of the groundset

cosimplify()

Return the cosimplification of the matroid.

A matroid is cosimple if it contains no cocircuits of length 1 or 2. The cosimplification of a matroid is obtained by contracting all coloops (cocircuits of length 1) and contracting all but one element from each series class (a coclosed set of rank 1, that is, each pair in it forms a cocircuit of length 2).

OUTPUT:

A matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Fano().dual().delete('a')
sage: M.cosimplify().size()
3

delete(X)

Delete elements.

If $$e$$ is an element, then the matroid $$M \setminus e$$ is a matroid on groundset $$E(M) - e$$. A set $$X$$ is independent in $$M \setminus e$$ if and only if $$X$$ is independent in $$M$$. We say that $$M \setminus e$$ is the matroid obtained from $$M$$ by deleting $$e$$.

When deleting a set, the elements of that set are deleted one by one. It can be shown that the resulting matroid does not depend on the order of the deletions.

Sage supports the shortcut notation M \ X for M.delete(X).

INPUT:

• X – Either a single element of the groundset, or a collection of elements.

OUTPUT:

The matroid obtained by deleting the element(s) in X.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g']
sage: M.delete(['a', 'c'])
Binary matroid of rank 3 on 5 elements, type (1, 6)
sage: M.delete(['a']) == M \ ['a']
True


One can use a single element, rather than a set:

sage: M = matroids.CompleteGraphic(4)
sage: M.delete(1) == M.delete([1])
True
sage: M \ 1
Graphic matroid of rank 3 on 5 elements


Note that one can iterate over strings:

sage: M = matroids.named_matroids.Fano()
sage: M \ 'abc'
Binary matroid of rank 3 on 4 elements, type (0, 5)


The following is therefore ambiguous. Sage will delete the single element:

sage: M = Matroid(groundset=['a', 'b', 'c', 'abc'],
....:             bases=[['a', 'b', 'c'], ['a', 'b', 'abc']])
sage: sorted((M \ 'abc').groundset())
['a', 'b', 'c']

dependent_r_sets(r)

Return the list of dependent subsets of fixed size.

INPUT:

• r – a nonnegative integer.

OUTPUT:

An iterable containing all dependent subsets of size r.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.dependent_r_sets(3)
[]
sage: sorted([sorted(X) for X in
....: matroids.named_matroids.Vamos().dependent_r_sets(4)])
[['a', 'b', 'c', 'd'], ['a', 'b', 'e', 'f'], ['a', 'b', 'g', 'h'],
['c', 'd', 'e', 'f'], ['e', 'f', 'g', 'h']]


ALGORITHM:

Test all subsets of the groundset of cardinality r

dual()

Return the dual of the matroid.

Let $$M$$ be a matroid with ground set $$E$$. If $$B$$ is the set of bases of $$M$$, then the set $$\{E - b : b \in B\}$$ is the set of bases of another matroid, the dual of $$M$$.

Note

This function wraps self in a DualMatroid object. For more efficiency, subclasses that can, should override this method.

OUTPUT:

The dual matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: N = M.dual()
sage: N.rank()
6
sage: N
Dual of 'Pappus: Matroid of rank 3 on 9 elements with
circuit-closures
{2: {{'a', 'b', 'c'}, {'a', 'e', 'i'}, {'a', 'f', 'h'},
{'b', 'd', 'i'}, {'b', 'f', 'g'}, {'c', 'd', 'h'},
{'c', 'e', 'g'}, {'d', 'e', 'f'}, {'g', 'h', 'i'}},
3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'}}}'

equals(other)

Test for matroid equality.

Two matroids $$M$$ and $$N$$ are equal if they have the same groundset and a subset $$X$$ is independent in $$M$$ if and only if it is independent in $$N$$.

INPUT:

• other – A matroid.

OUTPUT:

Boolean.

Note

This method tests abstract matroid equality. The == operator takes a more restricted view: M == N returns True only if

1. the internal representations are of the same type,
2. those representations are equivalent (for an appropriate meaning of “equivalent” in that class), and
3. M.equals(N).

EXAMPLES:

A BinaryMatroid and BasisMatroid use different representations of the matroid internally, so == yields False, even if the matroids are equal:

sage: from sage.matroids.advanced import *
sage: M = matroids.named_matroids.Fano()
sage: M
Fano: Binary matroid of rank 3 on 7 elements, type (3, 0)
sage: M1 = BasisMatroid(M)
sage: M2 = Matroid(groundset='abcdefg', reduced_matrix=[
....:      [0, 1, 1, 1], [1, 0, 1, 1], [1, 1, 0, 1]], field=GF(2))
sage: M.equals(M1)
True
sage: M.equals(M2)
True
sage: M == M1
False
sage: M == M2
True


LinearMatroid instances M and N satisfy M == N if the representations are equivalent up to row operations and column scaling:

sage: M1 = LinearMatroid(groundset='abcd', matrix=Matrix(GF(7),
....:                               [[1, 0, 1, 1], [0, 1, 1, 2]]))
sage: M2 = LinearMatroid(groundset='abcd', matrix=Matrix(GF(7),
....:                               [[1, 0, 1, 1], [0, 1, 1, 3]]))
sage: M3 = LinearMatroid(groundset='abcd', matrix=Matrix(GF(7),
....:                               [[2, 6, 1, 0], [6, 1, 0, 1]]))
sage: M1.equals(M2)
True
sage: M1.equals(M3)
True
sage: M1 == M2
False
sage: M1 == M3
True

extension(element=None, subsets=None)

Return an extension of the matroid.

An extension of $$M$$ by an element $$e$$ is a matroid $$M'$$ such that $$M' \setminus e = M$$. The element element is placed such that it lies in the closure of each set in subsets, and otherwise as freely as possible. More precisely, the extension is defined by the modular cut generated by the sets in subsets.

INPUT:

• element – (default: None) the label of the new element. If not specified, a new label will be generated automatically.
• subsets – (default: None) a set of subsets of the matroid. The extension should be such that the new element is in the span of each of these. If not specified, the element is assumed to be in the span of the full groundset.

OUTPUT:

A matroid.

Note

Internally, sage uses the notion of a linear subclass for matroid extension. If subsets already consists of a linear subclass (i.e. the set of hyperplanes of a modular cut) then the faster method M._extension() can be used.

EXAMPLES:

First we add an element in general position:

sage: M = matroids.Uniform(3, 6)
sage: N = M.extension(6)
sage: N.is_isomorphic(matroids.Uniform(3, 7))
True


Next we add one inside the span of a specified hyperplane:

sage: M = matroids.Uniform(3, 6)
sage: H = [frozenset([0, 1])]
sage: N = M.extension(6, H)
sage: N
Matroid of rank 3 on 7 elements with 34 bases
sage: [sorted(C) for C in N.circuits() if len(C) == 3]
[[0, 1, 6]]


Putting an element in parallel with another:

sage: M = matroids.named_matroids.Fano()
sage: N = M.extension('z', ['c'])
sage: N.rank('cz')
1

extensions(element=None, line_length=None, subsets=None)

Return an iterable set of single-element extensions of the matroid.

An extension of a matroid $$M$$ by element $$e$$ is a matroid $$M'$$ such that $$M' \setminus e = M$$. By default, this method returns an iterable containing all extensions, but it can be restricted in two ways. If line_length is specified, the output is restricted to those matroids not containing a line minor of length $$k$$ greater than line_length. If subsets is specified, then the output is restricted to those matroids for which the new element lies in the closure of each member of subsets.

INPUT:

• element – (optional) the name of the newly added element in each extension.
• line_length – (optional) a natural number. If given, restricts the output to extensions that do not contain a $$U_{2, k}$$ minor where k > line_length.
• subsets – (optional) a collection of subsets of the ground set. If given, restricts the output to extensions where the new element is contained in all hyperplanes that contain an element of subsets.

OUTPUT:

An iterable containing matroids.

Note

The extension by a loop will always occur. The extension by a coloop will never occur.

EXAMPLES:

sage: M = matroids.named_matroids.P8()
sage: len(list(M.extensions()))
1705
sage: len(list(M.extensions(line_length=4)))
41
sage: sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h']
sage: len(list(M.extensions(subsets=[{'a', 'b'}], line_length=4)))
5

f_vector()

Return the $$f$$-vector of the matroid.

The $$f$$-vector is a vector $$(f_0, ..., f_r)$$, where $$f_i$$ is the number of flats of rank $$i$$, and $$r$$ is the rank of the matroid.

