# Matroid construction¶

## Theory¶

Matroids are combinatorial structures that capture the abstract properties of (linear/algebraic/…) dependence. Formally, a matroid is a pair $$M = (E, I)$$ of a finite set $$E$$, the groundset, and a collection of subsets $$I$$, the independent sets, subject to the following axioms:

• $$I$$ contains the empty set
• If $$X$$ is a set in $$I$$, then each subset of $$X$$ is in $$I$$
• If two subsets $$X$$, $$Y$$ are in $$I$$, and $$|X| > |Y|$$, then there exists $$x \in X - Y$$ such that $$Y + \{x\}$$ is in $$I$$.

See the Wikipedia article on matroids for more theory and examples. Matroids can be obtained from many types of mathematical structures, and Sage supports a number of them.

There are two main entry points to Sage’s matroid functionality. The object matroids. contains a number of constructors for well-known matroids. The function Matroid() allows you to define your own matroids from a variety of sources. We briefly introduce both below; follow the links for more comprehensive documentation.

Each matroid object in Sage comes with a number of built-in operations. An overview can be found in the documentation of the abstract matroid class.

## Built-in matroids¶

For built-in matroids, do the following:

• Within a Sage session, type matroids. (Do not press “Enter”, and do not forget the final period “.”)
• Hit “tab”.

You will see a list of methods which will construct matroids. For example:

sage: M = matroids.Wheel(4)
sage: M.is_connected()
True


or:

sage: U36 = matroids.Uniform(3, 6)
sage: U36.equals(U36.dual())
True


A number of special matroids are collected under a named_matroids submenu. To see which, type matroids.named_matroids.<tab> as above:

sage: F7 = matroids.named_matroids.Fano()
sage: len(F7.nonspanning_circuits())
7


## Constructing matroids¶

To define your own matroid, use the function Matroid(). This function attempts to interpret its arguments to create an appropriate matroid. The input arguments are documented in detail below.

EXAMPLES:

sage: A = Matrix(GF(2), [[1, 0, 0, 0, 1, 1, 1],
....:                    [0, 1, 0, 1, 0, 1, 1],
....:                    [0, 0, 1, 1, 1, 0, 1]])
sage: M = Matroid(A)
sage: M.is_isomorphic(matroids.named_matroids.Fano())
True

sage: M = Matroid(graphs.PetersenGraph())
sage: M.rank()
9


AUTHORS:

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

## Functions¶

sage.matroids.constructor.Matroid(groundset=None, data=None, **kwds)

Construct a matroid.

Matroids are combinatorial structures that capture the abstract properties of (linear/algebraic/…) dependence. Formally, a matroid is a pair $$M = (E, I)$$ of a finite set $$E$$, the groundset, and a collection of subsets $$I$$, the independent sets, subject to the following axioms:

• $$I$$ contains the empty set
• If $$X$$ is a set in $$I$$, then each subset of $$X$$ is in $$I$$
• If two subsets $$X$$, $$Y$$ are in $$I$$, and $$|X| > |Y|$$, then there exists $$x \in X - Y$$ such that $$Y + \{x\}$$ is in $$I$$.

See the Wikipedia article on matroids for more theory and examples. Matroids can be obtained from many types of mathematical structures, and Sage supports a number of them.

There are two main entry points to Sage’s matroid functionality. For built-in matroids, do the following:

• Within a Sage session, type “matroids.” (Do not press “Enter”, and do not forget the final period “.”)
• Hit “tab”.

You will see a list of methods which will construct matroids. For example:

sage: F7 = matroids.named_matroids.Fano()
sage: len(F7.nonspanning_circuits())
7


or:

sage: U36 = matroids.Uniform(3, 6)
sage: U36.equals(U36.dual())
True


To define your own matroid, use the function Matroid(). This function attempts to interpret its arguments to create an appropriate matroid. The following named arguments are supported:

INPUT:

• groundset – (optional) If provided, the groundset of the matroid. Otherwise, the function attempts to determine a groundset from the data.

Exactly one of the following inputs must be given (where data must be a positional argument and anything else must be a keyword argument):

• data – a graph or a matrix or a RevLex-Index string or a list of independent sets containing all bases or a matroid.
• bases – The list of bases (maximal independent sets) of the matroid.
• independent_sets – The list of independent sets of the matroid.
• circuits – The list of circuits of the matroid.
• graph – A graph, whose edges form the elements of the matroid.
• matrix – A matrix representation of the matroid.
• reduced_matrix – A reduced representation of the matroid: if reduced_matrix = A then the matroid is represented by $$[I\ \ A]$$ where $$I$$ is an appropriately sized identity matrix.
• rank_function – A function that computes the rank of each subset. Can only be provided together with a groundset.
• circuit_closures – Either a list of tuples (k, C) with C the closure of a circuit, and k the rank of C, or a dictionary D with D[k] the set of closures of rank-k circuits.
• revlex – the encoding as a string of 0 and * symbols. Used by [MatroidDatabase] and explained in [MMIB2012].
• matroid – An object that is already a matroid. Useful only with the regular option.

