Documentation for the matroids in the catalog

This module contains implementations for many of the functions accessible through matroids. and matroids.named_matroids. (type those lines in Sage and hit tab for a list).

The docstrings include educational information about each named matroid with the hopes that this class can be used as a reference. However, for a more comprehensive list of properties we refer to the appendix of [Oxl2011].

Todo

Add optional argument groundset to each method so users can customize the groundset of the matroid. We probably want some means of relabeling to accomplish that.

Add option to specify the field for represented matroids.

AUTHORS:

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

Functions

sage.matroids.catalog.AG(n, q, x=None)

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

OUTPUT:

A linear matroid whose elements are the points of \(AG(n, q)\).

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

EXAMPLES:

sage: M = matroids.AG(2, 3) \ 8
sage: M.is_isomorphic(matroids.named_matroids.AG23minus())
True
sage: matroids.AG(5, 4, 'z').size() == ((4 ^ 6 - 1) / (4 - 1) -                 # needs sage.rings.finite_rings
....:                                             (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)

sage.matroids.catalog.AG23minus()

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. See [Oxl2011], p. 653.

EXAMPLES:

sage: M = matroids.named_matroids.AG23minus()
sage: M.is_valid()
True

sage.matroids.catalog.AG32prime()

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

The matroid \(AG(3, 2)'\) is a 8-element matroid of rank-4. It is a smallest non-representable matroid. It is the unique relaxation of \(AG(3, 2)\). See [Oxl2011], p. 646.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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: M.contract('c').is_isomorphic(matroids.named_matroids.Fano())
True
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()                      # long time, needs sage.rings.finite_rings
True

sage.matroids.catalog.BetsyRoss()

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.named_matroids.BetsyRoss()
sage: len(M.circuit_closures()[2])
10
sage: M.is_valid() # long time
True

sage.matroids.catalog.Block_10_5()

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

EXAMPLES:

sage: M = matroids.named_matroids.Block_10_5()
sage: M.is_valid() # long time
True
sage: BD = BlockDesign(M.groundset(), M.nonspanning_circuits())                 # needs sage.graphs
sage: BD.is_t_design(return_parameters=True)                                    # needs sage.graphs
(True, (3, 10, 5, 3))

sage.matroids.catalog.Block_9_4()

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

EXAMPLES:

sage: M = matroids.named_matroids.Block_9_4()
sage: M.is_valid() # long time
True
sage: BD = BlockDesign(M.groundset(), M.nonspanning_circuits())                 # needs sage.graphs
sage: BD.is_t_design(return_parameters=True)                                    # needs sage.graphs
(True, (2, 9, 4, 3))

sage.matroids.catalog.CompleteGraphic(n)

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

sage.matroids.catalog.D16()

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 \(D_{16}\). See [CMO2012].

EXAMPLES:

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

sage.matroids.catalog.ExtendedBinaryGolayCode()

Return the matroid of the extended binary Golay code.

See GolayCode documentation for more on this code.

EXAMPLES:

sage: M = matroids.named_matroids.ExtendedBinaryGolayCode()
sage: C = LinearCode(M.representation())
sage: C.is_permutation_equivalent(codes.GolayCode(GF(2)))       # long time, needs sage.rings.finite_rings
True
sage: M.is_valid()
True

sage.matroids.catalog.ExtendedTernaryGolayCode()

Return the matroid of the extended ternary Golay code.

See GolayCode

EXAMPLES:

sage: M = matroids.named_matroids.ExtendedTernaryGolayCode()
sage: C = LinearCode(M.representation())
sage: C.is_permutation_equivalent(codes.GolayCode(GF(3)))       # long time, needs sage.rings.finite_rings
True
sage: M.is_valid()
True

sage.matroids.catalog.F8()

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

The matroid \(F_8\) is a 8-element matroid of rank-4. It is a smallest non-representable matroid. See [Oxl2011], p. 647.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.named_matroids.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.named_matroids.Fano()) for N in D]
[...True...]
sage: [N.is_isomorphic(matroids.named_matroids.NonFano()) for N in D]
[...True...]
sage: M.is_valid()                      # long time, needs sage.rings.finite_rings
True

sage.matroids.catalog.Fano()

