Database of matroids#

This module contains the implementation and documentation for all matroids in the database, accessible through matroids. and matroids.catalog. (type those lines followed by Tab for a list).

AUTHORS:

  • Michael Welsh, Stefan van Zwam (2013-04-01): initial version

  • Giorgos Mousa, Andreas Triantafyllos (2023-12-08): more matroids

REFERENCES:

For more information on the matroids that belong to Oxley’s matroid collection, as well as for most parametrized matroids, see [Oxl2011].

For more information on the matroids that belong to Brettell’s matroid collection, see the associated publications, [Bre2023] and [BP2023].

Note

The grouping of the matroids into collections can be viewed in the catalog of matroids.

sage.matroids.database_matroids.A9(groundset=None)[source]#

Return the matroid \(A9\).

An excluded minor for \(K_2\)-representable matroids. The UPF is \(P_4\). Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.A9(); M
A9: Quaternary matroid of rank 3 on 9 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.A9(); M
A9: Quaternary matroid of rank 3 on 9 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.AF12(groundset=None)[source]#

Return the matroid \(AF12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.AF12(); M
AF12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.AF12(); M
AF12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.AG(n, q, x=None, groundset=None)[source]#

Return the affine geometry of dimension n over the finite field of order q.

The affine geometry can be obtained from the projective geometry by removing a hyperplane.

INPUT:

  • n – a positive integer; the dimension of the projective space. This is one less than the rank of the resulting matroid.

  • q – a positive integer that is a prime power; the order of the finite field

  • x – a string (default: None); the name of the generator of a non-prime field, used for non-prime fields. If not supplied, 'x' is used.

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: a linear matroid whose elements are the points of \(AG(n, q)\)

EXAMPLES:

sage: M = matroids.AG(2, 3).delete(8)
sage: M.is_isomorphic(matroids.catalog.AG23minus())
True
sage: matroids.AG(5, 4, 'z').size() == ((4 ^ 6 - 1) / (4 - 1) - (4 ^ 5 - 1)/(4 - 1))
True
sage: M = matroids.AG(4, 2); M
AG(4, 2): Binary matroid of rank 5 on 16 elements, type (5, 0)

>>> from sage.all import *
>>> M = matroids.AG(Integer(2), Integer(3)).delete(Integer(8))
>>> M.is_isomorphic(matroids.catalog.AG23minus())
True
>>> matroids.AG(Integer(5), Integer(4), 'z').size() == ((Integer(4) ** Integer(6) - Integer(1)) / (Integer(4) - Integer(1)) - (Integer(4) ** Integer(5) - Integer(1))/(Integer(4) - Integer(1)))
True
>>> M = matroids.AG(Integer(4), Integer(2)); M
AG(4, 2): Binary matroid of rank 5 on 16 elements, type (5, 0)

REFERENCES:

[Oxl2011], p. 661.

sage.matroids.database_matroids.AG23(groundset='abcdefghi')[source]#

Return the matroid \(AG(2, 3)\).

The ternary affine plane. A matroid which is not graphic, not cographic, and not regular.

EXAMPLES:

sage: M = matroids.catalog.AG23(); M
AG(2, 3): Ternary matroid of rank 3 on 9 elements, type 3+
sage: M.is_valid() and M.is_3connected() and M.is_ternary()
True
sage: M.has_minor(matroids.catalog.K4())
False
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.delete(e).is_isomorphic(matroids.catalog.AG23minus())
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.AG23(); M
AG(2, 3): Ternary matroid of rank 3 on 9 elements, type 3+
>>> M.is_valid() and M.is_3connected() and M.is_ternary()
True
>>> M.has_minor(matroids.catalog.K4())
False
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.delete(e).is_isomorphic(matroids.catalog.AG23minus())
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 653.

sage.matroids.database_matroids.AG23minus(groundset=None)[source]#

Return the ternary affine plane minus a point.

This is a sixth-roots-of-unity matroid, and an excluded minor for the class of near-regular matroids.

EXAMPLES:

sage: M = matroids.catalog.AG23minus(); M
AG23minus: Matroid of rank 3 on 8 elements with circuit-closures
{2: {{'a', 'b', 'c'}, {'a', 'd', 'f'}, {'a', 'e', 'g'},
{'b', 'd', 'h'}, {'b', 'e', 'f'}, {'c', 'd', 'g'},
{'c', 'e', 'h'}, {'f', 'g', 'h'}},
3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.AG23minus(); M
AG23minus: Matroid of rank 3 on 8 elements with circuit-closures
{2: {{'a', 'b', 'c'}, {'a', 'd', 'f'}, {'a', 'e', 'g'},
{'b', 'd', 'h'}, {'b', 'e', 'f'}, {'c', 'd', 'g'},
{'c', 'e', 'h'}, {'f', 'g', 'h'}},
3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> M.is_valid()
True

REFERENCES:

[Oxl2011], p. 653.

sage.matroids.database_matroids.AG23minusDY(groundset=None)[source]#

Return the matroid \(AG23minusDY\).

The matroid obtained from a \(AG(2, 3)\setminus e\) by a single \(\delta-Y\) exchange on a triangle. An excluded minor for near-regular matroids. UPF is \(S\).

EXAMPLES:

sage: M = matroids.catalog.AG23minusDY(); M
Delta-Y of AG(2,3)\e: Ternary matroid of rank 4 on 8 elements, type 0-
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.AG23minusDY(); M
Delta-Y of AG(2,3)\e: Ternary matroid of rank 4 on 8 elements, type 0-
>>> M.is_valid()
True
sage.matroids.database_matroids.AG32(groundset='abcdefgh')[source]#

Return the matroid \(AG(3, 2)\).

The binary affine cube.

EXAMPLES:

sage: M = matroids.catalog.AG32(); M
AG(3, 2): Binary matroid of rank 4 on 8 elements, type (4, 0)
sage: M.is_valid() and M.is_3connected()
True

>>> from sage.all import *
>>> M = matroids.catalog.AG32(); M
AG(3, 2): Binary matroid of rank 4 on 8 elements, type (4, 0)
>>> M.is_valid() and M.is_3connected()
True

\(AG(3, 2)\) is isomorphic to the unique tipless binary \(4\)-spike:

sage: M.is_isomorphic(matroids.Z(4, False))
True

>>> from sage.all import *
>>> M.is_isomorphic(matroids.Z(Integer(4), False))
True

and it is identically self-dual:

sage: M.equals(M.dual())
True

>>> from sage.all import *
>>> M.equals(M.dual())
True

Every single-element deletion is isomorphic to \(F_7^*\) and every single-element contraction is isomorphic to \(F_7\):

sage: F7 = matroids.catalog.Fano()
sage: F7D = matroids.catalog.FanoDual()
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.delete(e).is_isomorphic(F7D)
True
sage: M.contract(e).is_isomorphic(F7)
True

>>> from sage.all import *
>>> F7 = matroids.catalog.Fano()
>>> F7D = matroids.catalog.FanoDual()
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.delete(e).is_isomorphic(F7D)
True
>>> M.contract(e).is_isomorphic(F7)
True

REFERENCES:

[Oxl2011], p. 645.

sage.matroids.database_matroids.AG32prime(groundset=None)[source]#

Return the matroid \(AG(3, 2)'\), represented as circuit closures.

The matroid \(AG(3, 2)'\) is an \(8\)-element matroid of rank-\(4\). It is a smallest non-representable matroid. It is the unique relaxation of \(AG(3, 2)\).

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.AG32prime(); M
AG(3, 2)': Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'c', 'e', 'g'}, {'a', 'd', 'g', 'h'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'},
     {'b', 'e', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'},
     {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: setprint(M.noncospanning_cocircuits())
[{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
 {'a', 'c', 'd', 'f'}, {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'},
 {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'b', 'd', 'f', 'h'},
 {'b', 'e', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'},
 {'d', 'e', 'f', 'g'}]
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.AG32prime(); M
AG(3, 2)': Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'c', 'e', 'g'}, {'a', 'd', 'g', 'h'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'},
     {'b', 'e', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'},
     {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> setprint(M.noncospanning_cocircuits())
[{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
 {'a', 'c', 'd', 'f'}, {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'},
 {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'b', 'd', 'f', 'h'},
 {'b', 'e', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'},
 {'d', 'e', 'f', 'g'}]
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
False

Self-dual but not identically self-dual:

sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

>>> from sage.all import *
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

Every single-element deletion is isomorphic to \(F_7^*\) or \((F_7^-)^*\) and every single-element contraction is isomorphic to \(F_7\) or \(F_7^-\):

sage: F7 = matroids.catalog.Fano()
sage: F7D = matroids.catalog.FanoDual()
sage: F7m = matroids.catalog.NonFano()
sage: F7mD = matroids.catalog.NonFanoDual()
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.delete(e).is_isomorphic(F7D) or M.delete(e).is_isomorphic(F7mD)
True
sage: Me = M.contract(e)
sage: Me.is_isomorphic(F7) or Me.is_isomorphic(F7m)
True

>>> from sage.all import *
>>> F7 = matroids.catalog.Fano()
>>> F7D = matroids.catalog.FanoDual()
>>> F7m = matroids.catalog.NonFano()
>>> F7mD = matroids.catalog.NonFanoDual()
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.delete(e).is_isomorphic(F7D) or M.delete(e).is_isomorphic(F7mD)
True
>>> Me = M.contract(e)
>>> Me.is_isomorphic(F7) or Me.is_isomorphic(F7m)
True

REFERENCES:

[Oxl2011], p. 646.

sage.matroids.database_matroids.AK12(groundset=None)[source]#

Return the matroid \(AK12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Not self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.AK12(); M
AK12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.AK12(); M
AK12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.BB9(groundset=None)[source]#

Return the matroid \(BB9\).

An excluded minor for \(K_2\)-representable matroids, and a restriction of the Betsy Ross matroid. The UPF is \(G\). Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: BB = matroids.catalog.BB9(); BB
BB9: Quaternary matroid of rank 3 on 9 elements
sage: BR = matroids.catalog.BetsyRoss()
sage: from itertools import combinations
sage: pairs = combinations(sorted(BR.groundset()), 2)
sage: for pair in pairs:
....:     if BR.delete(pair).is_isomorphic(BB):
....:         print(pair)
('a', 'h')
('b', 'i')
('c', 'j')
('d', 'f')
('e', 'g')

>>> from sage.all import *
>>> BB = matroids.catalog.BB9(); BB
BB9: Quaternary matroid of rank 3 on 9 elements
>>> BR = matroids.catalog.BetsyRoss()
>>> from itertools import combinations
>>> pairs = combinations(sorted(BR.groundset()), Integer(2))
>>> for pair in pairs:
...     if BR.delete(pair).is_isomorphic(BB):
...         print(pair)
('a', 'h')
('b', 'i')
('c', 'j')
('d', 'f')
('e', 'g')
sage.matroids.database_matroids.BB9gDY(groundset=None)[source]#

Return the matroid \(BB9gDY\).

An excluded minor for \(K_2\)-representable matroids. The UPF is \(G\). In a \(DY^*\)-equivalence class of 4 matroids, one of which can be obtained from BB9 by a segment-cosegment exchange on \(\{a,d,i,j\}\). Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.BB9gDY(); M
Segment cosegment exchange on BB9: Quaternary matroid of rank 5 on 9 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.BB9gDY(); M
Segment cosegment exchange on BB9: Quaternary matroid of rank 5 on 9 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.BetsyRoss(groundset=None)[source]#

Return the Betsy Ross matroid, represented by circuit closures.

An extremal golden-mean matroid. That is, if \(M\) is simple, rank \(3\), has the Betsy Ross matroid as a restriction and is a Golden Mean matroid, then \(M\) is the Betsy Ross matroid.

EXAMPLES:

sage: M = matroids.catalog.BetsyRoss(); M
BetsyRoss: Matroid of rank 3 on 11 elements with 25 nonspanning
circuits
sage: len(M.circuit_closures()[2])
10
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.BetsyRoss(); M
BetsyRoss: Matroid of rank 3 on 11 elements with 25 nonspanning
circuits
>>> len(M.circuit_closures()[Integer(2)])
10
>>> M.is_valid()
True
sage.matroids.database_matroids.Block_10_5(groundset=None)[source]#

Return the paving matroid whose nonspanning circuits form the blocks of a \(3-(10, 5, 3)\) design.

EXAMPLES:

sage: M = matroids.catalog.Block_10_5(); M
Block(10, 5): Matroid of rank 5 on 10 elements with 36 nonspanning
circuits
sage: M.is_valid() and M.is_paving()
True
sage: BD = BlockDesign(M.groundset(), list(M.nonspanning_circuits()))
sage: BD.is_t_design(return_parameters=True)
(True, (3, 10, 5, 3))

>>> from sage.all import *
>>> M = matroids.catalog.Block_10_5(); M
Block(10, 5): Matroid of rank 5 on 10 elements with 36 nonspanning
circuits
>>> M.is_valid() and M.is_paving()
True
>>> BD = BlockDesign(M.groundset(), list(M.nonspanning_circuits()))
>>> BD.is_t_design(return_parameters=True)
(True, (3, 10, 5, 3))
sage.matroids.database_matroids.Block_9_4(groundset=None)[source]#

Return the paving matroid whose nonspanning circuits form the blocks of a \(2-(9, 4, 3)\) design.

EXAMPLES:

sage: M = matroids.catalog.Block_9_4(); M
Block(9, 4): Matroid of rank 4 on 9 elements with 18 nonspanning
circuits
sage: M.is_valid() and M.is_paving()
True
sage: BD = BlockDesign(M.groundset(), list(M.nonspanning_circuits()))
sage: BD.is_t_design(return_parameters=True)
(True, (2, 9, 4, 3))

>>> from sage.all import *
>>> M = matroids.catalog.Block_9_4(); M
Block(9, 4): Matroid of rank 4 on 9 elements with 18 nonspanning
circuits
>>> M.is_valid() and M.is_paving()
True
>>> BD = BlockDesign(M.groundset(), list(M.nonspanning_circuits()))
>>> BD.is_t_design(return_parameters=True)
(True, (2, 9, 4, 3))
sage.matroids.database_matroids.CompleteGraphic(n, groundset=None)[source]#

Return the cycle matroid of the complete graph on \(n\) vertices.

INPUT:

  • n – an integer, the number of vertices of the underlying complete graph.

OUTPUT: The graphic matroid associated with the \(n\)-vertex complete graph. This matroid has rank \(n - 1\).

EXAMPLES:

sage: # needs sage.graphs
sage: from sage.matroids.advanced import setprint
sage: M = matroids.CompleteGraphic(5); M
M(K5): Graphic matroid of rank 4 on 10 elements
sage: M.has_minor(matroids.Uniform(2, 4))
False
sage: simplify(M.contract(randrange(0,
....:                 10))).is_isomorphic(matroids.CompleteGraphic(4))
True
sage: setprint(M.closure([0, 2, 4, 5]))
{0, 1, 2, 4, 5, 7}
sage: M.is_valid()
True

>>> from sage.all import *
>>> # needs sage.graphs
>>> from sage.matroids.advanced import setprint
>>> M = matroids.CompleteGraphic(Integer(5)); M
M(K5): Graphic matroid of rank 4 on 10 elements
>>> M.has_minor(matroids.Uniform(Integer(2), Integer(4)))
False
>>> simplify(M.contract(randrange(Integer(0),
...                 Integer(10)))).is_isomorphic(matroids.CompleteGraphic(Integer(4)))
True
>>> setprint(M.closure([Integer(0), Integer(2), Integer(4), Integer(5)]))
{0, 1, 2, 4, 5, 7}
>>> M.is_valid()
True
sage.matroids.database_matroids.D10(groundset=None)[source]#

Return the matroid \(D10\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(G\). Has a TQ8-minor. In a \(DY^*\)-equivalence class of \(13\) matroids.