Further options:

• regular – (default: False) boolean. If True, output a RegularMatroid instance such that, if the input defines a valid regular matroid, then the output represents this matroid. Note that this option can be combined with any type of input.
• ring – any ring. If provided, and the input is a matrix or reduced_matrix, output will be a linear matroid over the ring or field ring.
• field – any field. Same as ring, but only fields are allowed.
• check – (default: True) boolean. If True and regular is true, the output is checked to make sure it is a valid regular matroid.

Warning

Except for regular matroids, the input is not checked for validity. If your data does not correspond to an actual matroid, the behavior of the methods is undefined and may cause strange errors. To ensure you have a matroid, run M.is_valid().

Note

The Matroid() method will return instances of type BasisMatroid, CircuitClosuresMatroid, LinearMatroid, BinaryMatroid, TernaryMatroid, QuaternaryMatroid, RegularMatroid, or RankMatroid. To import these classes (and other useful functions) directly into Sage’s main namespace, type:

sage: from sage.matroids.advanced import *


EXAMPLES:

Note that in these examples we will often use the fact that strings are iterable in these examples. So we type 'abcd' to denote the list ['a', 'b', 'c', 'd'].

1. List of bases:

All of the following inputs are allowed, and equivalent:

sage: M1 = Matroid(groundset='abcd', bases=['ab', 'ac', 'ad',
....:                                       'bc', 'bd', 'cd'])
sage: M2 = Matroid(bases=['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M3 = Matroid(['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M4 = Matroid('abcd', ['ab', 'ac', 'ad', 'bc', 'bd', 'cd'])
sage: M5 = Matroid('abcd', bases=[['a', 'b'], ['a', 'c'],
....:                             ['a', 'd'], ['b', 'c'],
....:                             ['b', 'd'], ['c', 'd']])
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True
sage: M1 == M5
True


We do not check if the provided input forms an actual matroid:

sage: M1 = Matroid(groundset='abcd', bases=['ab', 'cd'])
sage: M1.full_rank()
2
sage: M1.is_valid()
False


Bases may be repeated:

sage: M1 = Matroid(['ab', 'ac'])
sage: M2 = Matroid(['ab', 'ac', 'ab'])
sage: M1 == M2
True

2. List of independent sets:

sage: M1 = Matroid(groundset='abcd',
....:              independent_sets=['', 'a', 'b', 'c', 'd', 'ab',
....:                               'ac', 'ad', 'bc', 'bd', 'cd'])


We only require that the list of independent sets contains each basis of the matroid; omissions of smaller independent sets and repetitions are allowed:

sage: M1 = Matroid(bases=['ab', 'ac'])
sage: M2 = Matroid(independent_sets=['a', 'ab', 'b', 'ab', 'a',
....:                                'b', 'ac'])
sage: M1 == M2
True

3. List of circuits:

sage: M1 = Matroid(groundset='abc', circuits=['bc'])
sage: M2 = Matroid(bases=['ab', 'ac'])
sage: M1 == M2
True


A matroid specified by a list of circuits gets converted to a BasisMatroid internally:

sage: M = Matroid(groundset='abcd', circuits=['abc', 'abd', 'acd',
....:                                         'bcd'])
sage: type(M)
<type 'sage.matroids.basis_matroid.BasisMatroid'>


Strange things can happen if the input does not satisfy the circuit axioms, and these are not always caught by the is_valid() method. So always check whether your input makes sense!

sage: M = Matroid('abcd', circuits=['ab', 'acd'])
sage: M.is_valid()
True
sage: [sorted(C) for C in M.circuits()]
[['a']]

4. Graph:

Sage has great support for graphs, see sage.graphs.graph.

sage: G = graphs.PetersenGraph()
sage: Matroid(G)
Graphic matroid of rank 9 on 15 elements


If each edge has a unique label, then those are used as the ground set labels:

sage: G = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'c')])
sage: M = Matroid(G)
sage: sorted(M.groundset())
['a', 'b', 'c']


If there are parallel edges, then integers are used for the ground set. If there are no edges in parallel, and is not a complete list of labels, or the labels are not unique, then vertex tuples are used:

sage: G = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'b')])
sage: M = Matroid(G)
sage: sorted(M.groundset())
[(0, 1), (0, 2), (1, 2)]
sage: H = Graph([(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'b'), (1, 2, 'c')], multiedges=True)
sage: N = Matroid(H)
sage: sorted(N.groundset())
[0, 1, 2, 3]