OUTPUT:

List of integers.

EXAMPLES:

sage: M = matroids.named_matroids.BetsyRoss()
sage: M.f_vector()
[1, 11, 20, 1]

flat_cover(solver=None, verbose=0)

Return a minimum-size cover of the nonbases by non-spanning flats.

A nonbasis is a subset that has the size of a basis, yet is dependent. A flat is a closed set.

INPUT:

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve() of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity of the LP solver. Set to 0 by default, which means quiet.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano()
sage: setprint(M.flat_cover())
[{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'},
{'b', 'c', 'd'}, {'b', 'e', 'g'}, {'c', 'f', 'g'},
{'d', 'e', 'f'}]

flats(r)

Return the collection of flats of the matroid of specified rank.

A flat is a closed set.

INPUT:

• r – A natural number.

OUTPUT:

An iterable containing all flats of rank r.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted([sorted(F) for F in M.flats(2)])
[['a', 'b', 'f'], ['a', 'c', 'e'], ['a', 'd', 'g'],
['b', 'c', 'd'], ['b', 'e', 'g'], ['c', 'f', 'g'],
['d', 'e', 'f']]

full_corank()

Return the corank of the matroid.

The corank of the matroid equals the rank of the dual matroid. It is given by M.size() - M.full_rank().

OUTPUT:

Integer.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.full_corank()
4

full_rank()

Return the rank of the matroid.

The rank of the matroid is the size of the largest independent subset of the groundset.

OUTPUT:

Integer.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.full_rank()
4
sage: M.dual().full_rank()
4

fundamental_circuit(B, e)

Return the $$B$$-fundamental circuit using $$e$$.

If $$B$$ is a basis, and $$e$$ an element not in $$B$$, then the $$B$$-fundamental circuit using $$e$$ is the unique matroid circuit contained in $$B\cup e$$.

INPUT:

• B – a basis of the matroid.
• e – an element not in B.

OUTPUT:

A set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.fundamental_circuit('defg', 'c'))
['c', 'd', 'e', 'f']

fundamental_cocircuit(B, e)

Return the $$B$$-fundamental cocircuit using $$e$$.

If $$B$$ is a basis, and $$e$$ an element of $$B$$, then the $$B$$-fundamental cocircuit using $$e$$ is the unique matroid cocircuit that intersects $$B$$ only in $$e$$.

This is equal to M.dual().fundamental_circuit(M.groundset().difference(B), e).

INPUT:

• B – a basis of the matroid.
• e – an element of B.

OUTPUT:

A set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: sorted(M.fundamental_cocircuit('abch', 'c'))
['c', 'd', 'e', 'f']

groundset()

Return the groundset of the matroid.

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

OUTPUT:

A set.

Note

Subclasses should implement this method. The return type should be frozenset or any type with compatible interface.

EXAMPLES:

sage: M = sage.matroids.matroid.Matroid()
sage: M.groundset()
Traceback (most recent call last):
...
NotImplementedError: subclasses need to implement this.

has_line_minor(k, hyperlines=None, certificate=False)

Test if the matroid has a $$U_{2, k}$$-minor.

The matroid $$U_{2, k}$$ is a matroid on $$k$$ elements in which every subset of at most 2 elements is independent, and every subset of more than two elements is dependent.

The optional argument hyperlines restricts the search space: this method returns True if $$si(M/F)$$ is isomorphic to $$U_{2, l}$$ with $$l \geq k$$ for some $$F$$ in hyperlines, and False otherwise.

INPUT:

• k – the length of the line minor
• hyperlines – (default: None) a set of flats of codimension 2. Defaults to the set of all flats of codimension 2.
• certificate – (default: False) if True returns (True, F), where F is a flat and self.minor(contractions=F) has a $$U_{2,k}$$ restriction or (False, None).

OUTPUT:

Boolean or tuple.

EXAMPLES:

sage: M = matroids.named_matroids.N1()
sage: M.has_line_minor(4)
True
sage: M.has_line_minor(5)
False
sage: M.has_line_minor(k=4, hyperlines=[['a', 'b', 'c']])
False
sage: M.has_line_minor(k=4, hyperlines=[['a', 'b', 'c'],
....:                                   ['a', 'b', 'd' ]])
True
sage: M.has_line_minor(4, certificate=True)
(True, frozenset({'a', 'b', 'd'}))
sage: M.has_line_minor(5, certificate=True)
(False, None)
sage: M.has_line_minor(k=4, hyperlines=[['a', 'b', 'c'],
....:                                   ['a', 'b', 'd' ]], certificate=True)
(True, frozenset({'a', 'b', 'd'}))

has_minor(N, certificate=False)

Check if self has a minor isomorphic to N, and optionally return frozensets X and Y so that N is isomorphic to self.minor(X, Y).

INPUT:

• N – An instance of a Matroid object,
• certificate – Boolean (Defalt: False) If True, returns True, (X, Y, dic) where N is isomorphic to self.minor(X, Y), and dic is an isomorphism between N and self.minor(X, Y).

OUTPUT:

boolean or tuple.

Todo

This important method can (and should) be optimized considerably. See [Hli2006] p.1219 for hints to that end.

EXAMPLES:

sage: M = matroids.Whirl(3)
sage: matroids.named_matroids.Fano().has_minor(M)
False
sage: matroids.named_matroids.NonFano().has_minor(M)
True
sage: matroids.named_matroids.NonFano().has_minor(M, certificate=True)
(True, (frozenset(), frozenset({'g'}),
{0: 'b', 1: 'c', 2: 'a', 3: 'd', 4: 'e', 5: 'f'}))
sage: M = matroids.named_matroids.Fano()
sage: M.has_minor(M, True)
(True,
(frozenset(),
frozenset(),
{'a': 'a', 'b': 'b', 'c': 'c', 'd': 'd', 'e': 'e', 'f': 'f', 'g': 'g'}))

hyperplanes()

Return the set of hyperplanes of the matroid.

A hyperplane is a flat of rank self.full_rank() - 1. A flat is a closed set.

OUTPUT:

An iterable containing all hyperplanes of the matroid.

EXAMPLES:

sage: M = matroids.Uniform(2, 3)
sage: sorted([sorted(F) for F in M.hyperplanes()])
[[0], [1], [2]]

independence_matroid_polytope()

Return the independence matroid polytope of self.

This is defined as the convex hull of the vertices

$\sum_{i \in I} e_i$

over all independent sets $$I$$ of the matroid. Here $$e_i$$ are the standard basis vectors of $$\RR^n$$. An arbitrary labelling of the groundset by $${0,\dots,n-1}$$ is chosen.

EXAMPLES:

sage: M = matroids.Whirl(4)
sage: M.independence_matroid_polytope()
A 8-dimensional polyhedron in ZZ^8 defined as the convex hull
of 135 vertices

sage: M = matroids.named_matroids.NonFano()
sage: M.independence_matroid_polytope()
A 7-dimensional polyhedron in ZZ^7 defined as the convex hull
of 58 vertices


REFERENCE:

[DLHK2007]

independent_r_sets(r)

Return the list of size-r independent subsets of the matroid.

INPUT:

• r – a nonnegative integer.

OUTPUT:

An iterable containing all independent subsets of the matroid of cardinality r.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: M.independent_r_sets(4)
[]
sage: S = M.independent_r_sets(3)
sage: len(S)
75
sage: frozenset({'a', 'c', 'e'}) in S
True


ALGORITHM:

Test all subsets of the groundset of cardinality r

independent_sets()

Return the list of independent subsets of the matroid.

OUTPUT:

An iterable containing all independent subsets of the matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: I = M.independent_sets()
sage: len(I)
121

intersection(other, weights=None)

Return a maximum-weight common independent set.

A common independent set of matroids $$M$$ and $$N$$ with the same groundset $$E$$ is a subset of $$E$$ that is independent both in $$M$$ and $$N$$. The weight of a subset S is sum(weights(e) for e in S).

INPUT:

• other – a second matroid with the same groundset as this matroid.
• weights – (default: None) a dictionary which specifies a weight for each element of the common groundset. Defaults to the all-1 weight function.

OUTPUT:

A subset of the groundset.