EXAMPLES:

sage: M = matroids.catalog.D10(); M
D10: Quaternary matroid of rank 4 on 10 elements
sage: M.has_minor(matroids.catalog.TQ8())
True

>>> from sage.all import *
>>> M = matroids.catalog.D10(); M
D10: Quaternary matroid of rank 4 on 10 elements
>>> M.has_minor(matroids.catalog.TQ8())
True
sage.matroids.database_matroids.D16(groundset='abcdefghijklmnop')[source]#

Return the matroid \(D_{16}\).

Let \(M\) be a \(4\)-connected binary matroid and \(N\) an internally \(4\)-connected proper minor of \(M\) with at least 7 elements. Then some element of \(M\) can be deleted or contracted preserving an \(N\)-minor, unless \(M\) is \(D_{16}\).

EXAMPLES:

sage: M = matroids.catalog.D16(); M
D16: Binary matroid of rank 8 on 16 elements, type (0, 0)
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.D16(); M
D16: Binary matroid of rank 8 on 16 elements, type (0, 0)
>>> M.is_valid()
True

REFERENCES:

[CMO2012]

sage.matroids.database_matroids.ExtendedBinaryGolayCode(groundset='abcdefghijklmnopqrstuvwx')[source]#

Return the matroid of the extended binary Golay code.

EXAMPLES:

sage: M = matroids.catalog.ExtendedBinaryGolayCode(); M
Extended Binary Golay Code: Binary matroid of rank 12 on 24 elements, type (12, 0)
sage: C = LinearCode(M.representation())
sage: C.is_permutation_equivalent(codes.GolayCode(GF(2)))
True
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.ExtendedBinaryGolayCode(); M
Extended Binary Golay Code: Binary matroid of rank 12 on 24 elements, type (12, 0)
>>> C = LinearCode(M.representation())
>>> C.is_permutation_equivalent(codes.GolayCode(GF(Integer(2))))
True
>>> M.is_valid()
True

See also

GolayCode

sage.matroids.database_matroids.ExtendedTernaryGolayCode(groundset='abcdefghijkl')[source]#

Return the matroid of the extended ternary Golay code.

This is the unique Steiner system \(S(5, 6, 12)\).

EXAMPLES:

sage: M = matroids.catalog.ExtendedTernaryGolayCode(); M
Extended Ternary Golay Code: Ternary matroid of rank 6 on 12 elements,
type 6+
sage: C = LinearCode(M.representation())
sage: C.is_permutation_equivalent(codes.GolayCode(GF(3)))
True
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.ExtendedTernaryGolayCode(); M
Extended Ternary Golay Code: Ternary matroid of rank 6 on 12 elements,
type 6+
>>> C = LinearCode(M.representation())
>>> C.is_permutation_equivalent(codes.GolayCode(GF(Integer(3))))
True
>>> M.is_valid()
True

The automorphism group is the \(5\)-transitive Mathieu group \(M12\):

sage: # long time sage: G = M.automorphism_group() sage: G.is_transitive() True sage: G.structure_description() ‘M12’

\(S(5, 6, 12)\) is an identically self-dual matroid:

sage: M.equals(M.dual())
True

>>> from sage.all import *
>>> M.equals(M.dual())
True

Every contraction of three elements is isomorphic to \(AG(2, 3)\); every contraction of two elements and deletion of two elements is isomorphic to \(P8\):

sage: import random, itertools
sage: C = list(itertools.combinations(M.groundset(), 3))
sage: elements = random.choice(C)
sage: N = M.contract(elements)
sage: N.is_isomorphic(matroids.catalog.AG23())
True
sage: C = list(itertools.combinations(M.groundset(), 4))
sage: elements = list(random.choice(C))
sage: random.shuffle(elements)
sage: N = M.contract(elements[:2]).delete(elements[2:4])
sage: N.is_isomorphic(matroids.catalog.P8())
True

>>> from sage.all import *
>>> import random, itertools
>>> C = list(itertools.combinations(M.groundset(), Integer(3)))
>>> elements = random.choice(C)
>>> N = M.contract(elements)
>>> N.is_isomorphic(matroids.catalog.AG23())
True
>>> C = list(itertools.combinations(M.groundset(), Integer(4)))
>>> elements = list(random.choice(C))
>>> random.shuffle(elements)
>>> N = M.contract(elements[:Integer(2)]).delete(elements[Integer(2):Integer(4)])
>>> N.is_isomorphic(matroids.catalog.P8())
True

See also

GolayCode

REFERENCES:

[Oxl2011], p. 658.

sage.matroids.database_matroids.F8(groundset=None)[source]#

Return the matroid \(F_8\), represented as circuit closures.

The matroid \(F_8\) is an \(8\)-element matroid of rank-\(4\). It is a smallest non-representable matroid.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.catalog.F8(); M
F8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'c', 'e', 'g'}, {'a', 'd', 'g', 'h'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'},
     {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: D = get_nonisomorphic_matroids([M.contract(i) for i in M.groundset()])
sage: len(D)
3
sage: [N.is_isomorphic(matroids.catalog.Fano()) for N in D]
[...True...]
sage: [N.is_isomorphic(matroids.catalog.NonFano()) for N in D]
[...True...]
sage: M.is_valid() and M.is_paving()
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import *
>>> M = matroids.catalog.F8(); M
F8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'c', 'e', 'g'}, {'a', 'd', 'g', 'h'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'},
     {'c', 'd', 'e', 'h'}, {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> D = get_nonisomorphic_matroids([M.contract(i) for i in M.groundset()])
>>> len(D)
3
>>> [N.is_isomorphic(matroids.catalog.Fano()) for N in D]
[...True...]
>>> [N.is_isomorphic(matroids.catalog.NonFano()) for N in D]
[...True...]
>>> M.is_valid() and M.is_paving()
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 647.

sage.matroids.database_matroids.FA11(groundset=None)[source]#

Return the matroid \(FA11\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(PT\). In a \(DY^*\)-equivalence class of \(6\) matroids. Has an FF10-minor (delete \(10\)).

EXAMPLES:

sage: FA11 = matroids.catalog.FA11(); FA11
FA11: Quaternary matroid of rank 5 on 11 elements
sage: FF10 = matroids.catalog.FF10()
sage: FF10.is_isomorphic(FA11.delete(10))
True

>>> from sage.all import *
>>> FA11 = matroids.catalog.FA11(); FA11
FA11: Quaternary matroid of rank 5 on 11 elements
>>> FF10 = matroids.catalog.FF10()
>>> FF10.is_isomorphic(FA11.delete(Integer(10)))
True
sage.matroids.database_matroids.FA15(groundset=None)[source]#

Return the matroid \(FA15\).

An excluded minor for \(O\)-representable matroids. UPF is \(PT\). In a \(DY^*\)-equivalence class of \(6\) matroids. Has an SQ14-minor.

EXAMPLES:

sage: M = matroids.catalog.FA15(); M
FA15: Quaternary matroid of rank 7 on 15 elements
sage: M.has_minor(matroids.catalog.N3pp())
True

>>> from sage.all import *
>>> M = matroids.catalog.FA15(); M
FA15: Quaternary matroid of rank 7 on 15 elements
>>> M.has_minor(matroids.catalog.N3pp())
True
sage.matroids.database_matroids.FF10(groundset=None)[source]#

Return the matroid \(FF10\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(P_4\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.FF10(); M
FF10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FF10(); M
FF10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FF12(groundset=None)[source]#

Return the matroid \(FF12\).

An excluded minor for \(P_4\)-representable matroids. Self-dual. UPF is \((Q(a,b),<-1,a,b,a-2,a-1,a+1,b-1,ab-a+b,ab-a-b,ab-a-2b>)\). Has an FF10-minor (contract ‘c’ and delete ‘d’).

EXAMPLES:

sage: M = matroids.catalog.FF12(); M
FF12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True
sage: FF10 = matroids.catalog.FF10()
sage: FF10.is_isomorphic(M.contract('c').delete('d'))
True

>>> from sage.all import *
>>> M = matroids.catalog.FF12(); M
FF12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
>>> FF10 = matroids.catalog.FF10()
>>> FF10.is_isomorphic(M.contract('c').delete('d'))
True
sage.matroids.database_matroids.FK10(groundset=None)[source]#

Return the matroid \(FK10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.FK10(); M
FK10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FK10(); M
FK10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FK12(groundset=None)[source]#

Return the matroid \(FK12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.FK12(); M
FK12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FK12(); M
FK12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FM14(groundset=None)[source]#

Return the matroid \(FM14\).

An excluded minor for \(P_4\)-representable matroids. Self-dual. UPF is \(PT\).

EXAMPLES:

sage: M = matroids.catalog.FM14(); M
FM14: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FM14(); M
FM14: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FN9(groundset=None)[source]#

Return the matroid \(FN9\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids. In a \(DY^*\)-equivalence class of \(10\) matroids. UPF is \(U_1^{(2)}\). (An excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.FN9(); M
FN9: Quaternary matroid of rank 3 on 9 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.FN9(); M
FN9: Quaternary matroid of rank 3 on 9 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.FP10(groundset=None)[source]#

Return the matroid \(FP10\).

An excluded minor for \(K_2\)- and \(G\)-representable matroids (and \(H_2\)- and \(GF(5)\)-representable matroids). UPF is \(U_1^{(2)}\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.FP10(); M
FP10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FP10(); M
FP10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FP12(groundset=None)[source]#

Return the matroid \(FP12\).

An excluded minor for \(K_2\)- and \(G\)-representable matroids (and \(H_2\)- and \(GF(5)\)-representable matroids). UPF is \(W\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.FP12(); M
FP12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FP12(); M
FP12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FQ12(groundset=None)[source]#

Return the matroid \(FQ12\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(PT\). Has` a PP9-minor (contract \(4\) and \(7\), delete \(6\)) and FF10-minor (contract ‘c’ and delete ‘d’).

EXAMPLES:

sage: FQ12 = matroids.catalog.FQ12(); FQ12
FQ12: Quaternary matroid of rank 6 on 12 elements
sage: PP9 = matroids.catalog.PP9()
sage: PP9.is_isomorphic(FQ12.contract([4,7]).delete(6))
True
sage: FF10 = matroids.catalog.FF10()
sage: FF10.is_isomorphic(FQ12.contract('c').delete('d'))
True

>>> from sage.all import *
>>> FQ12 = matroids.catalog.FQ12(); FQ12
FQ12: Quaternary matroid of rank 6 on 12 elements
>>> PP9 = matroids.catalog.PP9()
>>> PP9.is_isomorphic(FQ12.contract([Integer(4),Integer(7)]).delete(Integer(6)))
True
>>> FF10 = matroids.catalog.FF10()
>>> FF10.is_isomorphic(FQ12.contract('c').delete('d'))
True
sage.matroids.database_matroids.FR12(groundset=None)[source]#

Return the matroid \(FR12\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(P_4\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.FR12(); M
FR12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FR12(); M
FR12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FS12(groundset=None)[source]#

Return the matroid \(FS12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Rank \(5\). UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.FS12(); M
FS12: Quaternary matroid of rank 5 on 12 elements
sage: M.rank()
5

>>> from sage.all import *
>>> M = matroids.catalog.FS12(); M
FS12: Quaternary matroid of rank 5 on 12 elements
>>> M.rank()
5
sage.matroids.database_matroids.FT10(groundset=None)[source]#

Return the matroid \(FT10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.FT10(); M
FT10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FT10(); M
FT10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FU10(groundset=None)[source]#

Return the matroid \(FU10\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(G\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.FU10(); M
FU10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.FU10(); M
FU10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.FV14(groundset=None)[source]#

Return the matroid \(FV14\)

An excluded minor for \(P_4\)-representable matroids. Not self-dual. UPF is \(PT\).

EXAMPLES:

sage: M = matroids.catalog.FV14(); M
FV14: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.FV14(); M
FV14: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.FX9(groundset=None)[source]#

Return the matroid \(FX9\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids. UPF is \((Q(a,b), <-1,a,b,a-1,b-1,a-b,a+b,a+b-2,a+b-2ab>)\). (An excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.FX9(); M
FX9: Quaternary matroid of rank 4 on 9 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.FX9(); M
FX9: Quaternary matroid of rank 4 on 9 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.FY10(groundset=None)[source]#

Return the matroid \(FY10\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(G\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.FY10(); M
FY10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.FY10(); M
FY10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.FZ10(groundset=None)[source]#

Return the matroid \(FZ10\).

An excluded minor for \(K_2\)- and \(G\)-representable matroids (and \(H_2\)- and \(GF(5)\)-representable matroids). UPF is \(W\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.FZ10(); M
FZ10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.FZ10(); M
FZ10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.FZ12(groundset=None)[source]#

Return the matroid \(FZ12\).

An excluded minor for \(K_2\)- and \(G\)-representable matroids (and \(H_2\)- and \(GF(5)\)-representable matroids). UPF is \(W\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.FZ12(); M
FZ12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.FZ12(); M
FZ12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.Fano(groundset='abcdefg')[source]#

Return the Fano matroid, represented over \(GF(2)\).

The Fano matroid, or Fano plane, or \(F_7\), is a \(7\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has characteristic two.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.Fano(); M
Fano: Binary matroid of rank 3 on 7 elements, type (3, 0)
sage: M.automorphism_group().is_transitive()
True
sage: M.automorphism_group().structure_description()
'PSL(3,2)'

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.Fano(); M
Fano: Binary matroid of rank 3 on 7 elements, type (3, 0)
>>> M.automorphism_group().is_transitive()
True
>>> M.automorphism_group().structure_description()
'PSL(3,2)'

Every single-element deletion of \(F_7\) is isomorphic to \(M(K_4)\):

sage: setprint(sorted(M.nonspanning_circuits()))
[{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'}, {'b', 'c', 'd'},
 {'b', 'e', 'g'}, {'c', 'f', 'g'}, {'d', 'e', 'f'}]
sage: M.delete(M.groundset_list()[randrange(0,
....:                  7)]).is_isomorphic(matroids.CompleteGraphic(4))
True

>>> from sage.all import *
>>> setprint(sorted(M.nonspanning_circuits()))
[{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'}, {'b', 'c', 'd'},
 {'b', 'e', 'g'}, {'c', 'f', 'g'}, {'d', 'e', 'f'}]
>>> M.delete(M.groundset_list()[randrange(Integer(0),
...                  Integer(7))]).is_isomorphic(matroids.CompleteGraphic(Integer(4)))
True

It is also the projective plane of order two, i.e. \(PG(2, 2)\):

sage: M.is_isomorphic(matroids.PG(2, 2))
True

>>> from sage.all import *
>>> M.is_isomorphic(matroids.PG(Integer(2), Integer(2)))
True

\(F_7\) is isomorphic to the unique binary \(3\)-spike:

sage: M.is_isomorphic(matroids.Z(3))
True

>>> from sage.all import *
>>> M.is_isomorphic(matroids.Z(Integer(3)))
True

REFERENCES:

[Oxl2011], p. 643.

sage.matroids.database_matroids.FanoDual(groundset='abcdefg')[source]#

Return the dual of the Fano matroid.