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. It is also the projective plane of order two, i.e. \(\mathrm{PG}(2, 2)\). See [Oxl2011], p. 643.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Fano(); M
Fano: Binary matroid of rank 3 on 7 elements, type (3, 0)
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,                                  # needs sage.graphs
....:                  7)]).is_isomorphic(matroids.CompleteGraphic(4))
True

sage.matroids.catalog.J()

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

The matroid \(J\) is a 8-element matroid of rank-4. It is representable over a field if and only if that field has at least three elements. See [Oxl2011], p. 650.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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())
True
sage: M.has_minor(matroids.CompleteGraphic(4))                                  # needs sage.graphs
False
sage: M.is_valid()
True

sage.matroids.catalog.K33dual()

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. See [Oxl2011], p. 652.

EXAMPLES:

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

sage.matroids.catalog.L8()

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

The matroid \(L_8\) is a 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. See [Oxl2011], p. 648.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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.matroids.catalog.N1()

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

\(N_1\) is an excluded minor for the dyadic matroids. See [Oxl2011], p. 554.

EXAMPLES:

sage: M = matroids.named_matroids.N1()
sage: M.is_field_isomorphic(M.dual())
True
sage: M.is_valid()
True

sage.matroids.catalog.N2()

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

\(N_2\) is an excluded minor for the dyadic matroids. See [Oxl2011], p. 554.

EXAMPLES:

sage: M = matroids.named_matroids.N2()
sage: M.is_field_isomorphic(M.dual())
True
sage: M.is_valid()
True

sage.matroids.catalog.NonFano()

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 \(F_7\). See [Oxl2011], p. 643.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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))                  # needs sage.graphs
True
sage: M.delete('g').is_isomorphic(matroids.CompleteGraphic(4))                  # needs sage.graphs
False

sage.matroids.catalog.NonPappus()

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. See [Oxl2011], p. 655.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.NonPappus(); M
NonPappus: Matroid of rank 3 on 9 elements with circuit-closures
{2: {{'a', 'b', 'c'}, {'a', 'e', 'i'}, {'a', 'f', 'h'},
     {'b', 'd', 'i'}, {'b', 'f', 'g'}, {'c', 'd', 'h'},
     {'c', 'e', 'g'}, {'g', 'h', 'i'}},
 3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'}}}
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'}, {'g', 'h', 'i'}]
sage: M.is_dependent(['d', 'e', 'f'])
False
sage: M.is_valid() # long time
True

sage.matroids.catalog.NonVamos()

Return the non-Vamos matroid.

The non-Vamos matroid, or \(V_8^+\) is an 8-element matroid of rank 4. It is a tightening of the Vamos matroid. It is representable over some field. See [Oxl2011], p. 72, 84.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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

sage.matroids.catalog.NotP8()

Return the matroid NotP8.

This is a matroid that is not \(P_8\), found on page 512 of [Oxl1992] (the first edition).

EXAMPLES:

sage: M = matroids.named_matroids.P8()
sage: N = matroids.named_matroids.NotP8()
sage: M.is_isomorphic(N)
False
sage: M.is_valid()
True

sage.matroids.catalog.O7()

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 \(M(K_4)\). See [Oxl2011], p. 644

EXAMPLES:

sage: M = matroids.named_matroids.O7(); M
O7: Ternary matroid of rank 3 on 7 elements, type 0+
sage: M.delete('e').is_isomorphic(matroids.CompleteGraphic(4))                  # needs sage.graphs
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

sage.matroids.catalog.P6()

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. See [Oxl2011], p. 641.

EXAMPLES:

sage: M = matroids.named_matroids.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,                         # needs sage.rings.finite_rings
....:                                          nrows=5)).has_minor(M)
False
sage: M.is_valid()
True

sage.matroids.catalog.P7()

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 3 elements. It is one of two ternary 3-spikes, with the other being \(F_7^-\). See [Oxl2011], p. 644.

EXAMPLES:

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

sage.matroids.catalog.P8()

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

The matroid \(P_8\) is a 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. See [Oxl2011], p. 650.