EXAMPLES:

sage: M = matroids.named_matroids.T12()
sage: N = matroids.named_matroids.ExtendedTernaryGolayCode()
sage: w = {'a':30, 'b':10, 'c':11, 'd':20, 'e':70, 'f':21, 'g':90,
....:      'h':12, 'i':80, 'j':13, 'k':40, 'l':21}
sage: Y = M.intersection(N, w)
sage: sorted(Y)
['a', 'd', 'e', 'g', 'i', 'k']
sage: sum([w[y] for y in Y])
330
sage: M = matroids.named_matroids.Fano()
sage: N = matroids.Uniform(4, 7)
sage: M.intersection(N)
Traceback (most recent call last):
...
ValueError: matroid intersection requires equal groundsets.

intersection_unweighted(other)

Return a maximum-cardinality common independent set.

A common independent set of matroids $$M$$ and $$N$$ with the same groundset $$E$$ is a subset of $$E$$ that is independent both in $$M$$ and $$N$$.

INPUT:

• other – a second matroid with the same groundset as this matroid.

OUTPUT:

A subset of the groundset.

EXAMPLES:

sage: M = matroids.named_matroids.T12()
sage: N = matroids.named_matroids.ExtendedTernaryGolayCode()
sage: len(M.intersection_unweighted(N))
6
sage: M = matroids.named_matroids.Fano()
sage: N = matroids.Uniform(4, 7)
sage: M.intersection_unweighted(N)
Traceback (most recent call last):
...
ValueError: matroid intersection requires equal groundsets.

is_3connected(certificate=False, algorithm=None, separation=False)

Return True if the matroid is 3-connected, False otherwise. It can optionally return a separator as a witness.

A $$k$$-separation in a matroid is a partition $$(X, Y)$$ of the groundset with $$|X| \geq k, |Y| \geq k$$ and $$r(X) + r(Y) - r(M) < k$$. A matroid is $$k$$-connected if it has no $$l$$-separations for $$l < k$$.

INPUT:

• certificate – (default: False) a boolean; if True, then return True, None if the matroid is 3-connected, and False, $$X$$ otherwise, where $$X$$ is a $$<3$$-separation
• algorithm – (default: None); specify which algorithm to compute 3-connectivity:
• None – The most appropriate algorithm is chosen automatically.
• "bridges" – Bixby and Cunningham’s algorithm, based on bridges [BC1977]. Note that this cannot return a separator.
• "intersection" – An algorithm based on matroid intersection.
• "shifting" – An algorithm based on the shifting algorithm [Raj1987].

OUTPUT:

boolean, or a tuple (boolean, frozenset)

ALGORITHM:

• Bridges based: The 3-connectivity algorithm from [BC1977] which runs in $$O((r(E))^2|E|)$$ time.
• Matroid intersection based: Evaluates the connectivity between $$O(|E|^2)$$ pairs of disjoint sets $$S$$, $$T$$ with $$|S| = |T| = 2$$.
• Shifting algorithm: The shifting algorithm from [Raj1987] which runs in $$O((r(E))^2|E|)$$ time.

EXAMPLES:

sage: matroids.Uniform(2, 3).is_3connected()
True
sage: M = Matroid(ring=QQ, matrix=[[1, 0, 0, 1, 1, 0],
....:                              [0, 1, 0, 1, 2, 0],
....:                              [0, 0, 1, 0, 0, 1]])
sage: M.is_3connected()
False
sage: M.is_3connected() == M.is_3connected(algorithm="bridges")
True
sage: M.is_3connected() == M.is_3connected(algorithm="intersection")
True
sage: N = Matroid(circuit_closures={2: ['abc', 'cdef'],
....:                               3: ['abcdef']},
....:             groundset='abcdef')
sage: N.is_3connected()
False
sage: matroids.named_matroids.BetsyRoss().is_3connected()
True
sage: M = matroids.named_matroids.R6()
sage: M.is_3connected()
False
sage: B, X = M.is_3connected(True)
sage: M.connectivity(X)
1

is_4connected(certificate=False, algorithm=None)

Return True if the matroid is 4-connected, False otherwise. It can optionally return a separator as a witness.

INPUT:

• certificate – (default: False) a boolean; if True, then return True, None if the matroid is 4-connected, and False, $$X$$ otherwise, where $$X$$ is a $$<4$$-separation
• algorithm – (default: None); specify which algorithm to compute 4-connectivity:
• None – The most appropriate algorithm is chosen automatically.
• "intersection" – an algorithm based on matroid intersection, equivalent to calling is_kconnected(4,certificate).
• "shifting" – an algorithm based on the shifting algorithm [Raj1987].

OUTPUT:

boolean, or a tuple (boolean, frozenset)

EXAMPLES:

sage: M = matroids.Uniform(2, 6)
sage: B, X = M.is_4connected(True)
sage: (B, M.connectivity(X)<=3)
(False, True)
sage: matroids.Uniform(4, 8).is_4connected()
True
sage: M = Matroid(field=GF(2), matrix=[[1,0,0,1,0,1,1,0,0,1,1,1],
....:                                  [0,1,0,1,0,1,0,1,0,0,0,1],
....:                                  [0,0,1,1,0,0,1,1,0,1,0,1],
....:                                  [0,0,0,0,1,1,1,1,0,0,1,1],
....:                                  [0,0,0,0,0,0,0,0,1,1,1,1]])
sage: M.is_4connected() == M.is_4connected(algorithm="shifting")
True
sage: M.is_4connected() == M.is_4connected(algorithm="intersection")
True

is_basis(X)

Check if a subset is a basis of the matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_basis('abc')
False
sage: M.is_basis('abce')
True
sage: M.is_basis('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_binary(randomized_tests=1)

Decide if self is a binary matroid.

INPUT:

• randomized_tests – (default: 1) an integer; the number of times a certain necessary condition for being binary is tested, using randomization

OUTPUT:

A Boolean.

ALGORITHM:

First, compare the binary matroids local to two random bases. If these matroids are not isomorphic, return False. This test is performed randomized_tests times. Next, test if a binary matroid local to some basis is isomorphic to self.

EXAMPLES:

sage: N = matroids.named_matroids.Fano()
sage: N.is_binary()
True
sage: N = matroids.named_matroids.NonFano()
sage: N.is_binary()
False

is_chordal(k1=4, k2=None, certificate=False)

Return if a matroid is [k1, k2]-chordal.

A matroid $$M$$ is $$[k_1, k_2]$$-chordal if every circuit of length $$\ell$$ with $$k_1 \leq \ell \leq k_2$$ has a chord. We say $$M$$ is $$k$$-chordal if $$k_1 = k$$ and $$k_2 = \infty$$. We call $$M$$ chordal if it is $$4$$-chordal.

INPUT:

• k1 – (optional) the integer $$k_1$$
• k2 – (optional) the integer $$k_2$$; if not specified, then this method returns if self is $$k_1$$-chordal
• certificate – (default: False) boolean; if True return True, C, where C is a non k1 k2 circuit

Output:

• boolean or tuple

EXAMPLES:

sage: M = matroids.Uniform(2,4)
sage: [M.is_chordal(i) for i in range(4, 8)]
[True, True, True, True]
sage: M = matroids.named_matroids.NonFano()
sage: [M.is_chordal(i) for i in range(4, 8)]
[False, True, True, True]
sage: M = matroids.named_matroids.N2()
sage: [M.is_chordal(i) for i in range(4, 10)]
[False, False, False, False, True, True]
sage: M.is_chordal(4, 5)
False
sage: M.is_chordal(4, 5, certificate=True)
(False, frozenset({'a', 'b', 'e', 'f', 'g'}))

is_circuit(X)

Test if a subset is a circuit of the matroid.

A circuit is an inclusionwise minimal dependent subset.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_circuit('abc')
False
sage: M.is_circuit('abcd')
True
sage: M.is_circuit('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_circuit_chordal(C, certificate=False)

Check if the circuit C has a chord.

A circuit $$C$$ in a matroid $$M$$ has a chord $$x \in E$$ if there exists sets $$A, B$$ such that $$C = A \sqcup B$$ and $$A + x$$ and $$B + x$$ are circuits.

INPUT:

• C – a circuit
• certificate – (default: False) a boolean, if True return True, (x, Ax, Bx), where x is a chord and Ax and Bx are circuits whose union is the elements of C together with x, if False return False, None

OUTPUT:

• boolean or tuple

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.is_circuit_chordal(['b','c','d'])
False
sage: M.is_circuit_chordal(['b','c','d'], certificate=True)
(False, None)
sage: M.is_circuit_chordal(['a','b','d','e'])
True
sage: X = M.is_circuit_chordal(frozenset(['a','b','d','e']), certificate=True)[1]
sage: X # py2
('c', frozenset({'b', 'c', 'd'}), frozenset({'a', 'c', 'e'}))
sage: M.is_circuit(X[1]) and M.is_circuit(X[2])
True
sage: X[1].intersection(X[2]) == frozenset([X[0]])
True

is_closed(X)

Test if a subset is a closed set of the matroid.