\(F_7^*\) is a \(7\)-element matroid of rank-\(3\).

EXAMPLES:

sage: F7 = matroids.catalog.Fano()
sage: F7D = matroids.catalog.FanoDual(); F7D
F7*: Binary matroid of rank 4 on 7 elements, type (3, 7)
sage: F7.is_isomorphic(F7D.dual())
True
sage: F7D.automorphism_group().is_transitive()
True
sage: F7D.automorphism_group().structure_description()
'PSL(3,2)'

>>> from sage.all import *
>>> F7 = matroids.catalog.Fano()
>>> F7D = matroids.catalog.FanoDual(); F7D
F7*: Binary matroid of rank 4 on 7 elements, type (3, 7)
>>> F7.is_isomorphic(F7D.dual())
True
>>> F7D.automorphism_group().is_transitive()
True
>>> F7D.automorphism_group().structure_description()
'PSL(3,2)'

Every single-element deletion of \(F_7^*\) is isomorphic to \(M(K_{2, 3})\):

sage: K2_3 = Matroid(graphs.CompleteBipartiteGraph(2, 3))
sage: import random
sage: e = random.choice(list(F7D.groundset()))
sage: F7D.delete(e).is_isomorphic(K2_3)
True

>>> from sage.all import *
>>> K2_3 = Matroid(graphs.CompleteBipartiteGraph(Integer(2), Integer(3)))
>>> import random
>>> e = random.choice(list(F7D.groundset()))
>>> F7D.delete(e).is_isomorphic(K2_3)
True

REFERENCES:

[Oxl2011], p. 643.

sage.matroids.database_matroids.GK10(groundset=None)[source]#

Return the matroid \(GK10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Not self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.GK10(); M
GK10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.GK10(); M
GK10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.GP10(groundset=None)[source]#

Return the matroid \(GP10\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(G\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.GP10(); M
GP10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.GP10(); M
GP10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.GP12(groundset=None)[source]#

Return the matroid \(GP12\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(G\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.GP12(); M
GP12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.GP12(); M
GP12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.J(groundset='abcdefgh')[source]#

Return the matroid \(J\), represented over \(GF(3)\).

The matroid \(J\) is an \(8\)-element matroid of rank-\(4\). It is representable over a field if and only if that field has at least three elements.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.J(); M
J: Ternary matroid of rank 4 on 8 elements, type 0-
sage: setprint(M.truncation().nonbases())
[{'a', 'b', 'f'}, {'a', 'c', 'g'}, {'a', 'd', 'h'}]
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.has_minor(matroids.CompleteGraphic(4))
False
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.J(); M
J: Ternary matroid of rank 4 on 8 elements, type 0-
>>> setprint(M.truncation().nonbases())
[{'a', 'b', 'f'}, {'a', 'c', 'g'}, {'a', 'd', 'h'}]
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.has_minor(matroids.CompleteGraphic(Integer(4)))
False
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 650.

sage.matroids.database_matroids.K33(groundset='abcdefghi')[source]#

Return the graphic matroid \(M(K_{3,3})\).

\(M(K_{3,3})\) is an excluded minor for the class of cographic matroids.

EXAMPLES:

sage: # needs sage.graphs
sage: M = matroids.catalog.K33(); M
M(K3, 3): Regular matroid of rank 5 on 9 elements with 81 bases
sage: M.is_valid()
True
sage: G1 = M.automorphism_group()
sage: G2 = matroids.catalog.K33dual().automorphism_group()
sage: G1.is_isomorphic(G2)
True

>>> from sage.all import *
>>> # needs sage.graphs
>>> M = matroids.catalog.K33(); M
M(K3, 3): Regular matroid of rank 5 on 9 elements with 81 bases
>>> M.is_valid()
True
>>> G1 = M.automorphism_group()
>>> G2 = matroids.catalog.K33dual().automorphism_group()
>>> G1.is_isomorphic(G2)
True

REFERENCES:

[Oxl2011], p. 652-3.

sage.matroids.database_matroids.K33dual(groundset='abcdefghi')[source]#

Return the matroid \(M*(K_{3, 3})\), represented over the regular partial field.

The matroid \(M*(K_{3, 3})\) is a \(9\)-element matroid of rank-\(4\). It is an excluded minor for the class of graphic matroids. It is the graft matroid of the \(4\)-wheel with every vertex except the hub being coloured.

EXAMPLES:

sage: # needs sage.graphs
sage: M = matroids.catalog.K33dual(); M
M*(K3, 3): Regular matroid of rank 4 on 9 elements with 81 bases
sage: any(N.is_3connected() for N in M.linear_extensions(simple=True))
False
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> # needs sage.graphs
>>> M = matroids.catalog.K33dual(); M
M*(K3, 3): Regular matroid of rank 4 on 9 elements with 81 bases
>>> any(N.is_3connected() for N in M.linear_extensions(simple=True))
False
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 652-3.

sage.matroids.database_matroids.K4(groundset='abcdef')[source]#

The graphic matroid of the complete graph \(K_4\).

EXAMPLES:

sage: M = matroids.catalog.K4(); M
M(K4): Graphic matroid of rank 3 on 6 elements
sage: M.is_graphic()
True

>>> from sage.all import *
>>> M = matroids.catalog.K4(); M
M(K4): Graphic matroid of rank 3 on 6 elements
>>> M.is_graphic()
True

\(M(K_4)\) is isomorphic to \(M(\mathcal{W}_3)\), the rank-\(3\) wheel:

sage: W3 = matroids.Wheel(3)
sage: M.is_isomorphic(W3)
True

>>> from sage.all import *
>>> W3 = matroids.Wheel(Integer(3))
>>> M.is_isomorphic(W3)
True

and to the tipless binary \(3\)-spike:

sage: Z = matroids.Z(3, False)
sage: M.is_isomorphic(Z)
True

>>> from sage.all import *
>>> Z = matroids.Z(Integer(3), False)
>>> M.is_isomorphic(Z)
True

It has a transitive automorphism group:

sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 640.

sage.matroids.database_matroids.K5(groundset='abcdefghij')[source]#

Return the graphic matroid \(M(K_5)\).

\(M(K_5)\) is an excluded minor for the class of cographic matroids. It is the \(3\)-dimensional Desargues configuration.

EXAMPLES:

sage: M = matroids.catalog.K5(); M
M(K5): Graphic matroid of rank 4 on 10 elements
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.K5(); M
M(K5): Graphic matroid of rank 4 on 10 elements
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 656.

sage.matroids.database_matroids.K5dual(groundset='abcdefghij')[source]#

Return the matroid \(M^*(K_5)\).

\(M^*(K_5)\) is an excluded minor for the class of graphic matroids.

EXAMPLES:

sage: M = matroids.catalog.K5dual(); M
M*(K5): Dual of 'Graphic matroid of rank 4 on 10 elements'
sage: M.is_3connected()
True
sage: G1 = M.automorphism_group()
sage: G2 = matroids.catalog.K5().automorphism_group()
sage: G1.is_isomorphic(G2)
True

>>> from sage.all import *
>>> M = matroids.catalog.K5dual(); M
M*(K5): Dual of 'Graphic matroid of rank 4 on 10 elements'
>>> M.is_3connected()
True
>>> G1 = M.automorphism_group()
>>> G2 = matroids.catalog.K5().automorphism_group()
>>> G1.is_isomorphic(G2)
True

REFERENCES:

[Oxl2011], p. 656.

sage.matroids.database_matroids.KB12(groundset=None)[source]#

Return the matroid \(KB12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.KB12(); M
KB12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.KB12(); M
KB12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.KF10(groundset=None)[source]#

Return the matroid \(KF10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.KF10(); M
KF10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.KF10(); M
KF10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.KP8(groundset=None)[source]#

Return the matroid \(KP8\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(G\). Self-dual. Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.KP8(); M
KP8: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.KP8(); M
KP8: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.KQ9(groundset=None)[source]#

Return the matroid \(KQ9\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids.) Has a TQ8-minor` (delete \(6\)) and a KP8-minor (delete \(8\)). UPF is \(GF(4)\).

EXAMPLES:

sage: KQ9 = matroids.catalog.KQ9(); KQ9
KQ9: Quaternary matroid of rank 4 on 9 elements
sage: TQ8 = matroids.catalog.TQ8()
sage: TQ8.is_isomorphic(KQ9.delete(6))
True
sage: KP8 = matroids.catalog.KP8()
sage: KP8.is_isomorphic(KQ9.delete(8))
True

>>> from sage.all import *
>>> KQ9 = matroids.catalog.KQ9(); KQ9
KQ9: Quaternary matroid of rank 4 on 9 elements
>>> TQ8 = matroids.catalog.TQ8()
>>> TQ8.is_isomorphic(KQ9.delete(Integer(6)))
True
>>> KP8 = matroids.catalog.KP8()
>>> KP8.is_isomorphic(KQ9.delete(Integer(8)))
True
sage.matroids.database_matroids.KR9(groundset=None)[source]#

Return the matroid \(KR9\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids.) In a \(DY\)-equivalence class of \(4\) matroids. Has a KP8-minor (delete \(8\)). UPF is \(GF(4)\).

EXAMPLES:

sage: KR9 = matroids.catalog.KR9(); KR9
KR9: Quaternary matroid of rank 4 on 9 elements
sage: KP8 = matroids.catalog.KP8()
sage: KP8.is_isomorphic(KR9.delete(8))
True

>>> from sage.all import *
>>> KR9 = matroids.catalog.KR9(); KR9
KR9: Quaternary matroid of rank 4 on 9 elements
>>> KP8 = matroids.catalog.KP8()
>>> KP8.is_isomorphic(KR9.delete(Integer(8)))
True
sage.matroids.database_matroids.KT10(groundset=None)[source]#

Return the matroid \(KT10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.KT10(); M
KT10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.KT10(); M
KT10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.L8(groundset=None)[source]#

Return the matroid \(L_8\), represented as circuit closures.

The matroid \(L_8\) is an \(8\)-element matroid of rank-\(4\). It is representable over all fields with at least five elements. It is a cube, yet it is not a tipless spike.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.L8(); M
L8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'f', 'g'}, {'a', 'c', 'e', 'g'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'd', 'f', 'h'},
     {'c', 'd', 'e', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: M.equals(M.dual())
True
sage: M.is_valid()  # long time
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.L8(); M
L8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'f', 'g'}, {'a', 'c', 'e', 'g'},
     {'a', 'e', 'f', 'h'}, {'b', 'c', 'd', 'g'}, {'b', 'd', 'f', 'h'},
     {'c', 'd', 'e', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> M.equals(M.dual())
True
>>> M.is_valid()  # long time
True
>>> M.automorphism_group().is_transitive()
True

Every single-element contraction is isomorphic to the free extension of \(M(K_4)\):

sage: K4 = matroids.catalog.K4(range(6))
sage: Bext = [list(b) for b in K4.bases()] + [list(I)+[6] for I in
....:                                         K4.independent_sets(2)]
sage: K4ext = Matroid(bases=Bext)
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.contract(e).is_isomorphic(K4ext)
True

>>> from sage.all import *
>>> K4 = matroids.catalog.K4(range(Integer(6)))
>>> Bext = [list(b) for b in K4.bases()] + [list(I)+[Integer(6)] for I in
...                                         K4.independent_sets(Integer(2))]
>>> K4ext = Matroid(bases=Bext)
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.contract(e).is_isomorphic(K4ext)
True

REFERENCES:

[Oxl2011], p. 648.

sage.matroids.database_matroids.LP8(groundset=None)[source]#

Return the matroid \(LP8\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids. Self-dual. UPF is \(W\). (Also an excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.LP8(); M
LP8: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.LP8(); M
LP8: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.M8591(groundset=None)[source]#

Return the matroid \(M8591\).

An excluded minor for \(K_2\)-representable matroids. A \(Y-\delta\) exchange on the unique triad gives A9. The UPF is \(P_4\).

EXAMPLES:

sage: M = matroids.catalog.M8591(); M
M8591: Quaternary matroid of rank 4 on 9 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.M8591(); M
M8591: Quaternary matroid of rank 4 on 9 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.N1(groundset='abcdefghij')[source]#

Return the matroid \(N_1\), represented over \(\GF{3}\).

\(N_1\) is an excluded minor for the dyadic matroids.

EXAMPLES:

sage: M = matroids.catalog.N1(); M
N1: Ternary matroid of rank 5 on 10 elements, type 0+
sage: M.is_field_isomorphic(M.dual())
True
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.N1(); M
N1: Ternary matroid of rank 5 on 10 elements, type 0+
>>> M.is_field_isomorphic(M.dual())
True
>>> M.is_valid()
True

REFERENCES:

[Oxl2011], p. 554.

sage.matroids.database_matroids.N2(groundset='abcdefghijkl')[source]#

Return the matroid \(N_2\), represented over \(\GF{3}\).

\(N_2\) is an excluded minor for the dyadic matroids.

EXAMPLES:

sage: M = matroids.catalog.N2(); M
N2: Ternary matroid of rank 6 on 12 elements, type 0+
sage: M.is_field_isomorphic(M.dual())
True
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.N2(); M
N2: Ternary matroid of rank 6 on 12 elements, type 0+
>>> M.is_field_isomorphic(M.dual())
True
>>> M.is_valid()
True

REFERENCES:

[Oxl2011], p. 554.

sage.matroids.database_matroids.N3(groundset=None)[source]#

Return the matroid \(N3\).

An excluded minor for dyadic matroids (and \(GF(5)\)-representable matroids). UPF is \(GF(3)\). \(4\)- (but not \(5\)-) connected. Self-dual.

EXAMPLES:

sage: N3 = matroids.catalog.N3(); N3
N3: Ternary matroid of rank 7 on 14 elements, type 0+
sage: N3.is_isomorphic(N3.dual())
True
sage: N3.is_kconnected(4)
True
sage: N3.is_kconnected(5)
False

>>> from sage.all import *
>>> N3 = matroids.catalog.N3(); N3
N3: Ternary matroid of rank 7 on 14 elements, type 0+
>>> N3.is_isomorphic(N3.dual())
True
>>> N3.is_kconnected(Integer(4))
True
>>> N3.is_kconnected(Integer(5))
False
sage.matroids.database_matroids.N3pp(groundset=None)[source]#

Return the matroid \(N3pp\).

An excluded minor for \(K_2\)-representable matroids. Self-dual. Obtained by relaxing the two complementary circuit-hyperplanes of N4. Not \(P_4\)-representable, but \(O\)-representable, and hence representable over all fields of size at least four.