The GraphicMatroid object forces its graph to be connected. If a disconnected graph is used as input, it will connect the components.

sage: G1 = graphs.CycleGraph(3); G2 = graphs.DiamondGraph() sage: G = G1.disjoint_union(G2) sage: M = Matroid(G) sage: M Graphic matroid of rank 5 on 8 elements sage: M.graph() Looped multi-graph on 6 vertices sage: M.graph().is_connected() True sage: M.is_connected() False

If the keyword regular is set to True, the output will instead be an instance of RegularMatroid.

sage: G = Graph([(0, 1), (0, 2), (1, 2)])
sage: M = Matroid(G, regular=True); M
Regular matroid of rank 2 on 3 elements with 3 bases


Note: if a groundset is specified, we assume it is in the same order as G.edge_iterator() provides:

sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)], multiedges=True)
sage: M = Matroid('abcd', G)
sage: M.rank(['b', 'c'])
1


As before, if no edge labels are present and the graph is simple, we use the tuples (i, j) of endpoints. If that fails, we simply use a list [0..m-1]

sage: G = Graph([(0, 1), (0, 2), (1, 2)])
sage: M = Matroid(G, regular=True)
sage: sorted(M.groundset())
[(0, 1), (0, 2), (1, 2)]

sage: G = Graph([(0, 1), (0, 2), (0, 2), (1, 2)], multiedges=True)
sage: M = Matroid(G, regular=True)
sage: sorted(M.groundset())
[0, 1, 2, 3]


When the graph keyword is used, a variety of inputs can be converted to a graph automatically. The following uses a graph6 string (see the Graph method’s documentation):

sage: Matroid(graph=':IAKGsaOscI]Gb~')
Graphic matroid of rank 9 on 17 elements


However, this method is no more clever than Graph():

sage: Matroid(graph=41/2)
Traceback (most recent call last):
...
ValueError: This input cannot be turned into a graph

5. Matrix:

The basic input is a Sage matrix:

sage: A = Matrix(GF(2), [[1, 0, 0, 1, 1, 0],
....:                    [0, 1, 0, 1, 0, 1],
....:                    [0, 0, 1, 0, 1, 1]])
sage: M = Matroid(matrix=A)
sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True


Various shortcuts are possible:

sage: M1 = Matroid(matrix=[[1, 0, 0, 1, 1, 0],
....:                      [0, 1, 0, 1, 0, 1],
....:                      [0, 0, 1, 0, 1, 1]], ring=GF(2))
sage: M2 = Matroid(reduced_matrix=[[1, 1, 0],
....:                              [1, 0, 1],
....:                              [0, 1, 1]], ring=GF(2))
sage: M3 = Matroid(groundset=[0, 1, 2, 3, 4, 5],
....:              matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:              ring=GF(2))
sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M4 = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M1 == M2
True
sage: M1 == M3
True
sage: M1 == M4
True


However, with unnamed arguments the input has to be a Matrix instance, or the function will try to interpret it as a set of bases:

sage: Matroid([0, 1, 2], [[1, 0, 1], [0, 1, 1]])
Traceback (most recent call last):
...
ValueError: basis has wrong cardinality.


If the groundset size equals number of rows plus number of columns, an identity matrix is prepended. Otherwise the groundset size must equal the number of columns:

sage: A = Matrix(GF(2), [[1, 1, 0], [1, 0, 1], [0, 1, 1]])
sage: M = Matroid([0, 1, 2], A)
sage: N = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.rank()
2
sage: N.rank()
3


We automatically create an optimized subclass, if available:

sage: Matroid([0, 1, 2, 3, 4, 5],
....:         matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:         field=GF(2))
Binary matroid of rank 3 on 6 elements, type (2, 7)
sage: Matroid([0, 1, 2, 3, 4, 5],
....:         matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:         field=GF(3))
Ternary matroid of rank 3 on 6 elements, type 0-
sage: Matroid([0, 1, 2, 3, 4, 5],
....:         matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:         field=GF(4, 'x'))
Quaternary matroid of rank 3 on 6 elements
sage: Matroid([0, 1, 2, 3, 4, 5],
....:         matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:         field=GF(2), regular=True)
Regular matroid of rank 3 on 6 elements with 16 bases


Otherwise the generic LinearMatroid class is used:

sage: Matroid([0, 1, 2, 3, 4, 5],
....:         matrix=[[1, 1, 0], [1, 0, 1], [0, 1, 1]],
....:         field=GF(83))
Linear matroid of rank 3 on 6 elements represented over the Finite
Field of size 83