A set is closed if adding any element to it will increase the rank of the set.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_closed('abc')
False
sage: M.is_closed('abcd')
True
sage: M.is_closed('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_cobasis(X)

Check if a subset is a cobasis of the matroid.

A cobasis is the complement of a basis. It is a basis of the dual matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_cobasis('abc')
False
sage: M.is_cobasis('abce')
True
sage: M.is_cobasis('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_cocircuit(X)

Test if a subset is a cocircuit of the matroid.

A cocircuit is an inclusionwise minimal subset that is dependent in the dual matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_cocircuit('abc')
False
sage: M.is_cocircuit('abcd')
True
sage: M.is_cocircuit('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_coclosed(X)

Test if a subset is a coclosed set of the matroid.

A set is coclosed if it is a closed set of the dual matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_coclosed('abc')
False
sage: M.is_coclosed('abcd')
True
sage: M.is_coclosed('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_codependent(X)

Check if a subset is codependent in the matroid.

A set is codependent if it is dependent in the dual of the matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_codependent('abc')
False
sage: M.is_codependent('abcd')
True
sage: M.is_codependent('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_coindependent(X)

Check if a subset is coindependent in the matroid.

A set is coindependent if it is independent in the dual matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_coindependent('abc')
True
sage: M.is_coindependent('abcd')
False
sage: M.is_coindependent('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_connected(certificate=False)

Test if the matroid is connected.

A separation in a matroid is a partition $$(X, Y)$$ of the groundset with $$X, Y$$ nonempty and $$r(X) + r(Y) = r(X\cup Y)$$. A matroid is connected if it has no separations.

OUTPUT:

Boolean.

EXAMPLES:

sage: M = Matroid(ring=QQ, matrix=[[1, 0, 0, 1, 1, 0],
....:                              [0, 1, 0, 1, 2, 0],
....:                              [0, 0, 1, 0, 0, 1]])
sage: M.is_connected()
False
sage: matroids.named_matroids.Pappus().is_connected()
True

is_cosimple()

Test if the matroid is cosimple.

A matroid is cosimple if it contains no cocircuits of length 1 or 2.

Dual method of M.is_simple().

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Fano().dual()
sage: M.is_cosimple()
True
sage: N = M \ 'a'
sage: N.is_cosimple()
False

is_dependent(X)

Check if a subset X is dependent in the matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_dependent('abc')
False
sage: M.is_dependent('abcd')
True
sage: M.is_dependent('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_independent(X)

Check if a subset X is independent in the matroid.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_independent('abc')
True
sage: M.is_independent('abcd')
False
sage: M.is_independent('abcx')
Traceback (most recent call last):
...
ValueError: 'abcx' is not a subset of the groundset

is_isomorphic(other, certificate=False)

Test matroid isomorphism.

Two matroids $$M$$ and $$N$$ are isomorphic if there is a bijection $$f$$ from the groundset of $$M$$ to the groundset of $$N$$ such that a subset $$X$$ is independent in $$M$$ if and only if $$f(X)$$ is independent in $$N$$.

INPUT:

• other – A matroid,
• optional parameter certificate – Boolean.

OUTPUT:

Boolean, and, if certificate = True, a dictionary or None

EXAMPLES:

sage: M1 = matroids.Wheel(3)
sage: M2 = matroids.CompleteGraphic(4)
sage: M1.is_isomorphic(M2)
True
sage: M1.is_isomorphic(M2, certificate=True)
(True, {0: 0, 1: 1, 2: 2, 3: 3, 4: 5, 5: 4})
sage: G3 = graphs.CompleteGraph(4)
sage: M1.is_isomorphic(G3)
Traceback (most recent call last):
...
TypeError: can only test for isomorphism between matroids.

sage: M1 = matroids.named_matroids.Fano()
sage: M2 = matroids.named_matroids.NonFano()
sage: M1.is_isomorphic(M2)
False
sage: M1.is_isomorphic(M2, certificate=True)
(False, None)

is_isomorphism(other, morphism)

Test if a provided morphism induces a matroid isomorphism.

A morphism is a map from the groundset of self to the groundset of other.

INPUT:

• other – A matroid.
• morphism – A map. Can be, for instance, a dictionary, function, or permutation.

OUTPUT:

Boolean.

Note

If you know the input is valid, consider using the faster method self._is_isomorphism.

EXAMPLES:

sage: M = matroids.named_matroids.Pappus()
sage: N = matroids.named_matroids.NonPappus()
sage: N.is_isomorphism(M, {e:e for e in M.groundset()})
False

sage: M = matroids.named_matroids.Fano() \ ['g']
sage: N = matroids.Wheel(3)
sage: morphism = {'a':0, 'b':1, 'c': 2, 'd':4, 'e':5, 'f':3}
sage: M.is_isomorphism(N, morphism)
True


A morphism can be specified as a dictionary (above), a permutation, a function, and many other types of maps:

sage: M = matroids.named_matroids.Fano()
sage: P = PermutationGroup([[('a', 'b', 'c'),
....:                        ('d', 'e', 'f'), ('g')]]).gen()
sage: M.is_isomorphism(M, P)
True

sage: M = matroids.named_matroids.Pappus()
sage: N = matroids.named_matroids.NonPappus()
sage: def f(x):
....:     return x
....:
sage: N.is_isomorphism(M, f)
False
sage: N.is_isomorphism(N, f)
True


There is extensive checking for inappropriate input:

sage: M = matroids.CompleteGraphic(4)
sage: M.is_isomorphism(graphs.CompleteGraph(4), lambda x:x)
Traceback (most recent call last):
...
TypeError: can only test for isomorphism between matroids.

sage: M = matroids.CompleteGraphic(4)
sage: sorted(M.groundset())
[0, 1, 2, 3, 4, 5]
sage: M.is_isomorphism(M, {0: 1, 1: 2, 2: 3})
Traceback (most recent call last):
...
ValueError: domain of morphism does not contain groundset of this
matroid.

sage: M = matroids.CompleteGraphic(4)
sage: sorted(M.groundset())
[0, 1, 2, 3, 4, 5]
sage: M.is_isomorphism(M, {0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1})
Traceback (most recent call last):
...
ValueError: range of morphism does not contain groundset of other
matroid.

sage: M = matroids.CompleteGraphic(3)
sage: N = Matroid(bases=['ab', 'ac', 'bc'])
sage: f = [0, 1, 2]
sage: g = {'a': 0, 'b': 1, 'c': 2}
sage: N.is_isomorphism(M, f)
Traceback (most recent call last):
...
ValueError: the morphism argument does not seem to be an
isomorphism.

sage: N.is_isomorphism(M, g)
True

is_k_closed(k)

Return if self is a k-closed matroid.

We say a matroid is $$k$$-closed if all $$k$$-closed subsets are closed in M.

EXAMPLES:

sage: PR = RootSystem(['A',4]).root_lattice().positive_roots()
sage: m = matrix([x.to_vector() for x in PR]).transpose()
sage: M = Matroid(m)
sage: M.is_k_closed(3)
True
sage: M.is_k_closed(4)
True

sage: PR = RootSystem(['D',4]).root_lattice().positive_roots()
sage: m = matrix([x.to_vector() for x in PR]).transpose()
sage: M = Matroid(m)
sage: M.is_k_closed(3)
False
sage: M.is_k_closed(4)
True

is_kconnected(k, certificate=False)

Return True if the matroid is $$k$$-connected, False otherwise. It can optionally return a separator as a witness.

INPUT:

• k – a integer greater or equal to 1.
• certificate – (default: False) a boolean; if True, then return True, None if the matroid is k-connected, and False, X otherwise, where X is a $$<k$$-separation

OUTPUT:

boolean, or a tuple (boolean, frozenset)

ALGORITHM:

Apply linking algorithm to find small separator.