EXAMPLES:

sage: M = matroids.catalog.N3pp(); M
N3=: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.N3pp(); M
N3=: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.N4(groundset=None)[source]#

Return the matroid \(N4\).

An excluded minor for dyadic matroids (and \(GF(5)\)-representable matroids). UPF is \(GF(3)\). \(4\)- (but not \(5\)-) connected. Self-dual.

EXAMPLES:

sage: N4 = matroids.catalog.N4(); N4
N4: Ternary matroid of rank 8 on 16 elements, type 0+
sage: N4.is_isomorphic(N4.dual())
True
sage: N4.is_kconnected(4)
True
sage: N4.is_kconnected(5)
False

>>> from sage.all import *
>>> N4 = matroids.catalog.N4(); N4
N4: Ternary matroid of rank 8 on 16 elements, type 0+
>>> N4.is_isomorphic(N4.dual())
True
>>> N4.is_kconnected(Integer(4))
True
>>> N4.is_kconnected(Integer(5))
False
sage.matroids.database_matroids.NestOfTwistedCubes(groundset=None)[source]#

Return the NestOfTwistedCubes matroid.

A matroid with no \(U(2,4)\)-detachable pairs (only \(\{e_i,f_i\}\) pairs are detachable).

EXAMPLES:

sage: M = matroids.catalog.NestOfTwistedCubes(); M
NestOfTwistedCubes: Matroid of rank 6 on 12 elements with 57 circuits
sage: M.is_3connected()
True

>>> from sage.all import *
>>> M = matroids.catalog.NestOfTwistedCubes(); M
NestOfTwistedCubes: Matroid of rank 6 on 12 elements with 57 circuits
>>> M.is_3connected()
True
sage.matroids.database_matroids.NonDesargues(groundset=None)[source]#

Return the NonDesargues matroid.

The NonDesargues matroid is a \(10\)-element matroid of rank-\(3\). It is not representable over any division ring. It is not graphic, not cographic, and not regular.

EXAMPLES:

sage: M = matroids.catalog.NonDesargues(); M
NonDesargues: Matroid of rank 3 on 10 elements with 9 nonspanning circuits
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.NonDesargues(); M
NonDesargues: Matroid of rank 3 on 10 elements with 9 nonspanning circuits
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 657.

sage.matroids.database_matroids.NonFano(groundset='abcdefg')[source]#

Return the non-Fano matroid, represented over \(GF(3)\).

The non-Fano matroid, or \(F_7^-\), is a \(7\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has characteristic other than two. It is the unique relaxation of \(F_7\).

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.NonFano(); M
NonFano: Ternary matroid of rank 3 on 7 elements, type 0-
sage: setprint(M.nonbases())
[{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'},
 {'b', 'c', 'd'}, {'b', 'e', 'g'}, {'c', 'f', 'g'}]
sage: M.delete('f').is_isomorphic(matroids.CompleteGraphic(4))
True
sage: M.delete('g').is_isomorphic(matroids.CompleteGraphic(4))
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.NonFano(); M
NonFano: Ternary matroid of rank 3 on 7 elements, type 0-
>>> setprint(M.nonbases())
[{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'},
 {'b', 'c', 'd'}, {'b', 'e', 'g'}, {'c', 'f', 'g'}]
>>> M.delete('f').is_isomorphic(matroids.CompleteGraphic(Integer(4)))
True
>>> M.delete('g').is_isomorphic(matroids.CompleteGraphic(Integer(4)))
False

REFERENCES:

[Oxl2011], p. 643-4.

sage.matroids.database_matroids.NonFanoDual(groundset='abcdefg')[source]#

Return the dual of the non-Fano matroid.

\((F_7^-)^*\) is a \(7\)-element matroid of rank-\(3\). Every single-element contraction of \((F_7 )^*\) is isomorphic to \(M(K_4)\) or \(\mathcal{W}^3\).

EXAMPLES:

sage: M = matroids.catalog.NonFanoDual(); M
NonFano*: Ternary matroid of rank 4 on 7 elements, type 0-
sage: sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g']

>>> from sage.all import *
>>> M = matroids.catalog.NonFanoDual(); M
NonFano*: Ternary matroid of rank 4 on 7 elements, type 0-
>>> sorted(M.groundset())
['a', 'b', 'c', 'd', 'e', 'f', 'g']

Every single-element contraction of \((F_7^-)^*\) is isomorphic to \(M(K_4)\) or \(\mathcal{W}^3\):

sage: import random
sage: e = random.choice(list(M.groundset()))
sage: N = M.contract(e)
sage: K4 = matroids.catalog.K4()
sage: W3 = matroids.catalog.Whirl3()
sage: N.is_isomorphic(K4) or N.is_isomorphic(W3)
True

>>> from sage.all import *
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> N = M.contract(e)
>>> K4 = matroids.catalog.K4()
>>> W3 = matroids.catalog.Whirl3()
>>> N.is_isomorphic(K4) or N.is_isomorphic(W3)
True

REFERENCES:

[Oxl2011], p. 643-4.

sage.matroids.database_matroids.NonPappus(groundset=None)[source]#

Return the non-Pappus matroid.

The non-Pappus matroid is a \(9\)-element matroid of rank-\(3\). It is not representable over any commutative field. It is the unique relaxation of the Pappus matroid.

EXAMPLES:

sage: M = matroids.catalog.NonPappus(); M
NonPappus: Matroid of rank 3 on 9 elements with 8 nonspanning circuits
sage: NSC = set([('a', 'b', 'c'), ('a', 'e', 'i'), ('a', 'f', 'h'),
....:            ('b', 'd', 'i'), ('b', 'f', 'g'), ('c', 'd', 'h'),
....:            ('c', 'e', 'g'), ('g', 'h', 'i')])
sage: NSC == set(tuple(sorted(C)) for C in M.nonspanning_circuits())
True
sage: M.is_dependent(['d', 'e', 'f'])
False
sage: M.is_valid() and M.is_paving()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.NonPappus(); M
NonPappus: Matroid of rank 3 on 9 elements with 8 nonspanning circuits
>>> NSC = set([('a', 'b', 'c'), ('a', 'e', 'i'), ('a', 'f', 'h'),
...            ('b', 'd', 'i'), ('b', 'f', 'g'), ('c', 'd', 'h'),
...            ('c', 'e', 'g'), ('g', 'h', 'i')])
>>> NSC == set(tuple(sorted(C)) for C in M.nonspanning_circuits())
True
>>> M.is_dependent(['d', 'e', 'f'])
False
>>> M.is_valid() and M.is_paving()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 655.

sage.matroids.database_matroids.NonVamos(groundset=None)[source]#

Return the non-\(V\acute{a}mos\) matroid.

The non-\(V\acute{a}mos\) matroid, or \(V_8^+\) is an \(8\)-element matroid of rank \(4\). It is a tightening of the \(V\acute{a}mos\) matroid. It is representable over some field.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.NonVamos(); M
NonVamos: 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'}, {'c', 'd', 'g', 'h'}, {'e', 'f', 'g', 'h'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: setprint(M.nonbases())
[{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'}, {'a', 'b', 'g', 'h'},
 {'c', 'd', 'e', 'f'}, {'c', 'd', 'g', 'h'}, {'e', 'f', 'g', 'h'}]
sage: M.is_dependent(['c', 'd', 'g', 'h'])
True
sage: M.is_valid()  # long time
True

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.NonVamos(); M
NonVamos: 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'}, {'c', 'd', 'g', 'h'}, {'e', 'f', 'g', 'h'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> setprint(M.nonbases())
[{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'}, {'a', 'b', 'g', 'h'},
 {'c', 'd', 'e', 'f'}, {'c', 'd', 'g', 'h'}, {'e', 'f', 'g', 'h'}]
>>> M.is_dependent(['c', 'd', 'g', 'h'])
True
>>> M.is_valid()  # long time
True

REFERENCES:

[Oxl2011], p. 72, 84.

sage.matroids.database_matroids.NotP8(groundset='abcdefgh')[source]#

Return the matroid NotP8.

EXAMPLES:

sage: M = matroids.catalog.P8()
sage: N = matroids.catalog.NotP8(); N
NotP8: Ternary matroid of rank 4 on 8 elements, type 0-
sage: M.is_isomorphic(N)
False
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.P8()
>>> N = matroids.catalog.NotP8(); N
NotP8: Ternary matroid of rank 4 on 8 elements, type 0-
>>> M.is_isomorphic(N)
False
>>> M.is_valid()
True

REFERENCES:

[Oxl1992], p.512 (the first edition).

sage.matroids.database_matroids.O7(groundset='abcdefg')[source]#

Return the matroid \(O_7\), represented over \(GF(3)\).

The matroid \(O_7\) is a \(7\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has at least three elements. It is obtained by freely adding a point to any line of \(M(K_4)\).

EXAMPLES:

sage: M = matroids.catalog.O7(); M
O7: Ternary matroid of rank 3 on 7 elements, type 0+
sage: M.delete('e').is_isomorphic(matroids.CompleteGraphic(4))
True
sage: M.tutte_polynomial()
y^4 + x^3 + x*y^2 + 3*y^3 + 4*x^2 + 5*x*y + 5*y^2 + 4*x + 4*y

>>> from sage.all import *
>>> M = matroids.catalog.O7(); M
O7: Ternary matroid of rank 3 on 7 elements, type 0+
>>> M.delete('e').is_isomorphic(matroids.CompleteGraphic(Integer(4)))
True
>>> M.tutte_polynomial()
y^4 + x^3 + x*y^2 + 3*y^3 + 4*x^2 + 5*x*y + 5*y^2 + 4*x + 4*y

REFERENCES:

[Oxl2011], p. 644.

sage.matroids.database_matroids.OW14(groundset=None)[source]#

Return the matroid \(OW14\).

An excluded minor for \(P_4\)-representable matroids. Self-dual. UPF is \(Orthrus\).

EXAMPLES:

sage: M = matroids.catalog.OW14(); M
OW14: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.OW14(); M
OW14: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.P6(groundset=None)[source]#

Return the matroid \(P_6\), represented as circuit closures.

The matroid \(P_6\) is a \(6\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has at least five elements. It is the unique relaxation of \(Q_6\). It is an excluded minor for the class of quaternary matroids.

EXAMPLES:

sage: M = matroids.catalog.P6(); M
P6: Matroid of rank 3 on 6 elements with circuit-closures
{2: {{'a', 'b', 'c'}}, 3: {{'a', 'b', 'c', 'd', 'e', 'f'}}}
sage: len(set(M.nonspanning_circuits()).difference(M.nonbases())) == 0
True
sage: Matroid(matrix=random_matrix(GF(4, 'a'), ncols=5, nrows=5)).has_minor(M)
False
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.P6(); M
P6: Matroid of rank 3 on 6 elements with circuit-closures
{2: {{'a', 'b', 'c'}}, 3: {{'a', 'b', 'c', 'd', 'e', 'f'}}}
>>> len(set(M.nonspanning_circuits()).difference(M.nonbases())) == Integer(0)
True
>>> Matroid(matrix=random_matrix(GF(Integer(4), 'a'), ncols=Integer(5), nrows=Integer(5))).has_minor(M)
False
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 641-2.

sage.matroids.database_matroids.P7(groundset='abcdefg')[source]#

Return the matroid \(P_7\), represented over \(GF(3)\).

The matroid \(P_7\) is a \(7\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has at least three elements. It is one of two ternary \(3\)-spikes, with the other being \(F_7^-\).

EXAMPLES:

sage: M = matroids.catalog.P7(); M
P7: Ternary matroid of rank 3 on 7 elements, type 1+
sage: M.whitney_numbers2()
[1, 7, 11, 1]
sage: M.has_minor(matroids.CompleteGraphic(4))
False
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.P7(); M
P7: Ternary matroid of rank 3 on 7 elements, type 1+
>>> M.whitney_numbers2()
[1, 7, 11, 1]
>>> M.has_minor(matroids.CompleteGraphic(Integer(4)))
False
>>> M.is_valid()
True

REFERENCES:

[Oxl2011], p. 644-5.

sage.matroids.database_matroids.P8(groundset='abcdefgh')[source]#

Return the matroid \(P_8\), represented over \(GF(3)\).

The matroid \(P_8\) is an \(8\)-element matroid of rank-\(4\). It is uniquely representable over all fields of characteristic other than two. It is an excluded minor for all fields of characteristic two with four or more elements.

EXAMPLES:

sage: M = matroids.catalog.P8(); M
P8: Ternary matroid of rank 4 on 8 elements, type 2+
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: Matroid(matrix=random_matrix(GF(4, 'a'), ncols=5, nrows=5)).has_minor(M)
False
sage: M.bicycle_dimension()
2

>>> from sage.all import *
>>> M = matroids.catalog.P8(); M
P8: Ternary matroid of rank 4 on 8 elements, type 2+
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> Matroid(matrix=random_matrix(GF(Integer(4), 'a'), ncols=Integer(5), nrows=Integer(5))).has_minor(M)
False
>>> M.bicycle_dimension()
2

REFERENCES:

[Oxl2011], p. 650-1.

sage.matroids.database_matroids.P8p(groundset=None)[source]#

Return the matroid \(P8^-\).

\(P8^-\) is obtained by relaxing one of the disjoint circuit-hyperplanes of P8. An excluded minor for \(2\)-regular matroids. UPF is \(K_2\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.P8p(); M
P8-: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.P8p(); M
P8-: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.P8pp(groundset=None)[source]#

Return the matroid \(P_8^=\), represented as circuit closures.

The matroid \(P_8^=\) is an \(8\)-element matroid of rank-\(4\). It can be obtained from \(P_8\) by relaxing the unique pair of disjoint circuit-hyperplanes. It is an excluded minor for \(GF(4)\)-representability. It is representable over all fields with at least five elements.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.catalog.P8pp(); M
P8'': Matroid of rank 4 on 8 elements with 8 nonspanning circuits
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: len(get_nonisomorphic_matroids([M.contract(i) for i in M.groundset()]))
1
sage: M.is_valid() and M.is_paving()
True

>>> from sage.all import *
>>> from sage.matroids.advanced import *
>>> M = matroids.catalog.P8pp(); M
P8'': Matroid of rank 4 on 8 elements with 8 nonspanning circuits
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> len(get_nonisomorphic_matroids([M.contract(i) for i in M.groundset()]))
1
>>> M.is_valid() and M.is_paving()
True

REFERENCES:

[Oxl2011], p. 651.

sage.matroids.database_matroids.P9(groundset='abcdefghi')[source]#

Return the matroid \(P_9\).

EXAMPLES:

sage: M = matroids.catalog.P9(); M
P9: Binary matroid of rank 4 on 9 elements, type (1, 1)
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.P9(); M
P9: Binary matroid of rank 4 on 9 elements, type (1, 1)
>>> M.is_valid()
True

REFERENCES:

This is the matroid referred to as \(P_9\) by Oxley in his paper “The binary matroids with no 4-wheel minor”, [Oxl1987].

sage.matroids.database_matroids.PG(n, q, x=None, groundset=None)[source]#

Return the projective geometry of dimension n over the finite field of order q.

INPUT:

  • n – a positive integer; the dimension of the projective space. This is one less than the rank of the resulting matroid.