EXAMPLES:

sage: M = matroids.named_matroids.P8(); M
P8: Ternary matroid of rank 4 on 8 elements, type 2+
sage: M.is_isomorphic(M.dual())
True
sage: Matroid(matrix=random_matrix(GF(4, 'a'), ncols=5,                         # needs sage.rings.finite_rings
....:                              nrows=5)).has_minor(M)
False
sage: M.bicycle_dimension()
2

sage.matroids.catalog.P8pp()

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

The matroid \(P_8^=\) is a 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. See [Oxl2011], p. 651.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.named_matroids.P8pp(); M
P8'': Matroid of rank 4 on 8 elements with circuit-closures
{3: {{'a', 'b', 'f', 'h'}, {'a', 'c', 'e', 'f'}, {'a', 'c', 'g', 'h'},
     {'a', 'd', 'e', 'g'}, {'b', 'c', 'e', 'g'}, {'b', 'd', 'e', 'h'},
     {'b', 'd', 'f', 'g'}, {'c', 'd', 'f', 'h'}},
 4: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'}}}
sage: M.is_isomorphic(M.dual())
True
sage: len(get_nonisomorphic_matroids([M.contract(i)
....:                                        for i in M.groundset()]))
1
sage: M.is_valid() # long time
True

sage.matroids.catalog.P9()

Return the matroid \(P_9\).

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

EXAMPLES:

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

sage.matroids.catalog.PG(n, q, x=None)

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

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.named_matroids.Fano())
True
sage: matroids.PG(5, 4, 'z').size() == (4^6 - 1) / (4 - 1)                      # needs sage.rings.finite_rings
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

sage.matroids.catalog.Pappus()

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. See [Oxl2011], p. 655.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Pappus(); M
Pappus: Matroid of rank 3 on 9 elements with circuit-closures
{2: {{'a', 'b', 'c'}, {'a', 'e', 'i'}, {'a', 'f', 'h'},
     {'b', 'd', 'i'}, {'b', 'f', 'g'}, {'c', 'd', 'h'},
     {'c', 'e', 'g'}, {'d', 'e', 'f'}, {'g', 'h', 'i'}},
 3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'}}}
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() # long time
True

sage.matroids.catalog.Q10()

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.named_matroids.Q10()                                         # needs sage.rings.finite_rings
sage: M.is_isomorphic(M.dual())                                                 # needs sage.rings.finite_rings
True
sage: M.is_valid()                                                              # needs sage.rings.finite_rings
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.named_matroids.Q10().linear_extensions(simple=True) # needs sage.rings.finite_rings sage: [M for M in S if not M.has_line_minor(5)] # long time, needs sage.rings.finite_rings []

sage.matroids.catalog.Q6()

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. See [Oxl2011], p. 641.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.Q6(); M                                       # needs sage.rings.finite_rings
Q6: Quaternary matroid of rank 3 on 6 elements
sage: setprint(M.hyperplanes())                                                 # needs sage.rings.finite_rings
[{'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()                  # needs sage.rings.finite_rings
False

sage.matroids.catalog.Q8()

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

The matroid \(Q_8\) is a 8-element matroid of rank-4. It is a smallest non-representable matroid. See [Oxl2011], p. 647.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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.matroids.catalog.R10()

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

The matroid \(R_{10}\) is a 10-element regular 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. See [Oxl2011], p. 656.

EXAMPLES:

sage: M = matroids.named_matroids.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.equals(M.dual())
False
sage: M.is_isomorphic(M.dual())                                                 # needs sage.graphs
True
sage: M.is_valid()
True

Check the splitter property:

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

sage.matroids.catalog.R12()

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. See [Oxl2011], p. 657.

EXAMPLES:

sage: M = matroids.named_matroids.R12(); M
R12: Regular matroid of rank 6 on 12 elements with 441 bases
sage: M.equals(M.dual())
False
sage: M.is_isomorphic(M.dual())                                                 # needs sage.graphs
True
sage: M.is_valid()
True

sage.matroids.catalog.R6()

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 \(U_{2, 4}\). See [Oxl2011], p. 642.