EXAMPLES:

sage: matroids.Uniform(2, 3).is_kconnected(3)
True
sage: M = Matroid(ring=QQ, matrix=[[1, 0, 0, 1, 1, 0],
....:                              [0, 1, 0, 1, 2, 0],
....:                              [0, 0, 1, 0, 0, 1]])
sage: M.is_kconnected(3)
False
sage: N = Matroid(circuit_closures={2: ['abc', 'cdef'],
....:                               3: ['abcdef']},
....:             groundset='abcdef')
sage: N.is_kconnected(3)
False
sage: matroids.named_matroids.BetsyRoss().is_kconnected(3)
True
sage: matroids.AG(5,2).is_kconnected(4)
True
sage: M = matroids.named_matroids.R6()
sage: M.is_kconnected(3)
False
sage: B, X = M.is_kconnected(3,True)
sage: M.connectivity(X)<3
True

is_max_weight_coindependent_generic(X=None, weights=None)

Test if only one cobasis of the subset X has maximal weight.

The weight of a subset S is sum(weights(e) for e in S).

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset
• weights – a dictionary or function mapping the elements of X to nonnegative weights.

OUTPUT:

Boolean.

ALGORITHM:

The greedy algorithm. If a weight function is given, then sort the elements of X by increasing weight, and otherwise use the ordering in which X lists its elements. Then greedily select elements if they are coindependent of all that was selected before. If an element is not coindependent of the previously selected elements, then we check if it is coindependent with the previously selected elements with higher weight.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano()
sage: M.is_max_weight_coindependent_generic()
False

sage: def wt(x):
....:   return x
....:
sage: M = matroids.Uniform(2, 8)
sage: M.is_max_weight_coindependent_generic(weights=wt)
True
sage: M.is_max_weight_coindependent_generic(weights={x: x for x in M.groundset()})
True
sage: M.is_max_weight_coindependent_generic()
False

sage: M=matroids.Uniform(2,5)
sage: wt={0: 1, 1: 1, 2: 1, 3: 2, 4: 2}
sage: M.is_max_weight_independent_generic(weights=wt)
True
sage: M.dual().is_max_weight_coindependent_generic(weights=wt)
True


Here is an example from [GriRei18] (Example 7.4.12 in v5):

sage: A = Matrix(QQ, [[ 1,  1,  0,  0],
....:                 [-1,  0,  1,  1],
....:                 [ 0, -1, -1, -1]])
sage: M = Matroid(A)
sage: M.is_max_weight_coindependent_generic()
False
sage: M.is_max_weight_coindependent_generic(weights={0: 1, 1: 3, 2: 3, 3: 2})
True
sage: M.is_max_weight_coindependent_generic(weights={0: 1, 1: 3, 2: 2, 3: 2})
False
sage: M.is_max_weight_coindependent_generic(weights={0: 2, 1: 3, 2: 1, 3: 1})
False

sage: M.is_max_weight_coindependent_generic(weights={0: 2, 1: 3, 2: -1, 3: 1})
Traceback (most recent call last):
...
ValueError: nonnegative weights were expected.

is_max_weight_independent_generic(X=None, weights=None)

Test if only one basis of the subset X has maximal weight.

The weight of a subset S is sum(weights(e) for e in S).

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset
• weights – a dictionary or function mapping the elements of X to nonnegative weights.

OUTPUT:

Boolean.

ALGORITHM:

The greedy algorithm. If a weight function is given, then sort the elements of X by decreasing weight, and otherwise use the ordering in which X lists its elements. Then greedily select elements if they are independent of all that was selected before. If an element is not independent of the previously selected elements, then we check if it is independent with the previously selected elements with higher weight.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano()
sage: M.is_max_weight_independent_generic()
False

sage: def wt(x):
....:   return x
....:
sage: M = matroids.Uniform(2, 8)
sage: M.is_max_weight_independent_generic(weights=wt)
True
sage: M.is_max_weight_independent_generic(weights={x: x for x in M.groundset()})
True
sage: M.is_max_weight_independent_generic()
False


Here is an example from [GriRei18] (Example 7.4.12 in v5):

sage: A = Matrix(QQ, [[ 1,  1,  0,  0],
....:                 [-1,  0,  1,  1],
....:                 [ 0, -1, -1, -1]])
sage: M = Matroid(A)
sage: M.is_max_weight_independent_generic()
False
sage: M.is_max_weight_independent_generic(weights={0: 1, 1: 3, 2: 3, 3: 2})
True
sage: M.is_max_weight_independent_generic(weights={0: 1, 1: 3, 2: 2, 3: 2})
False
sage: M.is_max_weight_independent_generic(weights={0: 2, 1: 3, 2: 1, 3: 1})
True

sage: M.is_max_weight_independent_generic(weights={0: 2, 1: 3, 2: -1, 3: 1})
Traceback (most recent call last):
...
ValueError: nonnegative weights were expected.

is_simple()

Test if the matroid is simple.

A matroid is simple if it contains no circuits of length 1 or 2.

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.is_simple()
True
sage: N = M / 'a'
sage: N.is_simple()
False

is_subset_k_closed(X, k)

Test if X is a k-closed set of the matroid.

A set $$S$$ is $$k$$-closed if the closure of any $$k$$ element subsets is contained in $$S$$.

INPUT:

• X – a subset (or any iterable) of the groundset
• k – a positive integer

OUTPUT:

Boolean.

EXAMPLES:

sage: m = matrix([[1,2,5,2], [0,2,1,0]])
sage: M = Matroid(m)
sage: M.is_subset_k_closed({1,3}, 2)
False
sage: M.is_subset_k_closed({0,1}, 1)
False
sage: M.is_subset_k_closed({1,2}, 1)
True

sage: m = matrix([[1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1],
....:            [0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2],
....:            [0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1],
....:            [0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1]])
sage: M = Matroid(m)
sage: M.is_subset_k_closed({0,2,3,11}, 3)
True
sage: M.is_subset_k_closed({0,2,3,11}, 4)
False
sage: M.is_subset_k_closed({0,1}, 4)
False
sage: M.is_subset_k_closed({0,1,4}, 4)
True

is_ternary(randomized_tests=1)

Decide if self is a ternary matroid.

INPUT:

• randomized_tests – (default: 1) an integer; the number of times a certain necessary condition for being ternary is tested, using randomization

OUTPUT:

A Boolean.

ALGORITHM:

First, compare the ternary matroids local to two random bases. If these matroids are not isomorphic, return False. This test is performed randomized_tests times. Next, test if a ternary matroid local to some basis is isomorphic to self.

EXAMPLES:

sage: N = matroids.named_matroids.Fano()
sage: N.is_ternary()
False
sage: N = matroids.named_matroids.NonFano()
sage: N.is_ternary()
True

is_valid()

Test if the data obey the matroid axioms.

The default implementation checks the (disproportionately slow) rank axioms. If $$r$$ is the rank function of a matroid, we check, for all pairs $$X, Y$$ of subsets,

• $$0 \leq r(X) \leq |X|$$
• If $$X \subseteq Y$$ then $$r(X) \leq r(Y)$$
• $$r(X\cup Y) + r(X\cap Y) \leq r(X) + r(Y)$$

Certain subclasses may check other axioms instead.

OUTPUT:

Boolean.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.is_valid()
True


The following is the ‘Escher matroid’ by Brylawski and Kelly. See Example 1.5.5 in [Oxl2011]

sage: M = Matroid(circuit_closures={2: [[1, 2, 3], [1, 4, 5]],
....: 3: [[1, 2, 3, 4, 5], [1, 2, 3, 6, 7], [1, 4, 5, 6, 7]]})
sage: M.is_valid()
False

isomorphism(other)

Return a matroid isomorphism.

Two matroids $$M$$ and $$N$$ are isomorphic if there is a bijection $$f$$ from the groundset of $$M$$ to the groundset of $$N$$ such that a subset $$X$$ is independent in $$M$$ if and only if $$f(X)$$ is independent in $$N$$. This method returns one isomorphism $$f$$ from self to other, if such an isomorphism exists.

INPUT:

• other – A matroid.

OUTPUT:

A dictionary, or None.

EXAMPLES:

sage: M1 = matroids.Wheel(3)
sage: M2 = matroids.CompleteGraphic(4)
sage: morphism=M1.isomorphism(M2)
sage: M1.is_isomorphism(M2, morphism)
True
sage: G3 = graphs.CompleteGraph(4)
sage: M1.isomorphism(G3)
Traceback (most recent call last):
...
TypeError: can only give isomorphism between matroids.

sage: M1 = matroids.named_matroids.Fano()
sage: M2 = matroids.named_matroids.NonFano()
sage: M1.isomorphism(M2) is not None
False

k_closure(X, k)

Return the k-closure of X.