  • q – a positive integer that is a prime power; the order of the finite field

  • x – a string (default: None); the name of the generator of a non-prime field, used for non-prime fields. If not supplied, 'x' is used.

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: a linear matroid whose elements are the points of \(PG(n, q)\)

EXAMPLES:

sage: M = matroids.PG(2, 2)
sage: M.is_isomorphic(matroids.catalog.Fano())
True
sage: matroids.PG(5, 4, 'z').size() == (4^6 - 1) / (4 - 1)
True
sage: M = matroids.PG(4, 7); M
PG(4, 7): Linear matroid of rank 5 on 2801 elements represented over the Finite Field
 of size 7

>>> from sage.all import *
>>> M = matroids.PG(Integer(2), Integer(2))
>>> M.is_isomorphic(matroids.catalog.Fano())
True
>>> matroids.PG(Integer(5), Integer(4), 'z').size() == (Integer(4)**Integer(6) - Integer(1)) / (Integer(4) - Integer(1))
True
>>> M = matroids.PG(Integer(4), Integer(7)); M
PG(4, 7): Linear matroid of rank 5 on 2801 elements represented over the Finite Field
 of size 7

REFERENCES:

[Oxl2011], p. 660.

sage.matroids.database_matroids.PG23(groundset=None)[source]#

Return the matroid \(PG23\).

The second smallest projective plane. Not graphic, not cographic, not regular, not near-regular.

EXAMPLES:

sage: M = matroids.catalog.PG23(); M
PG(2, 3): Ternary matroid of rank 3 on 13 elements, type 3+
sage: M.is_3connected()
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.PG23(); M
PG(2, 3): Ternary matroid of rank 3 on 13 elements, type 3+
>>> M.is_3connected()
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 659.

sage.matroids.database_matroids.PK10(groundset=None)[source]#

Return the matroid \(PK10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Not self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.PK10(); M
PK10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.PK10(); M
PK10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.PP10(groundset=None)[source]#

Return the matroid \(PP10\).

An excluded minor for \(P_4\)-representable matroids. UPF is \(U_1^{(2)}\). Has a TQ8-minor (e.g. delete ‘a’ and contract ‘e’) and a PP9 (and hence P8p) minor (contract ‘x’).

EXAMPLES:

sage: PP10 = matroids.catalog.PP10(); PP10
PP10: Quaternary matroid of rank 5 on 10 elements
sage: M = PP10.delete('a').contract('e')
sage: M.is_isomorphic(matroids.catalog.TQ8())
True
sage: M = PP10.contract('x')
sage: M.is_isomorphic(matroids.catalog.PP9())
True

>>> from sage.all import *
>>> PP10 = matroids.catalog.PP10(); PP10
PP10: Quaternary matroid of rank 5 on 10 elements
>>> M = PP10.delete('a').contract('e')
>>> M.is_isomorphic(matroids.catalog.TQ8())
True
>>> M = PP10.contract('x')
>>> M.is_isomorphic(matroids.catalog.PP9())
True
sage.matroids.database_matroids.PP9(groundset=None)[source]#

Return the matroid \(PP9\).

An excluded minor for \(K_2\)-representable matroids. A single-element extension of \(P8^-\). The UPF is \(P_4\). Has a P8p-minor (delete \(z\)). Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: P8p = matroids.catalog.P8p()
sage: PP9 = matroids.catalog.PP9(); PP9
PP9: Quaternary matroid of rank 4 on 9 elements
sage: for M in P8p.extensions():
....:     if M.is_isomorphic(PP9):
....:         print(True)
....:         break
True
sage: M = PP9.delete('z')
sage: M.is_isomorphic(P8p)
True

>>> from sage.all import *
>>> P8p = matroids.catalog.P8p()
>>> PP9 = matroids.catalog.PP9(); PP9
PP9: Quaternary matroid of rank 4 on 9 elements
>>> for M in P8p.extensions():
...     if M.is_isomorphic(PP9):
...         print(True)
...         break
True
>>> M = PP9.delete('z')
>>> M.is_isomorphic(P8p)
True
sage.matroids.database_matroids.Pappus(groundset=None)[source]#

Return the Pappus matroid.

The Pappus matroid is a \(9\)-element matroid of rank-\(3\). It is representable over a field if and only if that field either has 4 elements or more than 7 elements. It is an excluded minor for the class of GF(5)-representable matroids.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.Pappus(); M
Pappus: Matroid of rank 3 on 9 elements with 9 nonspanning circuits
sage: setprint(M.nonspanning_circuits())
[{'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'}]
sage: M.is_dependent(['d', 'e', 'f'])
True
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.Pappus(); M
Pappus: Matroid of rank 3 on 9 elements with 9 nonspanning circuits
>>> setprint(M.nonspanning_circuits())
[{'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'}]
>>> M.is_dependent(['d', 'e', 'f'])
True
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 655.

sage.matroids.database_matroids.Psi(r, groundset=None)[source]#

Return the matroid \(\Psi_r\).

The rank-\(r\) free swirl; defined for all \(r \ge 3\).

INPUT:

  • r – an integer (\(r \ge 3\)); the rank of the matroid

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: matroid (\(\Psi_r\))

EXAMPLES:

sage: matroids.Psi(7)
Psi_7: Matroid of rank 7 on 14 elements with 105 nonspanning circuits

>>> from sage.all import *
>>> matroids.Psi(Integer(7))
Psi_7: Matroid of rank 7 on 14 elements with 105 nonspanning circuits

The matroid \(\Psi_r\) is \(3\)-connected but, for all \(r \ge 4\), not \(4\)-connected:

sage: M = matroids.Psi(3)
sage: M.is_4connected()
True
sage: import random
sage: r = random.choice(range(4, 8))
sage: M = matroids.Psi(r)
sage: M.is_4connected()
False

>>> from sage.all import *
>>> M = matroids.Psi(Integer(3))
>>> M.is_4connected()
True
>>> import random
>>> r = random.choice(range(Integer(4), Integer(8)))
>>> M = matroids.Psi(r)
>>> M.is_4connected()
False

\(\Psi_3 \cong U_{3, 6}\):

sage: M = matroids.Psi(3)
sage: M.is_isomorphic(matroids.catalog.U36())
True

>>> from sage.all import *
>>> M = matroids.Psi(Integer(3))
>>> M.is_isomorphic(matroids.catalog.U36())
True

It is identically self-dual with a transitive automorphism group:

sage: M = matroids.Psi(r)
sage: M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()  # long time
True

>>> from sage.all import *
>>> M = matroids.Psi(r)
>>> M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()  # long time
True

REFERENCES:

[Oxl2011], p. 664.

sage.matroids.database_matroids.Q10(groundset='abcdefghij')[source]#

Return the matroid \(Q_{10}\), represented over \(\GF{4}\).

\(Q_{10}\) is a \(10\)-element, rank-\(5\), self-dual matroid. It is representable over \(\GF{3}\) and \(\GF{4}\), and hence is a sixth-roots-of-unity matroid. \(Q_{10}\) is a splitter for the class of sixth-root-of-unity matroids.

EXAMPLES:

sage: M = matroids.catalog.Q10(); M
Q10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.Q10(); M
Q10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
>>> M.is_valid()
True

Check the splitter property. By Seymour’s Theorem, and using self-duality, we only need to check that all 3-connected single-element extensions have an excluded minor for sixth-roots-of-unity. The only excluded minors that are quaternary are \(U_{2, 5}, U_{3, 5}, F_7, F_7^*\). As it happens, it suffices to check for \(U_{2, 5}\):

sage: S = matroids.catalog.Q10().linear_extensions(simple=True)
sage: [M for M in S if not M.has_line_minor(5)]
[]

>>> from sage.all import *
>>> S = matroids.catalog.Q10().linear_extensions(simple=True)
>>> [M for M in S if not M.has_line_minor(Integer(5))]
[]
sage.matroids.database_matroids.Q6(groundset='abcdef')[source]#

Return the matroid \(Q_6\), represented over \(GF(4)\).

The matroid \(Q_6\) is a \(6\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has at least four elements. It is the unique relaxation of the rank-\(3\) whirl.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.Q6(); M
Q6: Quaternary matroid of rank 3 on 6 elements
sage: setprint(M.hyperplanes())
[{'a', 'b', 'd'}, {'a', 'c'}, {'a', 'e'}, {'a', 'f'}, {'b', 'c', 'e'},
 {'b', 'f'}, {'c', 'd'}, {'c', 'f'}, {'d', 'e'}, {'d', 'f'},
 {'e', 'f'}]
sage: M.nonspanning_circuits() == M.noncospanning_cocircuits()
False
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.Q6(); M
Q6: Quaternary matroid of rank 3 on 6 elements
>>> setprint(M.hyperplanes())
[{'a', 'b', 'd'}, {'a', 'c'}, {'a', 'e'}, {'a', 'f'}, {'b', 'c', 'e'},
 {'b', 'f'}, {'c', 'd'}, {'c', 'f'}, {'d', 'e'}, {'d', 'f'},
 {'e', 'f'}]
>>> M.nonspanning_circuits() == M.noncospanning_cocircuits()
False
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 641.

sage.matroids.database_matroids.Q8(groundset=None)[source]#

Return the matroid \(Q_8\), represented as circuit closures.

The matroid \(Q_8\) is an \(8\)-element matroid of rank-\(4\). It is a smallest non-representable matroid.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.Q8(); M
Q8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'},
     {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'c', 'd', 'e', 'h'},
     {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: setprint(M.flats(3))
[{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
 {'a', 'c', 'd', 'f'}, {'a', 'c', 'e'}, {'a', 'c', 'g'},
 {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'}, {'a', 'e', 'g'},
 {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'b', 'd', 'f'},
 {'b', 'd', 'h'}, {'b', 'e', 'g'}, {'b', 'e', 'h'}, {'b', 'f', 'h'},
 {'b', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'e', 'g'},
 {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}, {'d', 'f', 'h'},
 {'e', 'g', 'h'}]
sage: M.is_valid()  # long time
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.Q8(); M
Q8: Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
     {'a', 'c', 'd', 'f'}, {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'},
     {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'c', 'd', 'e', 'h'},
     {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
>>> setprint(M.flats(Integer(3)))
[{'a', 'b', 'c', 'h'}, {'a', 'b', 'd', 'e'}, {'a', 'b', 'f', 'g'},
 {'a', 'c', 'd', 'f'}, {'a', 'c', 'e'}, {'a', 'c', 'g'},
 {'a', 'd', 'g', 'h'}, {'a', 'e', 'f', 'h'}, {'a', 'e', 'g'},
 {'b', 'c', 'd', 'g'}, {'b', 'c', 'e', 'f'}, {'b', 'd', 'f'},
 {'b', 'd', 'h'}, {'b', 'e', 'g'}, {'b', 'e', 'h'}, {'b', 'f', 'h'},
 {'b', 'g', 'h'}, {'c', 'd', 'e', 'h'}, {'c', 'e', 'g'},
 {'c', 'f', 'g', 'h'}, {'d', 'e', 'f', 'g'}, {'d', 'f', 'h'},
 {'e', 'g', 'h'}]
>>> M.is_valid()  # long time
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 647.

sage.matroids.database_matroids.R10(groundset='abcdefghij')[source]#

Return the matroid \(R_{10}\), represented over the regular partial field.

The NonDesargues matroid is a \(10\)-element matroid of rank-5. It is the unique splitter for the class of regular matroids. It is the graft matroid of \(K_{3, 3}\) in which every vertex is coloured.

EXAMPLES:

sage: M = matroids.catalog.R10(); M
R10: Regular matroid of rank 5 on 10 elements with 162 bases
sage: cct = []
sage: for i in M.circuits():
....:     cct.append(len(i))
sage: Set(cct)
{4, 6}
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.R10(); M
R10: Regular matroid of rank 5 on 10 elements with 162 bases
>>> cct = []
>>> for i in M.circuits():
...     cct.append(len(i))
>>> Set(cct)
{4, 6}
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
True

Every single-element deletion is isomorphic to \(M(K_{3, 3})\), and every single-element contraction is isomorphic to \(M^*(K_{3, 3})\):

sage: K33 = matroids.catalog.K33()
sage: K33D = matroids.catalog.K33dual()
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.delete(e).is_isomorphic(K33)
True
sage: M.contract(e).is_isomorphic(K33D)
True

>>> from sage.all import *
>>> K33 = matroids.catalog.K33()
>>> K33D = matroids.catalog.K33dual()
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.delete(e).is_isomorphic(K33)
True
>>> M.contract(e).is_isomorphic(K33D)
True

Check the splitter property:

sage: matroids.catalog.R10().linear_extensions(simple=True)
[]

>>> from sage.all import *
>>> matroids.catalog.R10().linear_extensions(simple=True)
[]

REFERENCES:

[Oxl2011], p. 656-7.

sage.matroids.database_matroids.R12(groundset='abcdefghijkl')[source]#

Return the matroid \(R_{12}\), represented over the regular partial field.

The matroid \(R_{12}\) is a \(12\)-element regular matroid of rank-\(6\). It induces a \(3\)-separation in its \(3\)-connected majors within the class of regular matroids. An excluded minor for the class of graphic or cographic matroids.

EXAMPLES:

sage: M = matroids.catalog.R12(); M
R12: Regular matroid of rank 6 on 12 elements with 441 bases
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.is_valid()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.R12(); M
R12: Regular matroid of rank 6 on 12 elements with 441 bases
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.is_valid()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 657.

sage.matroids.database_matroids.R6(groundset='abcdef')[source]#

Return the matroid \(R_6\), represented over \(GF(3)\).

The matroid \(R_6\) is a \(6\)-element matroid of rank-\(3\). It is representable over a field if and only if that field has at least three elements. It is isomorphic to the \(2\)-sum of two copies of \(U_{2, 4}\).

EXAMPLES:

sage: M = matroids.catalog.R6(); M
R6: Ternary matroid of rank 3 on 6 elements, type 2+
sage: M.equals(M.dual())
True
sage: M.is_connected()
True
sage: M.is_3connected()
False
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.R6(); M
R6: Ternary matroid of rank 3 on 6 elements, type 2+
>>> M.equals(M.dual())
True
>>> M.is_connected()
True
>>> M.is_3connected()
False
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 642.

sage.matroids.database_matroids.R8(groundset='abcdefgh')[source]#

Return the matroid \(R_8\), represented over \(GF(3)\).

The matroid \(R_8\) is an \(8\)-element matroid of rank-\(4\). It is representable over a field if and only if the characteristic of that field is not two. It is the real affine cube.