EXAMPLES:

sage: M = matroids.named_matroids.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()                                                         # needs sage.graphs
False

sage.matroids.catalog.R8()

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

The matroid \(R_8\) is a 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. See [Oxl2011], p. 646.

EXAMPLES:

sage: M = matroids.named_matroids.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.named_matroids.NonFano())
True
sage: M.equals(M.dual())
True
sage: M.has_minor(matroids.named_matroids.Fano())
False

sage.matroids.catalog.R9A()

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.named_matroids.R9A()
sage: M.is_valid() # long time
True

sage.matroids.catalog.R9B()

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.named_matroids.R9B()
sage: M.is_valid() # long time
True

sage.matroids.catalog.S8()

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

The matroid \(S_8\) is a 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. See [Oxl2011], p. 648.

EXAMPLES:

sage: from sage.matroids.advanced import *
sage: M = matroids.named_matroids.S8(); M
S8: Binary matroid of rank 4 on 8 elements, type (2, 0)
sage: M.contract('d').is_isomorphic(matroids.named_matroids.Fano())
True
sage: M.delete('d').is_isomorphic(
....:                           matroids.named_matroids.Fano().dual())
False
sage: M.is_graphic()
False
sage: D = get_nonisomorphic_matroids(
....:       list(matroids.named_matroids.Fano().linear_coextensions(
....:                                                 cosimple=True)))
sage: len(D)
2
sage: [N.is_isomorphic(M) for N in D]
[...True...]

sage.matroids.catalog.T12()

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. See [Oxl2011], p. 658.

EXAMPLES:

sage: M = matroids.named_matroids.T12()
sage: M
T12: Binary matroid of rank 6 on 12 elements, type (2, None)
sage: M.is_valid()
True

sage.matroids.catalog.T8()

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

The matroid \(T_8\) is a 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. See [Oxl2011], p. 649.

EXAMPLES:

sage: M = matroids.named_matroids.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.named_matroids.P7())
True
sage: M.has_minor(matroids.Uniform(3, 8))
False

sage.matroids.catalog.TernaryDowling3()

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

The matroid \(Q_3(GF(3)^ imes)\) 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. See [Oxl2011], p. 654.

EXAMPLES:

sage: M = matroids.named_matroids.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.matroids.catalog.Terrahawk()

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. See [CMO2011].

EXAMPLES:

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

sage.matroids.catalog.TicTacToe()

Return the TicTacToe matroid.

The dual of the TicTacToe matroid is not algebraic; it is unknown whether the TicTacToe matroid itself is algebraic. See [Hoc].

EXAMPLES:

sage: M = matroids.named_matroids.TicTacToe()
sage: M.is_valid() # long time
True

sage.matroids.catalog.Uniform(r, n)

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

INPUT:

• r – a nonnegative integer. The rank of the uniform matroid.

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

OUTPUT:

The uniform matroid \(U_{r,n}\).

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. See [Oxl2011], p. 660.

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

Check that bug github issue #15292 was fixed:

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

sage.matroids.catalog.Vamos()

Return the Vamos matroid, represented as circuit closures.

The Vamos matroid, or Vamos cube, or \(V_8\) is a 8-element matroid of rank-4. It violates Ingleton’s condition for representability over a division ring. It is not algebraic. See [Oxl2011], p. 649.

EXAMPLES:

sage: from sage.matroids.advanced import setprint
sage: M = matroids.named_matroids.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() # long time
True

sage.matroids.catalog.Wheel(n, field=None, ring=None)

Return the rank-\(n\) wheel.

INPUT:

• n – 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.

OUTPUT:

The rank-\(n\) wheel matroid, represented as a regular matroid.

See [Oxl2011], p. 659.

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))                              # needs sage.graphs
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+

sage.matroids.catalog.Whirl(n)

Return the rank-\(n\) whirl.

INPUT:

• n – a positive integer. The rank of the desired matroid.

OUTPUT:

The rank-\(n\) whirl matroid, represented as a ternary matroid.

The whirl is the unique relaxation of the wheel. See [Oxl2011], p. 659.

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))                              # needs sage.graphs
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]