A subset $$S$$ of the groundset is $$k$$-closed if the closure of any subset $$T$$ of $$S$$ satisfying $$|T| \leq k$$ is contained in $$S$$. The $$k$$-closure of a set $$X$$ is the smallest $$k$$-closed set containing $$X$$.

INPUT:

• X – a subset (or any iterable) of the groundset
• k – a positive integer

EXAMPLES:

sage: m = matrix([[1,2,5,2], [0,2,1,0]])
sage: M = Matroid(m)
sage: sorted(M.k_closure({1,3}, 2))
[0, 1, 2, 3]
sage: sorted(M.k_closure({0,1}, 1))
[0, 1, 3]
sage: sorted(M.k_closure({1,2}, 1))
[1, 2]

sage: m = matrix([[1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1],
....:            [0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2],
....:            [0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1],
....:            [0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1]])
sage: M = Matroid(m)
sage: sorted(M.k_closure({0,2,3,11}, 3))
[0, 2, 3, 11]
sage: sorted(M.k_closure({0,2,3,11}, 4))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
sage: sorted(M.k_closure({0,1}, 4))
[0, 1, 4]

lattice_of_flats()

Return the lattice of flats of the matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.lattice_of_flats()
Finite lattice containing 16 elements

linear_subclasses(line_length=None, subsets=None)

Return an iterable set of linear subclasses of the matroid.

A linear subclass is a set of hyperplanes (i.e. closed sets of rank $$r(M) - 1$$) with the following property:

• If $$H_1$$ and $$H_2$$ are members, and $$r(H_1 \cap H_2) = r(M) - 2$$, then any hyperplane $$H_3$$ containing $$H_1 \cap H_2$$ is a member too.

A linear subclass is the set of hyperplanes of a modular cut and uniquely determines the modular cut. Hence the collection of linear subclasses is in 1-to-1 correspondence with the collection of single-element extensions of a matroid. See [Oxl2011], section 7.2.

INPUT:

• line_length – (default: None) a natural number. If given, restricts the output to modular cuts that generate an extension by $$e$$ that does not contain a minor $$N$$ isomorphic to $$U_{2, k}$$, where k > line_length, and such that $$e \in E(N)$$.
• subsets – (default: None) a collection of subsets of the ground set. If given, restricts the output to linear subclasses such that each hyperplane contains an element of subsets.

OUTPUT:

An iterable collection of linear subclasses.

Note

The line_length argument only checks for lines using the new element of the corresponding extension. It is still possible that a long line exists by contracting the new element!

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: len(list(M.linear_subclasses()))
16
sage: len(list(M.linear_subclasses(line_length=3)))
8
sage: len(list(M.linear_subclasses(subsets=[{'a', 'b'}])))
5


The following matroid has an extension by element $$e$$ such that contracting $$e$$ creates a 6-point line, but no 6-point line minor uses $$e$$. Consequently, this method returns the modular cut, but the M.extensions() method doesn’t return the corresponding extension:

sage: M = Matroid(circuit_closures={2: ['abc', 'def'],
....:                               3: ['abcdef']})
sage: len(list(M.extensions('g', line_length=5)))
43
sage: len(list(M.linear_subclasses(line_length=5)))
44


Given disjoint subsets $$S$$ and $$T$$, return a connector $$I$$ and a separation $$X$$, which are optimal dual solutions in Tutte’s Linking Theorem:

$\begin{split}\max \{ r_N(S) + r_N(T) - r(N) \mid N = M/I\setminus J, E(N) = S\cup T\}=\\ \min \{ r_M(X) + r_M(Y) - r_M(E) \mid X \subseteq S, Y \subseteq T, E = X\cup Y, X\cap Y = \emptyset \}.\end{split}$

Here $$M$$ denotes this matroid.

INPUT:

• S – a subset (or any iterable) of the groundset
• T – a subset (or any iterable) of the groundset disjoint from S

OUTPUT:

A tuple (I, X) containing a frozenset I and a frozenset X.

ALGORITHM:

Compute a maximum-cardinality common independent set $$I$$ of of $$M / S \setminus T$$ and $$M \setminus S / T$$.

EXAMPLES:

sage: M = matroids.named_matroids.BetsyRoss()
sage: S = set('ab')
sage: T = set('cd')
sage: I, X = M.link(S, T)
sage: M.connectivity(X)
2
sage: J = M.groundset()-(S|T|I)
sage: N = M/I\J
sage: N.connectivity(S)
2

loops()

Return the set of loops of the matroid.

A loop is an element $$u$$ of the groundset such that the one-element set $$\{ u \}$$ is dependent.

OUTPUT:

A set of elements.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.loops()
frozenset()
sage: (M / ['a', 'b']).loops()
frozenset({'f'})

matroid_polytope()

Return the matroid polytope of self.

This is defined as the convex hull of the vertices

$e_B = \sum_{i \in B} e_i$

over all bases $$B$$ of the matroid. Here $$e_i$$ are the standard basis vectors of $$\RR^n$$. An arbitrary labelling of the groundset by $${0,\dots,n-1}$$ is chosen.

EXAMPLES:

sage: M = matroids.Whirl(4)
sage: P = M.matroid_polytope(); P
A 7-dimensional polyhedron in ZZ^8 defined as the convex hull
of 46 vertices

sage: M = matroids.named_matroids.NonFano()
sage: M.matroid_polytope()
A 6-dimensional polyhedron in ZZ^7 defined as the convex hull
of 29 vertices


REFERENCE:

max_coindependent(X)

Compute a maximal coindependent subset of X.

A set is coindependent if it is independent in the dual matroid. A set is coindependent if and only if the complement is spanning (i.e. contains a basis of the matroid).

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

A subset of X.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: X = M.max_coindependent(['a', 'c', 'd', 'e', 'f'])
sage: sorted(X) # py2
['a', 'c', 'd', 'e']
sage: sorted(X) # py3 random
['a', 'c', 'd', 'f']
sage: M.is_coindependent(X)
True
sage: all(M.is_codependent(X.union([y])) for y in M.groundset() if y not in X)
True
sage: M.max_coindependent(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset

max_independent(X)

Compute a maximal independent subset of X.

INPUT:

• X – a subset (or any iterable) of the groundset

OUTPUT:

Subset of X.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: X = M.max_independent(['a', 'c', 'd', 'e', 'f'])
sage: M.is_independent(X)
True
sage: all(M.is_dependent(X.union([y])) for y in M.groundset() if y not in X)
True
sage: M.max_independent(['x'])
Traceback (most recent call last):
...
ValueError: ['x'] is not a subset of the groundset

max_weight_coindependent(X=None, weights=None)

Return a maximum-weight coindependent set contained in X.

The weight of a subset S is sum(weights(e) for e in S).

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset
• weights – a dictionary or function mapping the elements of X to nonnegative weights.

OUTPUT:

A subset of X.

ALGORITHM:

The greedy algorithm. If a weight function is given, then sort the elements of X by decreasing weight, and otherwise use the ordering in which X lists its elements. Then greedily select elements if they are coindependent of all that was selected before.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano()
sage: X = M.max_weight_coindependent()
sage: M.is_cobasis(X)
True

sage: wt = {'a': 1, 'b': 2, 'c': 2, 'd': 1/2, 'e': 1, 'f': 2,
....:       'g': 2}
sage: setprint(M.max_weight_coindependent(weights=wt))
{'b', 'c', 'f', 'g'}
sage: wt = {'a': 1, 'b': -10, 'c': 2, 'd': 1/2, 'e': 1, 'f': 2,
....:       'g': 2}
sage: setprint(M.max_weight_coindependent(weights=wt))
Traceback (most recent call last):
...
ValueError: nonnegative weights were expected.

sage: def wt(x):
....:   return x
....:
sage: M = matroids.Uniform(2, 8)
sage: setprint(M.max_weight_coindependent(weights=wt))
{2, 3, 4, 5, 6, 7}
sage: setprint(M.max_weight_coindependent())
{0, 1, 2, 3, 4, 5}
sage: M.max_weight_coindependent(X=[], weights={})
frozenset()

max_weight_independent(X=None, weights=None)

Return a maximum-weight independent set contained in a subset.

The weight of a subset S is sum(weights(e) for e in S).

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset
• weights – a dictionary or function mapping the elements of X to nonnegative weights.

OUTPUT:

A subset of X.