EXAMPLES:

sage: M = matroids.catalog.R8(); M
R8: Ternary matroid of rank 4 on 8 elements, type 0+
sage: M.contract(M.groundset_list()[randrange(0,
....:            8)]).is_isomorphic(matroids.catalog.NonFano())
True
sage: M.equals(M.dual())
True
sage: M.has_minor(matroids.catalog.Fano())
False
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.R8(); M
R8: Ternary matroid of rank 4 on 8 elements, type 0+
>>> M.contract(M.groundset_list()[randrange(Integer(0),
...            Integer(8))]).is_isomorphic(matroids.catalog.NonFano())
True
>>> M.equals(M.dual())
True
>>> M.has_minor(matroids.catalog.Fano())
False
>>> M.automorphism_group().is_transitive()
True

Every single-element deletion is isomorphic to (F_7^-)^* and every single-element contraction is isomorphic to F_7^-:

sage: F7m = matroids.catalog.NonFano()
sage: F7mD = matroids.catalog.NonFanoDual()
sage: import random
sage: e = random.choice(list(M.groundset()))
sage: M.delete(e).is_isomorphic(F7mD)
True
sage: M.contract(e).is_isomorphic(F7m)
True

>>> from sage.all import *
>>> F7m = matroids.catalog.NonFano()
>>> F7mD = matroids.catalog.NonFanoDual()
>>> import random
>>> e = random.choice(list(M.groundset()))
>>> M.delete(e).is_isomorphic(F7mD)
True
>>> M.contract(e).is_isomorphic(F7m)
True

REFERENCES:

[Oxl2011], p. 646.

sage.matroids.database_matroids.R9(groundset=None)[source]#

Return the matroid \(R_9\).

The ternary Reid geometry. The only \(9\)-element rank-\(3\) simple ternary matroids are \(R_9\), \(Q_3(GF(3)^ imes)\), and \(AG(2, 3)\). It is not graphic, not cographic, and not regular.

EXAMPLES:

sage: M = matroids.catalog.R9(); M
R9: Matroid of rank 3 on 9 elements with 15 nonspanning circuits
sage: M.is_valid()
True
sage: len(M.nonspanning_circuits())
15
sage: M.is_simple() and M.is_ternary()
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.R9(); M
R9: Matroid of rank 3 on 9 elements with 15 nonspanning circuits
>>> M.is_valid()
True
>>> len(M.nonspanning_circuits())
15
>>> M.is_simple() and M.is_ternary()
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 654.

sage.matroids.database_matroids.R9A(groundset=None)[source]#

Return the matroid \(R_9^A\).

The matroid \(R_9^A\) is not representable over any field, yet none of the cross-ratios in its Tuttegroup equal 1. It is one of the 4 matroids on at most 9 elements with this property, the others being \({R_9^A}^*\), \(R_9^B\) and \({R_9^B}^*\).

EXAMPLES:

sage: M = matroids.catalog.R9A(); M
R9A: Matroid of rank 4 on 9 elements with 13 nonspanning circuits
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.R9A(); M
R9A: Matroid of rank 4 on 9 elements with 13 nonspanning circuits
>>> M.is_valid()
True
sage.matroids.database_matroids.R9B(groundset=None)[source]#

Return the matroid \(R_9^B\).

The matroid \(R_9^B\) is not representable over any field, yet none of the cross-ratios in its Tuttegroup equal 1. It is one of the 4 matroids on at most 9 elements with this property, the others being \({R_9^B}^*\), \(R_9^A\) and \({R_9^A}^*\).

EXAMPLES:

sage: M = matroids.catalog.R9B(); M
R9B: Matroid of rank 4 on 9 elements with 13 nonspanning circuits
sage: M.is_valid() and M.is_paving()
True

>>> from sage.all import *
>>> M = matroids.catalog.R9B(); M
R9B: Matroid of rank 4 on 9 elements with 13 nonspanning circuits
>>> M.is_valid() and M.is_paving()
True
sage.matroids.database_matroids.RelaxedNonFano(groundset=None)[source]#

Return the relaxed NonFano matroid.

An excluded minor for \(2\)-regular matroids. UPF is \(K_2\).

EXAMPLES:

sage: M = matroids.catalog.RelaxedNonFano(); M
F7=: Quaternary matroid of rank 3 on 7 elements
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.RelaxedNonFano(); M
F7=: Quaternary matroid of rank 3 on 7 elements
>>> M.is_valid()
True
sage.matroids.database_matroids.S8(groundset='abcdefgh')[source]#

Return the matroid \(S_8\), represented over \(GF(2)\).

The matroid \(S_8\) is an \(8\)-element matroid of rank-\(4\). It is representable over a field if and only if that field has characteristic two. It is the unique deletion of a non-tip element from the binary \(4\)-spike.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.catalog.S8(); M
S8: Binary matroid of rank 4 on 8 elements, type (2, 0)
sage: M.contract('d').is_isomorphic(matroids.catalog.Fano())
True
sage: M.delete('d').is_isomorphic(matroids.catalog.FanoDual())
False
sage: M.delete('h').is_isomorphic(matroids.catalog.FanoDual())
True
sage: M.is_graphic()
False
sage: D = get_nonisomorphic_matroids(
....:       list(matroids.catalog.Fano().linear_coextensions(cosimple=True)))
sage: len(D)
2
sage: [N.is_isomorphic(M) for N in D]
[...True...]
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import *
>>> M = matroids.catalog.S8(); M
S8: Binary matroid of rank 4 on 8 elements, type (2, 0)
>>> M.contract('d').is_isomorphic(matroids.catalog.Fano())
True
>>> M.delete('d').is_isomorphic(matroids.catalog.FanoDual())
False
>>> M.delete('h').is_isomorphic(matroids.catalog.FanoDual())
True
>>> M.is_graphic()
False
>>> D = get_nonisomorphic_matroids(
...       list(matroids.catalog.Fano().linear_coextensions(cosimple=True)))
>>> len(D)
2
>>> [N.is_isomorphic(M) for N in D]
[...True...]
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 648.

sage.matroids.database_matroids.Sp8(groundset=None)[source]#

Return the matroid \(Sp8\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids. UPF is \(U_1^{(2)}\). Self-dual. (An excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.Sp8(); M
Sp8: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.Sp8(); M
Sp8: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.Sp8pp(groundset=None)[source]#

Return the matroid \(Sp8=\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids. UPF is \((GF(2)(a,b),<a,b,a+1,b+1,ab+a+b>)\). Self-dual. (An excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: M = matroids.catalog.Sp8pp(); M
Sp8=: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.Sp8pp(); M
Sp8=: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.Spike(r, t=True, C3=[], groundset=None)[source]#

Return a rank-\(r\) spike.

Defined for all \(r \ge 3\); a rank-\(r\) spike with tip \(t\) and legs \(L_1, L_2, \ldots, L_r\), where \(L_i = \{t, x_i , y_i\}\). Deleting \(t\) gives a tipless rank-\(r\) spike.

The groundset is \(E = \{t, x_1, x_2, \ldots, x_r, y_1, y_2, \ldots, y_r\}\) with \(r(E) = r\).

The nonspanning circuits are \(\{L_1, L_2, \ldots, L_r\}\), all sets of the form \((L_i \cup L_j) \setminus t\) for \(1 \le i < j \le r\), and some (possibly empty) collection \(C_3\) of sets of the form \(\{z_1, z_2, \ldots, z_r\}\) where \(z_i \in \{x_i, y_i\}\) for all \(i\), and no two members of \(C_3\) have more than \(r-2\) common elements.

INPUT:

  • r – an integer (\(r \ge 3\)); the rank of the spike

  • t – boolean (default: True); whether the spike is tipped

  • C3 – a list (default: []); a list of extra nonspanning circuits. The default (i.e. the empty list) results in a free \(r\)-spike

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: matroid; a rank-\(r\) spike (tipped or tipless)

EXAMPLES:

sage: M = matroids.Spike(3, False); M
Free 3-spike\t: M \ {'t'}, where M is Matroid of rank 3 on 7 elements with 3
 nonspanning circuits
sage: M.is_isomorphic(matroids.Uniform(3, 6))
True
sage: len(matroids.Spike(8).bases())
4864
sage: import random
sage: r = random.choice(range(3, 20))
sage: M = matroids.Spike(r)
sage: M.is_3connected()
True

>>> from sage.all import *
>>> M = matroids.Spike(Integer(3), False); M
Free 3-spike\t: M \ {'t'}, where M is Matroid of rank 3 on 7 elements with 3
 nonspanning circuits
>>> M.is_isomorphic(matroids.Uniform(Integer(3), Integer(6)))
True
>>> len(matroids.Spike(Integer(8)).bases())
4864
>>> import random
>>> r = random.choice(range(Integer(3), Integer(20)))
>>> M = matroids.Spike(r)
>>> M.is_3connected()
True

Each of \(F_7\), \(F_7^-\), and \(P_7\), is a 3-spike. After inspection of the nonspanning circuits of these matroids, it becomes clear that they indeed constitute tipped 3-spikes. This can be verified by using an appropriate choice of extra circuits in \(C_3\):

sage: M = matroids.Spike(3, C3=[['x1', 'x2', 'y3'],
....:                           ['x1', 'x3', 'y2'],
....:                           ['x2', 'x3', 'y1'],
....:                           ['y1', 'y2', 'y3']])
sage: M.is_isomorphic(matroids.catalog.Fano())
True
sage: M = matroids.Spike(3, C3=[['x1', 'x2', 'x3'],
....:                           ['x1', 'y2', 'y3'],
....:                           ['x2', 'y1', 'y3']])
sage: M.is_isomorphic(matroids.catalog.NonFano())
True
sage: M = matroids.Spike(3, C3=[['x1', 'x2', 'y3'],
....:                           ['x3', 'y1', 'y2']])
sage: M.is_isomorphic(matroids.catalog.P7())
True

>>> from sage.all import *
>>> M = matroids.Spike(Integer(3), C3=[['x1', 'x2', 'y3'],
...                           ['x1', 'x3', 'y2'],
...                           ['x2', 'x3', 'y1'],
...                           ['y1', 'y2', 'y3']])
>>> M.is_isomorphic(matroids.catalog.Fano())
True
>>> M = matroids.Spike(Integer(3), C3=[['x1', 'x2', 'x3'],
...                           ['x1', 'y2', 'y3'],
...                           ['x2', 'y1', 'y3']])
>>> M.is_isomorphic(matroids.catalog.NonFano())
True
>>> M = matroids.Spike(Integer(3), C3=[['x1', 'x2', 'y3'],
...                           ['x3', 'y1', 'y2']])
>>> M.is_isomorphic(matroids.catalog.P7())
True

Deleting any element gives a self-dual matroid. The tipless free spike (i.e., when \(C_3\) is empty) is identically self-dual:

sage: M = matroids.Spike(6)
sage: e = random.choice(list(M.groundset()))
sage: Minor = M.delete(e)
sage: Minor.is_isomorphic(Minor.dual())
True
sage: r = random.choice(range(3, 8))
sage: M = matroids.Spike(r, False)
sage: M.equals(M.dual())
True

>>> from sage.all import *
>>> M = matroids.Spike(Integer(6))
>>> e = random.choice(list(M.groundset()))
>>> Minor = M.delete(e)
>>> Minor.is_isomorphic(Minor.dual())
True
>>> r = random.choice(range(Integer(3), Integer(8)))
>>> M = matroids.Spike(r, False)
>>> M.equals(M.dual())
True

REFERENCES:

[Oxl2011], p. 662.

sage.matroids.database_matroids.T12(groundset='abcdefghijkl')[source]#

Return the matroid \(T_{12}\).

The edges of the Petersen graph can be labeled by the \(4\)-circuits of \(T_{12}\) so that two edges are adjacent if and only if the corresponding \(4\)-circuits overlap in exactly two elements. Relaxing a circuit-hyperplane yields an excluded minor for the class of matroids that are either binary or ternary.

EXAMPLES:

sage: M = matroids.catalog.T12(); M
T12: Binary matroid of rank 6 on 12 elements, type (2, None)
sage: M.is_valid()
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
True

>>> from sage.all import *
>>> M = matroids.catalog.T12(); M
T12: Binary matroid of rank 6 on 12 elements, type (2, None)
>>> M.is_valid()
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
True

REFERENCES:

[Oxl2011], p. 658-9.

sage.matroids.database_matroids.T8(groundset='abcdefgh')[source]#

Return the matroid \(T_8\), represented over \(GF(3)\).

The matroid \(T_8\) is an \(8\)-element matroid of rank-\(4\). It is representable over a field if and only if that field has characteristic three. It is an excluded minor for the dyadic matroids.

EXAMPLES:

sage: M = matroids.catalog.T8(); M
T8: Ternary matroid of rank 4 on 8 elements, type 0-
sage: M.truncation().is_isomorphic(matroids.Uniform(3, 8))
True
sage: M.contract('e').is_isomorphic(matroids.catalog.P7())
True
sage: M.has_minor(matroids.Uniform(3, 8))
False
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.T8(); M
T8: Ternary matroid of rank 4 on 8 elements, type 0-
>>> M.truncation().is_isomorphic(matroids.Uniform(Integer(3), Integer(8)))
True
>>> M.contract('e').is_isomorphic(matroids.catalog.P7())
True
>>> M.has_minor(matroids.Uniform(Integer(3), Integer(8)))
False
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 649.

sage.matroids.database_matroids.TK10(groundset=None)[source]#

Return the matroid \(TK10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.TK10(); M
TK10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.TK10(); M
TK10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.TQ10(groundset=None)[source]#

Return the matroid \(TQ10\).

An excluded minor for \(K_2\)-representable matroids. UPF is \(G\). Self-dual. Has TQ8 as a minor (delete ‘d’ and contract ‘c’).

EXAMPLES:

sage: M = matroids.catalog.TQ10(); M
TQ10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True
sage: N = M.delete('d').contract('c')
sage: N.is_isomorphic(matroids.catalog.TQ8())
True

>>> from sage.all import *
>>> M = matroids.catalog.TQ10(); M
TQ10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
>>> N = M.delete('d').contract('c')
>>> N.is_isomorphic(matroids.catalog.TQ8())
True
sage.matroids.database_matroids.TQ8(groundset=None)[source]#

Return the matroid \(TQ8\).

An excluded minor for \(2\)-regular matroids. UPF is \(K_2\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.TQ8(); M
TQ8: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.TQ8(); M
TQ8: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.TQ9(groundset=None)[source]#

Return the matroid \(TQ9\).

An excluded minor for \(K_2\)-representable matroids, and a single-element extension of TQ8. The UPF is \(G\). Uniquely \(GF(5)\)-representable. (An excluded minor for \(H_2\)-representable matroids.)

EXAMPLES:

sage: TQ8 = matroids.catalog.TQ8()
sage: TQ9 = matroids.catalog.TQ9(); TQ9
TQ9: Quaternary matroid of rank 4 on 9 elements
sage: for M in TQ8.extensions():
....:     if M.is_isomorphic(TQ9):
....:         print(True)
....:         break
True

>>> from sage.all import *
>>> TQ8 = matroids.catalog.TQ8()
>>> TQ9 = matroids.catalog.TQ9(); TQ9
TQ9: Quaternary matroid of rank 4 on 9 elements
>>> for M in TQ8.extensions():
...     if M.is_isomorphic(TQ9):
...         print(True)
...         break
True
sage.matroids.database_matroids.TQ9p(groundset=None)[source]#

Return the matroid \(TQ9^-\).

An excluded minor for \(G\)- and \(K_2\)-representable matroids, and a single-element extension of TQ8. UPF is \(U_1^{(2)}\). (An excluded minor for \(H_2\)- and \(GF(5)\)-representable matroids.)