ALGORITHM:

The greedy algorithm. If a weight function is given, then sort the elements of X by decreasing weight, and otherwise use the ordering in which X lists its elements. Then greedily select elements if they are independent of all that was selected before.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano()
sage: X = M.max_weight_independent()
sage: M.is_basis(X)
True

sage: wt = {'a': 1, 'b': 2, 'c': 2, 'd': 1/2, 'e': 1,
....:       'f': 2, 'g': 2}
sage: setprint(M.max_weight_independent(weights=wt))
{'b', 'f', 'g'}
sage: def wt(x):
....:   return x
....:
sage: M = matroids.Uniform(2, 8)
sage: setprint(M.max_weight_independent(weights=wt))
{6, 7}
sage: setprint(M.max_weight_independent())
{0, 1}
sage: M.max_weight_coindependent(X=[], weights={})
frozenset()

minor(contractions=None, deletions=None)

Return the minor of self obtained by contracting, respectively deleting, the element(s) of contractions and deletions.

A minor of a matroid is a matroid obtained by repeatedly removing elements in one of two ways: either contract or delete them. It can be shown that the final matroid does not depend on the order in which elements are removed.

INPUT:

• contractions – (default: None) an element or set of elements to be contracted.
• deletions – (default: None) an element or set of elements to be deleted.

OUTPUT:

A matroid.

Note

The output is either of the same type as self, or an instance of MinorMatroid.

EXAMPLES:

sage: M = matroids.Wheel(4)
sage: N = M.minor(contractions=[7], deletions=[0])
sage: N.is_isomorphic(matroids.Wheel(3))
True


The sets of contractions and deletions need not be independent, respectively coindependent:

sage: M = matroids.named_matroids.Fano()
sage: M.rank('abf')
2
sage: M.minor(contractions='abf')
Binary matroid of rank 1 on 4 elements, type (1, 0)


However, they need to be subsets of the groundset, and disjoint:

sage: M = matroids.named_matroids.Vamos()
sage: N = M.minor('abc', 'defg')
sage: N # py2
M / {'a', 'b', 'c'} \ {'d', 'e', 'f', 'g'}, where M is Vamos:
Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'},
{'a', 'b', 'g', 'h'}, {'c', 'd', 'e', 'f'},
{'e', 'f', 'g', 'h'}},
4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: N.groundset()
frozenset({'h'})

sage: N = M.minor('defgh', 'abc')
sage: N # py2
M / {'d', 'e', 'f', 'g'} \ {'a', 'b', 'c', 'h'}, where M is Vamos:
Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'},
{'a', 'b', 'g', 'h'}, {'c', 'd', 'e', 'f'},
{'e', 'f', 'g', 'h'}},
4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: N.groundset()
frozenset()

sage: M.minor([1, 2, 3], 'efg')
Traceback (most recent call last):
...
ValueError: [1, 2, 3] is not a subset of the groundset
sage: M.minor('efg', [1, 2, 3])
Traceback (most recent call last):
...
ValueError: [1, 2, 3] is not a subset of the groundset
Traceback (most recent call last):
...
ValueError: contraction and deletion sets are not disjoint.


Warning

There can be ambiguity if elements of the groundset are themselves iterable, and their elements are in the groundset. The main example of this is when an element is a string. See the documentation of the methods contract() and delete() for an example of this.

modular_cut(subsets)

Compute the modular cut generated by subsets.

A modular cut is a collection $$C$$ of flats such that

• If $$F \in C$$ and $$F'$$ is a flat containing $$F$$, then $$F' \in C$$
• If $$F_1, F_2 \in C$$ form a modular pair of flats, then $$F_1\cap F_2 \in C$$.

A flat is a closed set, a modular pair is a pair $$F_1, F_2$$ of flats with $$r(F_1) + r(F_2) = r(F_1\cup F_2) + r(F_1\cap F_2)$$, where $$r$$ is the rank function of the matroid.

The modular cut generated by subsets is the smallest modular cut $$C$$ for which closure(S) in C for all $$S$$ in subsets.

There is a one-to-one correspondence between the modular cuts of a matroid and the single-element extensions of the matroid. See [Oxl2011] Section 7.2 for more information.

Note

Sage uses linear subclasses, rather than modular cuts, internally for matroid extension. A linear subclass is the set of hyperplanes (flats of rank $$r(M) - 1$$) of a modular cut. It determines the modular cut uniquely (see [Oxl2011] Section 7.2).

INPUT:

• subsets – A collection of subsets of the groundset.

OUTPUT:

A collection of subsets.

EXAMPLES:

Any extension of the Vamos matroid where the new point is placed on the lines through elements $$\{a, b\}$$ and through $$\{c, d\}$$ is an extension by a loop:

sage: M = matroids.named_matroids.Vamos()
sage: frozenset([]) in M.modular_cut(['ab', 'cd'])
True


In any extension of the matroid $$S_8 \setminus h$$, a point on the lines through $$\{c, g\}$$ and $$\{a, e\}$$ also is on the line through $$\{b, f\}$$:

sage: M = matroids.named_matroids.S8()
sage: N = M \ 'h'
sage: frozenset('bf') in N.modular_cut(['cg', 'ae'])
True


The modular cut of the full groundset is equal to just the groundset:

sage: M = matroids.named_matroids.Fano()
sage: M.modular_cut([M.groundset()]).difference(
....:                               [frozenset(M.groundset())])
set()

no_broken_circuits_sets(ordering=None)

Return the no broken circuits (NBC) sets of self.

An NBC set is a subset $$A$$ of the ground set under some total ordering $$<$$ such that $$A$$ contains no broken circuit.

INPUT:

• ordering – a total ordering of the groundset given as a list

EXAMPLES:

sage: M = Matroid(circuits=[[1,2,3], [3,4,5], [1,2,4,5]])
sage: SimplicialComplex(M.no_broken_circuits_sets())
Simplicial complex with vertex set (1, 2, 3, 4, 5)
and facets {(1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5)}
sage: SimplicialComplex(M.no_broken_circuits_sets([5,4,3,2,1]))
Simplicial complex with vertex set (1, 2, 3, 4, 5)
and facets {(1, 3, 5), (1, 4, 5), (2, 3, 5), (2, 4, 5)}

sage: M = Matroid(circuits=[[1,2,3], [1,4,5], [2,3,4,5]])
sage: SimplicialComplex(M.no_broken_circuits_sets([5,4,3,2,1]))
Simplicial complex with vertex set (1, 2, 3, 4, 5)
and facets {(1, 3, 5), (2, 3, 5), (2, 4, 5), (3, 4, 5)}

nonbases()

Return the list of nonbases of the matroid.

A nonbasis is a set with cardinality self.full_rank() that is not a basis.

OUTPUT:

An iterable containing the nonbases of the matroid.

EXAMPLES:

sage: M = matroids.Uniform(2, 4)
sage: list(M.nonbases())
[]
sage: [sorted(X) for X in matroids.named_matroids.P6().nonbases()]
[['a', 'b', 'c']]


ALGORITHM:

Test all subsets of the groundset of cardinality self.full_rank()

noncospanning_cocircuits()

Return the list of noncospanning cocircuits of the matroid.

A noncospanning cocircuit is a cocircuit whose corank is strictly smaller than the corank of the matroid.

OUTPUT:

An iterable containing all nonspanning circuits.

EXAMPLES:

sage: M = matroids.named_matroids.Fano().dual()
sage: sorted([sorted(C) for C in M.noncospanning_cocircuits()])
[['a', 'b', 'f'], ['a', 'c', 'e'], ['a', 'd', 'g'],
['b', 'c', 'd'], ['b', 'e', 'g'], ['c', 'f', 'g'],
['d', 'e', 'f']]

nonspanning_circuit_closures()

Return the list of closures of nonspanning circuits of the matroid.

A nonspanning circuit closure is a closed set containing a nonspanning circuit.

OUTPUT:

A dictionary containing the nonspanning circuit closures of the matroid, indexed by their ranks.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: CC = M.nonspanning_circuit_closures()
sage: len(CC[2])
7
sage: len(CC[3])
Traceback (most recent call last):
...
KeyError: 3

nonspanning_circuits()

Return the list of nonspanning circuits of the matroid.

A nonspanning circuit is a circuit whose rank is strictly smaller than the rank of the matroid.

OUTPUT:

An iterable containing all nonspanning circuits.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: sorted([sorted(C) for C in M.nonspanning_circuits()])
[['a', 'b', 'f'], ['a', 'c', 'e'], ['a', 'd', 'g'],
['b', 'c', 'd'], ['b', 'e', 'g'], ['c', 'f', 'g'],
['d', 'e', 'f']]

orlik_solomon_algebra(R, ordering=None)

Return the Orlik-Solomon algebra of self.