EXAMPLES:

sage: TQ8 = matroids.catalog.TQ8()
sage: TQ9p = matroids.catalog.TQ9p(); TQ9p
TQ9': Quaternary matroid of rank 4 on 9 elements
sage: for M in TQ8.extensions():
....:     if M.is_isomorphic(TQ9p):
....:         print(True)
....:         break
True

>>> from sage.all import *
>>> TQ8 = matroids.catalog.TQ8()
>>> TQ9p = matroids.catalog.TQ9p(); TQ9p
TQ9': Quaternary matroid of rank 4 on 9 elements
>>> for M in TQ8.extensions():
...     if M.is_isomorphic(TQ9p):
...         print(True)
...         break
True
sage.matroids.database_matroids.TU10(groundset=None)[source]#

Return the matroid \(TU10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.TU10(); M
TU10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.TU10(); M
TU10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.TernaryDowling3(groundset='abcdefghi')[source]#

Return the matroid \(Q_3(GF(3)^\times)\), represented over \(GF(3)\).

The matroid \(Q_3(GF(3)^\times)\) is a \(9\)-element matroid of rank-\(3\). It is the rank-\(3\) ternary Dowling geometry. It is representable over a field if and only if that field does not have characteristic two.

EXAMPLES:

sage: M = matroids.catalog.TernaryDowling3(); M
Q3(GF(3)x): Ternary matroid of rank 3 on 9 elements, type 0-
sage: len(list(M.linear_subclasses()))
72
sage: M.fundamental_cycle('abc', 'd')
{'a': 2, 'b': 1, 'd': 1}
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.TernaryDowling3(); M
Q3(GF(3)x): Ternary matroid of rank 3 on 9 elements, type 0-
>>> len(list(M.linear_subclasses()))
72
>>> M.fundamental_cycle('abc', 'd')
{'a': 2, 'b': 1, 'd': 1}
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 654.

sage.matroids.database_matroids.Terrahawk(groundset='abcdefghijklmnop')[source]#

Return the Terrahawk matroid.

The Terrahawk is a binary matroid that is a sporadic exception in a chain theorem for internally \(4\)-connected binary matroids.

EXAMPLES:

sage: M = matroids.catalog.Terrahawk(); M
Terrahawk: Binary matroid of rank 8 on 16 elements, type (0, 4)
sage: M.is_valid()
True

>>> from sage.all import *
>>> M = matroids.catalog.Terrahawk(); M
Terrahawk: Binary matroid of rank 8 on 16 elements, type (0, 4)
>>> M.is_valid()
True

REFERENCES:

[CMO2011]

sage.matroids.database_matroids.Theta(n, groundset=None)[source]#

Return the matroid \(\Theta_n\).

Defined for all \(n \ge 2\). \(\Theta_2 \cong U_{1,2} \bigoplus U_{1,2}\) and \(\Theta_3 \cong M(K_4)\).

INPUT:

  • n – an integer (\(n \ge 2\)); the rank of the matroid

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: matroid (\(\Theta_n\))

EXAMPLES:

sage: matroids.Theta(30)
Theta_30: Matroid of rank 30 on 60 elements with 16270 circuits
sage: M = matroids.Theta(2)
sage: U12 = matroids.Uniform(1, 2)
sage: U = U12.direct_sum(U12)
sage: M.is_isomorphic(U)
True
sage: M = matroids.Theta(3)
sage: M.is_isomorphic(matroids.catalog.K4())
True

>>> from sage.all import *
>>> matroids.Theta(Integer(30))
Theta_30: Matroid of rank 30 on 60 elements with 16270 circuits
>>> M = matroids.Theta(Integer(2))
>>> U12 = matroids.Uniform(Integer(1), Integer(2))
>>> U = U12.direct_sum(U12)
>>> M.is_isomorphic(U)
True
>>> M = matroids.Theta(Integer(3))
>>> M.is_isomorphic(matroids.catalog.K4())
True

\(\Theta_n\) is self-dual; identically self-dual if and only if \(n = 2\):

sage: M = matroids.Theta(2)
sage: M.equals(M.dual())
True
sage: import random
sage: n = random.choice(range(3, 7))
sage: M = matroids.Theta(n)
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

>>> from sage.all import *
>>> M = matroids.Theta(Integer(2))
>>> M.equals(M.dual())
True
>>> import random
>>> n = random.choice(range(Integer(3), Integer(7)))
>>> M = matroids.Theta(n)
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

For \(n \le 3\), its automorphism group is transitive, while for \(n \ge 4\) it is not:

sage: n = random.choice(range(4, 8))
sage: M = matroids.Theta(2 + n % 2)
sage: M.automorphism_group().is_transitive()
True
sage: M = matroids.Theta(n)
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> n = random.choice(range(Integer(4), Integer(8)))
>>> M = matroids.Theta(Integer(2) + n % Integer(2))
>>> M.automorphism_group().is_transitive()
True
>>> M = matroids.Theta(n)
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 663-4.

sage.matroids.database_matroids.TicTacToe(groundset=None)[source]#

Return the TicTacToe matroid.

The dual of the TicTacToe matroid is not algebraic; it is unknown whether the TicTacToe matroid itself is algebraic.

EXAMPLES:

sage: M = matroids.catalog.TicTacToe(); M
TicTacToe: Matroid of rank 5 on 9 elements with 8 nonspanning circuits
sage: M.is_valid() and M.is_paving()
True

>>> from sage.all import *
>>> M = matroids.catalog.TicTacToe(); M
TicTacToe: Matroid of rank 5 on 9 elements with 8 nonspanning circuits
>>> M.is_valid() and M.is_paving()
True

REFERENCES:

[Hoc]

sage.matroids.database_matroids.TippedFree3spike(groundset=None)[source]#

Return the tipped free \(3\)-spike.

Unique 3-connected extension of U36. Stabilizer for \(K_2\).

EXAMPLES:

sage: M = matroids.catalog.TippedFree3spike(); M
Tipped rank-3 free spike: Quaternary matroid of rank 3 on 7 elements
sage: M.has_minor(matroids.Uniform(3,6))
True

>>> from sage.all import *
>>> M = matroids.catalog.TippedFree3spike(); M
Tipped rank-3 free spike: Quaternary matroid of rank 3 on 7 elements
>>> M.has_minor(matroids.Uniform(Integer(3),Integer(6)))
True
sage.matroids.database_matroids.U24(groundset='abcd')[source]#

The uniform matroid of rank \(2\) on \(4\) elements.

The \(4\)-point line; isomorphic to \(\mathcal{W}^2\) , the rank-\(2\) whirl. The unique excluded minor for the class of binary matroids.

EXAMPLES:

sage: M = matroids.catalog.U24(); M
U(2, 4): Matroid of rank 2 on 4 elements with circuit-closures
{2: {{'a', 'b', 'c', 'd'}}}
sage: N = matroids.Uniform(2, 4)
sage: M.is_isomorphic(N)
True
sage: M.automorphism_group().structure_description()
'S4'

>>> from sage.all import *
>>> M = matroids.catalog.U24(); M
U(2, 4): Matroid of rank 2 on 4 elements with circuit-closures
{2: {{'a', 'b', 'c', 'd'}}}
>>> N = matroids.Uniform(Integer(2), Integer(4))
>>> M.is_isomorphic(N)
True
>>> M.automorphism_group().structure_description()
'S4'

\(U_{2,4}\) is isomorphic to \(\mathcal{W}^2\):

sage: W2 = matroids.Whirl(2)
sage: W2.is_isomorphic(M)
True

>>> from sage.all import *
>>> W2 = matroids.Whirl(Integer(2))
>>> W2.is_isomorphic(M)
True

identically self-dual:

sage: M.equals(M.dual())
True

>>> from sage.all import *
>>> M.equals(M.dual())
True

and \(3\)-connected:

sage: M.is_3connected()
True

>>> from sage.all import *
>>> M.is_3connected()
True

REFERENCES:

[Oxl2011], p. 639.

sage.matroids.database_matroids.U25(groundset='abcde')[source]#

The uniform matroid of rank \(2\) on \(5\) elements.

\(U_{2,5}\) is the \(5\)-point line. Dual to \(U_{3,5}\).

EXAMPLES:

sage: U25 = matroids.catalog.U25(); U25
U(2, 5): Matroid of rank 2 on 5 elements with circuit-closures
{2: {{'a', 'b', 'c', 'd', 'e'}}}
sage: U25.is_graphic() or U25.is_regular()
False
sage: U35 = matroids.catalog.U35()
sage: U25.is_isomorphic(U35.dual())
True

>>> from sage.all import *
>>> U25 = matroids.catalog.U25(); U25
U(2, 5): Matroid of rank 2 on 5 elements with circuit-closures
{2: {{'a', 'b', 'c', 'd', 'e'}}}
>>> U25.is_graphic() or U25.is_regular()
False
>>> U35 = matroids.catalog.U35()
>>> U25.is_isomorphic(U35.dual())
True

REFERENCES:

[Oxl2011], p. 640.

sage.matroids.database_matroids.U35(groundset='abcde')[source]#

The uniform matroid of rank \(3\) on \(5\) elements.

\(U_{3,5}\) is five points freely placed in the plane. Dual to \(U_{2,5}\).

EXAMPLES:

sage: U35 = matroids.catalog.U35(); U35
U(3, 5): Matroid of rank 3 on 5 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd', 'e'}}}
sage: U35.is_graphic() or U35.is_regular()
False
sage: U25 = matroids.catalog.U25()
sage: U35.is_isomorphic(U25.dual())
True

>>> from sage.all import *
>>> U35 = matroids.catalog.U35(); U35
U(3, 5): Matroid of rank 3 on 5 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd', 'e'}}}
>>> U35.is_graphic() or U35.is_regular()
False
>>> U25 = matroids.catalog.U25()
>>> U35.is_isomorphic(U25.dual())
True

REFERENCES:

[Oxl2011], p. 640.

sage.matroids.database_matroids.U36(groundset='abcdef')[source]#

The uniform matroid of rank \(3\) on \(6\) elements.

Six points freely placed in the plane; the tipless free \(3\)-spike. Identically self-dual.

EXAMPLES:

sage: U36 = matroids.catalog.U36(); U36
U(3, 6): Matroid of rank 3 on 6 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd', 'e', 'f'}}}
sage: Z = matroids.Spike(3, False)
sage: U36.is_isomorphic(Z)
True
sage: U36.equals(U36.dual())
True
sage: U36.automorphism_group().structure_description()
'S6'

>>> from sage.all import *
>>> U36 = matroids.catalog.U36(); U36
U(3, 6): Matroid of rank 3 on 6 elements with circuit-closures
{3: {{'a', 'b', 'c', 'd', 'e', 'f'}}}
>>> Z = matroids.Spike(Integer(3), False)
>>> U36.is_isomorphic(Z)
True
>>> U36.equals(U36.dual())
True
>>> U36.automorphism_group().structure_description()
'S6'

REFERENCES:

[Oxl2011], p. 642.

sage.matroids.database_matroids.UA12(groundset=None)[source]#

Return the matroid \(UA12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Not self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.UA12(); M
UA12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.UA12(); M
UA12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.UG10(groundset=None)[source]#

Return the matroid \(UG10\).

An excluded minor for \(K_2\)- and \(P_4\)-representable matroids. Self-dual. An excluded minor for \(H_3\)- and \(H_2\)-representable matroids. Uniquely \(GF(5)\)-representable. Although not \(P_4\)-representable, it is \(O\)-representable, and hence is representable over all fields of size at least four.

EXAMPLES:

sage: M = matroids.catalog.UG10(); M
UG10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.UG10(); M
UG10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.UK10(groundset=None)[source]#

Return the matroid \(UK10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Not self-dual. UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.UK10(); M
UK10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.UK10(); M
UK10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.UK12(groundset=None)[source]#

Return the matroid \(UK12\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(I\).

EXAMPLES:

sage: M = matroids.catalog.UK12(); M
UK12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.UK12(); M
UK12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.UP14(groundset=None)[source]#

Return the matroid \(UP14\).

An excluded minor for \(K_2\)-representable matroids. Has disjoint circuit-hyperplanes. UPF is \(W\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.UP14(); M
UP14: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.UP14(); M
UP14: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.UQ10(groundset=None)[source]#

Return the matroid \(UQ10\).

An excluded minor for \(K_2\)- and \(G\)-representable matroids (and \(H_2\)- and \(GF(5)\)-representable matroids). Self-dual. UPF is \((Q(a,b), <-1,a,b,a-1,b-1,a-b,a+b,a+1,ab+b-1,ab-b+1>)\).

EXAMPLES:

sage: M = matroids.catalog.UQ10(); M
UQ10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.UQ10(); M
UQ10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.UQ12(groundset=None)[source]#

Return the matroid \(UQ12\).

An excluded minor for \(K_2\) and \(G\)-representable matroids (and \(H2\) and \(GF(5)\)-representable matroids). UPF is \(P_{pappus}\). Self-dual.

EXAMPLES:

sage: M = matroids.catalog.UQ12(); M
UQ12: Quaternary matroid of rank 6 on 12 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.UQ12(); M
UQ12: Quaternary matroid of rank 6 on 12 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.UT10(groundset=None)[source]#

Return the matroid \(UT10\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). Self-dual. UPF is \(I\).

EXAMPLES:

sage: M = matroids.catalog.UT10(); M
UT10: Quaternary matroid of rank 5 on 10 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.UT10(); M
UT10: Quaternary matroid of rank 5 on 10 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.Uniform(r, n, groundset=None)[source]#

Return the uniform matroid of rank \(r\) on \(n\) elements.

All subsets of size \(r\) or less are independent; all larger subsets are dependent. Representable when the field is sufficiently large. The precise bound is the subject of the MDS conjecture from coding theory.

INPUT:

  • r – a nonnegative integer; the rank of the uniform matroid

  • n – a nonnegative integer; the number of elements of the uniform matroid

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: the uniform matroid \(U_{r,n}\)

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.Uniform(2, 5); M
U(2, 5): Matroid of rank 2 on 5 elements with circuit-closures
{2: {{0, 1, 2, 3, 4}}}
sage: M.dual().is_isomorphic(matroids.Uniform(3, 5))
True
sage: setprint(M.hyperplanes())
[{0}, {1}, {2}, {3}, {4}]
sage: M.has_line_minor(6)
False
sage: M.is_valid()
True

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.Uniform(Integer(2), Integer(5)); M
U(2, 5): Matroid of rank 2 on 5 elements with circuit-closures
{2: {{0, 1, 2, 3, 4}}}
>>> M.dual().is_isomorphic(matroids.Uniform(Integer(3), Integer(5)))
True
>>> setprint(M.hyperplanes())
[{0}, {1}, {2}, {3}, {4}]
>>> M.has_line_minor(Integer(6))
False
>>> M.is_valid()
True

Check that bug Issue #15292 was fixed:

sage: M = matroids.Uniform(4,4)
sage: len(M.circuit_closures())
0

>>> from sage.all import *
>>> M = matroids.Uniform(Integer(4),Integer(4))
>>> len(M.circuit_closures())
0

REFERENCES:

[Oxl2011], p. 660.

sage.matroids.database_matroids.VP14(groundset=None)[source]#

Return the matroid \(VP14\).

An excluded minor for \(K_2\)-representable matroids. Has disjoint circuit-hyperplanes. UPF is \(W\). Not self-dual.

EXAMPLES:

sage: M = matroids.catalog.VP14(); M
VP14: Quaternary matroid of rank 7 on 14 elements
sage: M.is_isomorphic(M.dual())
False

>>> from sage.all import *
>>> M = matroids.catalog.VP14(); M
VP14: Quaternary matroid of rank 7 on 14 elements
>>> M.is_isomorphic(M.dual())
False
sage.matroids.database_matroids.Vamos(groundset=None)[source]#

Return the \(V\acute{a}mos\) matroid, represented as circuit closures.

The \(V\acute{a}mos\) matroid, or \(V\acute{a}mos\) cube, or \(V_8\) is an \(8\)-element matroid of rank-\(4\). It violates Ingleton’s condition for representability over a division ring. It is not algebraic.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.catalog.Vamos(); M
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: setprint(M.nonbases())
[{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'}, {'a', 'b', 'g', 'h'},
 {'c', 'd', 'e', 'f'}, {'e', 'f', 'g', 'h'}]
sage: M.is_dependent(['c', 'd', 'g', 'h'])
False
sage: M.is_valid() and M.is_paving()  # long time
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> from sage.matroids.advanced import setprint
>>> M = matroids.catalog.Vamos(); M
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'}}}
>>> setprint(M.nonbases())
[{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'}, {'a', 'b', 'g', 'h'},
 {'c', 'd', 'e', 'f'}, {'e', 'f', 'g', 'h'}]
>>> M.is_dependent(['c', 'd', 'g', 'h'])
False
>>> M.is_valid() and M.is_paving()  # long time
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 649.

sage.matroids.database_matroids.WQ8(groundset=None)[source]#

Return the matroid \(WQ8\).

An excluded minor for \(G\), \(K_2\), \(H_4\), and \(GF(5)\)-representable matroids. Self-dual. UPF is \((Z[\zeta,a], <\zeta,a-\zeta>)\) where \(\zeta\) is solution to \(x^2-x+1 = 0\) and \(a\) is an indeterminate.

EXAMPLES:

sage: M = matroids.catalog.WQ8(); M
WQ8: Quaternary matroid of rank 4 on 8 elements
sage: M.is_isomorphic(M.dual())
True

>>> from sage.all import *
>>> M = matroids.catalog.WQ8(); M
WQ8: Quaternary matroid of rank 4 on 8 elements
>>> M.is_isomorphic(M.dual())
True
sage.matroids.database_matroids.Wheel(r, field=None, ring=None, groundset=None)[source]#

Return the rank-\(r\) wheel.

INPUT:

  • r – a positive integer; the rank of the desired matroid

  • ring – any ring; if provided, output will be a linear matroid over the ring or field ring. If the ring is \(\ZZ\), then output will be a regular matroid.

  • field – any field; same as ring, but only fields are allowed

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: the rank-\(r\) wheel matroid, represented as a regular matroid

EXAMPLES:

sage: M = matroids.Wheel(5); M
Wheel(5): Regular matroid of rank 5 on 10 elements with 121 bases
sage: M.tutte_polynomial()
x^5 + y^5 + 5*x^4 + 5*x^3*y + 5*x^2*y^2 + 5*x*y^3 + 5*y^4 + 10*x^3 +
15*x^2*y + 15*x*y^2 + 10*y^3 + 10*x^2 + 16*x*y + 10*y^2 + 4*x + 4*y
sage: M.is_valid()
True
sage: M = matroids.Wheel(3)
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True
sage: M.is_isomorphic(matroids.Wheel(3, field=GF(3)))
True
sage: M = matroids.Wheel(3, field=GF(3)); M
Wheel(3): Ternary matroid of rank 3 on 6 elements, type 0+

>>> from sage.all import *
>>> M = matroids.Wheel(Integer(5)); M
Wheel(5): Regular matroid of rank 5 on 10 elements with 121 bases
>>> M.tutte_polynomial()
x^5 + y^5 + 5*x^4 + 5*x^3*y + 5*x^2*y^2 + 5*x*y^3 + 5*y^4 + 10*x^3 +
15*x^2*y + 15*x*y^2 + 10*y^3 + 10*x^2 + 16*x*y + 10*y^2 + 4*x + 4*y
>>> M.is_valid()
True
>>> M = matroids.Wheel(Integer(3))
>>> M.is_isomorphic(matroids.CompleteGraphic(Integer(4)))
True
>>> M.is_isomorphic(matroids.Wheel(Integer(3), field=GF(Integer(3))))
True
>>> M = matroids.Wheel(Integer(3), field=GF(Integer(3))); M
Wheel(3): Ternary matroid of rank 3 on 6 elements, type 0+

For \(r \ge 2\), the wheel is self-dual but not identically self-dual, and for \(r \ge 4\) it has a non-transitive automorphism group:

sage: import random
sage: r = random.choice(range(4, 8))
sage: M = matroids.Wheel(r)
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> import random
>>> r = random.choice(range(Integer(4), Integer(8)))
>>> M = matroids.Wheel(r)
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 659-60.

sage.matroids.database_matroids.Wheel4(groundset='abcdefgh')[source]#

Return the rank-\(4\) wheel.

A regular, graphic, and cographic matroid. Self-dual but not identically self-dual.

EXAMPLES:

sage: M = matroids.catalog.Wheel4(); M
Wheel(4): Regular matroid of rank 4 on 8 elements with 45 bases
sage: M.is_valid() and M.is_graphic() and M.dual().is_graphic()
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.Wheel4(); M
Wheel(4): Regular matroid of rank 4 on 8 elements with 45 bases
>>> M.is_valid() and M.is_graphic() and M.dual().is_graphic()
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 651-2.

sage.matroids.database_matroids.Whirl(r, groundset=None)[source]#

Return the rank-\(r\) whirl.

The whirl is the unique relaxation of the wheel.

INPUT:

  • r – a positive integer; the rank of the desired matroid.

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: the rank-\(r\) whirl matroid, represented as a ternary matroid

EXAMPLES:

sage: M = matroids.Whirl(5); M
Whirl(5): Ternary matroid of rank 5 on 10 elements, type 0-
sage: M.is_valid()
True
sage: M.tutte_polynomial()
x^5 + y^5 + 5*x^4 + 5*x^3*y + 5*x^2*y^2 + 5*x*y^3 + 5*y^4 + 10*x^3 + 15*x^2*y +
 15*x*y^2 + 10*y^3 + 10*x^2 + 15*x*y + 10*y^2 + 5*x + 5*y
sage: M.is_isomorphic(matroids.Wheel(5))
False
sage: M = matroids.Whirl(3)
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
False

>>> from sage.all import *
>>> M = matroids.Whirl(Integer(5)); M
Whirl(5): Ternary matroid of rank 5 on 10 elements, type 0-
>>> M.is_valid()
True
>>> M.tutte_polynomial()
x^5 + y^5 + 5*x^4 + 5*x^3*y + 5*x^2*y^2 + 5*x*y^3 + 5*y^4 + 10*x^3 + 15*x^2*y +
 15*x*y^2 + 10*y^3 + 10*x^2 + 15*x*y + 10*y^2 + 5*x + 5*y
>>> M.is_isomorphic(matroids.Wheel(Integer(5)))
False
>>> M = matroids.Whirl(Integer(3))
>>> M.is_isomorphic(matroids.CompleteGraphic(Integer(4)))
False

For \(r \ge 3\), the whirl is self-dual but not identically self-dual:

sage: import random
sage: r = random.choice(range(3, 10))
sage: M = matroids.Whirl(r)
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

>>> from sage.all import *
>>> import random
>>> r = random.choice(range(Integer(3), Integer(10)))
>>> M = matroids.Whirl(r)
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True

Except for \(\mathcal{W}^2\), which is isomorphic to \(U_{2, 4}\), these matroids have non-transitive automorphism groups:

sage: r = random.choice(range(3, 8))
sage: M = matroids.Whirl(r)
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> r = random.choice(range(Integer(3), Integer(8)))
>>> M = matroids.Whirl(r)
>>> M.automorphism_group().is_transitive()
False

Todo

Optional arguments ring and x, such that the resulting matroid is represented over ring by a reduced matrix like [-1  0  x] [ 1 -1  0] [ 0  1 -1]

REFERENCES:

[Oxl2011], p. 659-60.

sage.matroids.database_matroids.Whirl3(groundset='abcdef')[source]#

The rank-\(3\) whirl.

The unique relaxation of \(M(K_4)\). Self-dual but not identically self-dual.

EXAMPLES:

sage: W = matroids.catalog.Whirl3(); W
Whirl(3): Ternary matroid of rank 3 on 6 elements, type 0-
sage: W.equals(W.dual())
False
sage: W.is_isomorphic(W.dual())
True
sage: W.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> W = matroids.catalog.Whirl3(); W
Whirl(3): Ternary matroid of rank 3 on 6 elements, type 0-
>>> W.equals(W.dual())
False
>>> W.is_isomorphic(W.dual())
True
>>> W.automorphism_group().is_transitive()
False

For all elements \(e\), neither \(\mathcal{W}_3 \setminus \{e\}\) nor \(\mathcal{W}_3 / \{e\}\) is \(3\)-connected:

sage: import random
sage: e = random.choice(list(W.groundset()))
sage: W.delete(e).is_3connected()
False
sage: W.contract(e).is_3connected()
False

>>> from sage.all import *
>>> import random
>>> e = random.choice(list(W.groundset()))
>>> W.delete(e).is_3connected()
False
>>> W.contract(e).is_3connected()
False

REFERENCES:

[Oxl2011], p. 641.

sage.matroids.database_matroids.Whirl4(groundset='abcdefgh')[source]#

Return the rank-\(4\) whirl.

A matroid which is not graphic, not cographic, and not regular. Self-dual but not identically self-dual.

EXAMPLES:

sage: M = matroids.catalog.Whirl4(); M
Whirl(4): Ternary matroid of rank 4 on 8 elements, type 0+
sage: M.is_valid()
True
sage: M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
sage: M.automorphism_group().is_transitive()
False

>>> from sage.all import *
>>> M = matroids.catalog.Whirl4(); M
Whirl(4): Ternary matroid of rank 4 on 8 elements, type 0+
>>> M.is_valid()
True
>>> M.is_isomorphic(M.dual()) and not M.equals(M.dual())
True
>>> M.automorphism_group().is_transitive()
False

REFERENCES:

[Oxl2011], p. 652.

sage.matroids.database_matroids.XY13(groundset=None)[source]#

Return the matroid \(XY13\).

An excluded minor for \(G\)-representable matroids (and \(GF(5)\)-representable matroids). UPF is \(GF(4)\).

EXAMPLES:

sage: M = matroids.catalog.XY13(); M
XY13: Quaternary matroid of rank 6 on 13 elements
sage: M.is_3connected()
True

>>> from sage.all import *
>>> M = matroids.catalog.XY13(); M
XY13: Quaternary matroid of rank 6 on 13 elements
>>> M.is_3connected()
True
sage.matroids.database_matroids.Z(r, t=True, groundset=None)[source]#

Return the unique rank-\(r\) binary spike.

Defined for all \(r \ge 3\).

INPUT:

  • r – an integer (\(r \ge 3\)); the rank of the spike

  • t – boolean (default: True); whether the spike is tipped

  • groundset – a string (optional); the groundset of the matroid

OUTPUT: matroid; the unique rank-\(r\) binary spike (tipped or tipless)

EXAMPLES:

sage: matroids.Z(8)
Z_8: Binary matroid of rank 8 on 17 elements, type (7, 1)
sage: matroids.Z(9)
Z_9: Binary matroid of rank 9 on 19 elements, type (9, None)
sage: matroids.Z(20, False)
Z_20\t: Binary matroid of rank 20 on 40 elements, type (20, 0)

>>> from sage.all import *
>>> matroids.Z(Integer(8))
Z_8: Binary matroid of rank 8 on 17 elements, type (7, 1)
>>> matroids.Z(Integer(9))
Z_9: Binary matroid of rank 9 on 19 elements, type (9, None)
>>> matroids.Z(Integer(20), False)
Z_20\t: Binary matroid of rank 20 on 40 elements, type (20, 0)

It holds that \(Z_3 \setminus e \cong M(K4)\), for all \(e\):

sage: import random
sage: Z3 = matroids.Z(3)
sage: E = sorted(Z3.groundset()); E
['t', 'x1', 'x2', 'x3', 'y1', 'y2', 'y3']
sage: e = random.choice(E)
sage: Z3.delete(e).is_isomorphic(matroids.catalog.K4())
True

>>> from sage.all import *
>>> import random
>>> Z3 = matroids.Z(Integer(3))
>>> E = sorted(Z3.groundset()); E
['t', 'x1', 'x2', 'x3', 'y1', 'y2', 'y3']
>>> e = random.choice(E)
>>> Z3.delete(e).is_isomorphic(matroids.catalog.K4())
True

\(Z_3 \cong F_7\):

sage: F7 = matroids.catalog.Fano()
sage: Z3.is_isomorphic(F7)
True

>>> from sage.all import *
>>> F7 = matroids.catalog.Fano()
>>> Z3.is_isomorphic(F7)
True

\(Z_4 \setminus t \cong AG(3, 2)\):

sage: Z4 = matroids.Z(4, False)
sage: Z4.is_isomorphic(matroids.catalog.AG32())
True

>>> from sage.all import *
>>> Z4 = matroids.Z(Integer(4), False)
>>> Z4.is_isomorphic(matroids.catalog.AG32())
True

and \(Z_4 \setminus e \cong S_8\), for all \(e \neq t\):

sage: Z4 = matroids.Z(4)
sage: E = sorted(Z4.groundset())
sage: E.remove('t')
sage: e = random.choice(E)
sage: S8 = matroids.catalog.S8()
sage: Z4.delete(e).is_isomorphic(S8)
True
sage: Z4.delete('t').is_isomorphic(S8)
False

>>> from sage.all import *
>>> Z4 = matroids.Z(Integer(4))
>>> E = sorted(Z4.groundset())
>>> E.remove('t')
>>> e = random.choice(E)
>>> S8 = matroids.catalog.S8()
>>> Z4.delete(e).is_isomorphic(S8)
True
>>> Z4.delete('t').is_isomorphic(S8)
False

The tipless binary spike is self-dual; it is identically self-dual if and only if r is even. It also has a transitive automorphism group:

sage: r = random.choice(range(3, 8))
sage: Z = matroids.Z(r, False)
sage: Z.is_isomorphic(Z.dual())
True
sage: Z.equals(Z.dual()) != (r % 2 == 1)  # XOR
True
sage: Z.automorphism_group().is_transitive()  # long time
True

>>> from sage.all import *
>>> r = random.choice(range(Integer(3), Integer(8)))
>>> Z = matroids.Z(r, False)
>>> Z.is_isomorphic(Z.dual())
True
>>> Z.equals(Z.dual()) != (r % Integer(2) == Integer(1))  # XOR
True
>>> Z.automorphism_group().is_transitive()  # long time
True

REFERENCES:

[Oxl2011], p. 661-2.