INPUT:

• R – the base ring
• ordering – (optional) an ordering of the ground set

EXAMPLES:

sage: M = matroids.Uniform(3, 4)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: OS
Orlik-Solomon algebra of U(3, 4): Matroid of rank 3 on 4 elements
with circuit-closures
{3: {{0, 1, 2, 3}}}

partition()

Returns a minimum number of disjoint independent sets that covers the groundset.

OUTPUT:

A list of disjoint independent sets that covers the goundset.

EXAMPLES:

sage: M = matroids.named_matroids.Block_9_4()
sage: P = M.partition()
sage: all(map(M.is_independent,P))
True
sage: set.union(*P)==M.groundset()
True
sage: sum(map(len,P))==len(M.groundset())
True
sage: Matroid(matrix([])).partition()
[]


ALGORITHM:

Reduce partition to a matroid intersection between a matroid sum and a partition matroid. It’s known the direct method doesn’t gain much advantage over matroid intersection. [Cun1986]

plot(B=None, lineorders=None, pos_method=None, pos_dict=None, save_pos=False)

Return geometric representation as a sage graphics object.

INPUT:

• B – (optional) a list containing a basis. If internal point placement is used, these elements will be placed as vertices of a triangle.
• lineorders – (optional) A list of lists where each of the inner lists specify ground set elements in a certain order which will be used to draw the corresponding line in geometric representation (if it exists).
• pos_method – An integer specifying positioning method.
• 0: default positioning
• 1: use pos_dict if it is not None
• 2: Force directed (Not yet implemented).
• pos_dict: A dictionary mapping ground set elements to their (x,y) positions.
• save_pos: A boolean indicating that point placements (either internal or user provided) and line orders (if provided) will be cached in the matroid (M._cached_info) and can be used for reproducing the geometric representation during the same session

OUTPUT:

A sage graphics object of type <class ‘sage.plot.graphics.Graphics’> that corresponds to the geometric representation of the matroid

EXAMPLES:

sage: M=matroids.named_matroids.Fano()
sage: G=M.plot()
sage: type(G)
<class 'sage.plot.graphics.Graphics'>
sage: G.show()

rank(X=None)

Return the rank of X.

The rank of a subset $$X$$ is the size of the largest independent set contained in $$X$$.

If X is None, the rank of the groundset is returned.

INPUT:

• X – (default: the groundset) a subset (or any iterable) of the groundset

OUTPUT:

Integer.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.rank()
3
sage: M.rank(['a', 'b', 'f'])
2
sage: M.rank(['a', 'b', 'x'])
Traceback (most recent call last):
...
ValueError: ['a', 'b', 'x'] is not a subset of the groundset

show(B=None, lineorders=None, pos_method=None, pos_dict=None, save_pos=False, lims=None)

Show the geometric representation of the matroid.

INPUT:

• B – (optional) a list containing elements of the groundset not in any particular order. If internal point placement is used, these elements will be placed as vertices of a triangle.
• lineorders – (optional) A list of lists where each of the inner lists specify ground set elements in a certain order which will be used to draw the corresponding line in geometric representation (if it exists).
• pos_method – An integer specifying positioning method
• 0: default positioning
• 1: use pos_dict if it is not None
• 2: Force directed (Not yet implemented).
• pos_dict – A dictionary mapping ground set elements to their (x,y) positions.
• save_pos – A boolean indicating that point placements (either internal or user provided) and line orders (if provided) will be cached in the matroid (M._cached_info) and can be used for reproducing the geometric representation during the same session
• lims – A list of 4 elements [xmin,xmax,ymin,ymax]

EXAMPLES:

sage: M=matroids.named_matroids.TernaryDowling3()
sage: M.show(B=['a','b','c'])
sage: M.show(B=['a','b','c'],lineorders=[['f','e','i']])
sage: pos = {'a':(0,0), 'b': (0,1), 'c':(1,0), 'd':(1,1), 'e':(1,-1), 'f':(-1,1), 'g':(-1,-1),'h':(2,0), 'i':(0,2)}
sage: M.show(pos_method=1, pos_dict=pos,lims=[-3,3,-3,3])

simplify()

Return the simplification of the matroid.

A matroid is simple if it contains no circuits of length 1 or 2. The simplification of a matroid is obtained by deleting all loops (circuits of length 1) and deleting all but one element from each parallel class (a closed set of rank 1, that is, each pair in it forms a circuit of length 2).

OUTPUT:

A matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Fano().contract('a')
sage: M.size() - M.simplify().size()
3

size()

Return the size of the groundset.

OUTPUT:

Integer.

EXAMPLES:

sage: M = matroids.named_matroids.Vamos()
sage: M.size()
8

ternary_matroid(randomized_tests=1, verify=True)

Return a ternary matroid representing self, if such a representation exists.

INPUT:

• randomized_tests – (default: 1) an integer; the number of times a certain necessary condition for being ternary is tested, using randomization
• verify – (default: True), a Boolean; if True, any output will be a ternary matroid representing self; if False, any output will represent self if and only if the matroid is ternary

OUTPUT:

Either a TernaryMatroid, or None

ALGORITHM:

First, compare the ternary matroids local to two random bases. If these matroids are not isomorphic, return None. This test is performed randomized_tests times. Next, if verify is True, test if a ternary matroid local to some basis is isomorphic to self.

M._local_ternary_matroid()

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.ternary_matroid() is None
True
sage: N = matroids.named_matroids.NonFano()
sage: N.ternary_matroid()
NonFano: Ternary matroid of rank 3 on 7 elements, type 0-

truncation()

Return a rank-1 truncation of the matroid.

Let $$M$$ be a matroid of rank $$r$$. The truncation of $$M$$ is the matroid obtained by declaring all subsets of size $$r$$ dependent. It can be obtained by adding an element freely to the span of the matroid and then contracting that element.

OUTPUT:

A matroid.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: N = M.truncation()
sage: N.is_isomorphic(matroids.Uniform(2, 7))
True

tutte_polynomial(x=None, y=None)

Return the Tutte polynomial of the matroid.

The Tutte polynomial of a matroid is the polynomial

$T(x, y) = \sum_{A \subseteq E} (x - 1)^{r(E) - r(A)} (y - 1)^{r^*(E) - r^*(E\setminus A)},$

where $$E$$ is the groundset of the matroid, $$r$$ is the rank function, and $$r^*$$ is the corank function. Tutte defined his polynomial differently:

$T(x, y)=\sum_{B} x^i(B) y^e(B),$

where the sum ranges over all bases of the matroid, $$i(B)$$ is the number of internally active elements of $$B$$, and $$e(B)$$ is the number of externally active elements of $$B$$.

INPUT:

• x – (optional) a variable or numerical argument.
• y – (optional) a variable or numerical argument.

OUTPUT:

The Tutte-polynomial $$T(x, y)$$, where $$x$$ and $$y$$ are substituted with any values provided as input.

Todo

Make implementation more efficient, e.g. generalizing the approach from trac ticket #1314 from graphs to matroids.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: M.tutte_polynomial()
y^4 + x^3 + 3*y^3 + 4*x^2 + 7*x*y + 6*y^2 + 3*x + 3*y
sage: M.tutte_polynomial(1, 1) == M.bases_count()
True


ALGORITHM:

Enumerate the bases and compute the internal and external activities for each $$B$$.

union(matroids)

Return the matroid union with another matroid or a list of matroids.

Let $$(M_1, M_2, \ldots, M_k)$$ be a list of matroids where each $$M_i$$ has ground set $$E_i$$. The matroid union $$M$$ of $$(M_1, M_2, \ldots, M_k)$$ has ground set $$E = \cup E_i$$. Moreover, a set $$I \subseteq E$$ is independent in $$M$$ if and only if the restriction of $$I$$ to $$E_i$$ is independent in $$M_i$$ for every $$i$$.

INPUT:

• matroids - a matroid or a list of matroids

OUTPUT:

An instance of MatroidUnion.

EXAMPLES:

sage: M = matroids.named_matroids.Fano()
sage: N = M.union(matroids.named_matroids.NonFano()); N
Matroid of rank 6 on 7 elements as matroid union of
Binary matroid of rank 3 on 7 elements, type (3, 0)
Ternary matroid of rank 3 on 7 elements, type 0-