An integer matrix is automatically converted to a matrix over $$\QQ$$. If you really want integers, you can specify the ring explicitly:

sage: A = Matrix([[1, 1, 0], [1, 0, 1], [0, 1, -1]])
sage: A.base_ring()
Integer Ring
sage: M = Matroid([0, 1, 2, 3, 4, 5], A)
sage: M.base_ring()
Rational Field
sage: M = Matroid([0, 1, 2, 3, 4, 5], A, ring=ZZ)
sage: M.base_ring()
Integer Ring

6. Rank function:

Any function mapping subsets to integers can be used as input:

sage: def f(X):
....:     return min(len(X), 2)
sage: M = Matroid('abcd', rank_function=f)
sage: M
Matroid of rank 2 on 4 elements
sage: M.is_isomorphic(matroids.Uniform(2, 4))
True

7. Circuit closures:

This is often a really concise way to specify a matroid. The usual way is a dictionary of lists:

sage: M = Matroid(circuit_closures={3: ['edfg', 'acdg', 'bcfg',
....:     'cefh', 'afgh', 'abce', 'abdf', 'begh', 'bcdh', 'adeh'],
....:     4: ['abcdefgh']})
sage: M.equals(matroids.named_matroids.P8())
True


You can also input tuples $$(k, X)$$ where $$X$$ is the closure of a circuit, and $$k$$ the rank of $$X$$:

sage: M = Matroid(circuit_closures=[(2, 'abd'), (3, 'abcdef'),
....:                               (2, 'bce')])
sage: M.equals(matroids.named_matroids.Q6())
True

8. RevLex-Index:

This requires the groundset to be given and also needs a additional keyword argument rank to specify the rank of the matroid:

sage: M = Matroid("abcdef", "000000******0**", rank=4); M
Matroid of rank 4 on 6 elements with 8 bases
sage: list(M.bases())
[frozenset({'a', 'b', 'd', 'f'}),
frozenset({'a', 'c', 'd', 'f'}),
frozenset({'b', 'c', 'd', 'f'}),
frozenset({'a', 'b', 'e', 'f'}),
frozenset({'a', 'c', 'e', 'f'}),
frozenset({'b', 'c', 'e', 'f'}),
frozenset({'b', 'd', 'e', 'f'}),
frozenset({'c', 'd', 'e', 'f'})]


Only the 0 symbols really matter, any symbol can be used instead of *:

sage: Matroid(“abcdefg”, revlex=”0++++++++0++++0+++++0+–++—-+–++”, rank=4) Matroid of rank 4 on 7 elements with 31 bases

It is checked that the input makes sense (but not that it defines a matroid):

sage: Matroid("abcdef", "000000******0**")
Traceback (most recent call last):
...
TypeError: for RevLex-Index, the rank needs to be specified
sage: Matroid("abcdef", "000000******0**", rank=3)
Traceback (most recent call last):
...
ValueError: expected string of length 20 (6 choose 3), got 15
sage: M = Matroid("abcdef", "*0000000000000*", rank=4); M
Matroid of rank 4 on 6 elements with 2 bases
sage: M.is_valid()
False

9. Matroid:

Most of the time, the matroid itself is returned:

sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M)
sage: N is M
True


But it can be useful with the regular option:

sage: M = Matroid(circuit_closures={2:['adb', 'bec', 'cfa',
....:                                  'def'], 3:['abcdef']})
sage: N = Matroid(M, regular=True)
sage: N
Regular matroid of rank 3 on 6 elements with 16 bases
sage: Matrix(N)
[1 0 0 1 1 0]
[0 1 0 1 1 1]
[0 0 1 0 1 1]


The regular option:

sage: M = Matroid(reduced_matrix=[[1, 1, 0],
....:                             [1, 0, 1],
....:                             [0, 1, 1]], regular=True)
sage: M
Regular matroid of rank 3 on 6 elements with 16 bases

sage: M.is_isomorphic(matroids.CompleteGraphic(4))
True


By default we check if the resulting matroid is actually regular. To increase speed, this check can be skipped:

sage: M = matroids.named_matroids.Fano()
sage: N = Matroid(M, regular=True)
Traceback (most recent call last):
...
ValueError: input is not a valid regular matroid
sage: N = Matroid(M, regular=True, check=False)
sage: N
Regular matroid of rank 3 on 7 elements with 32 bases

sage: N.is_valid()
False


Sometimes the output is regular, but represents a different matroid from the one you intended:

sage: M = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]))
sage: N = Matroid(Matrix(GF(3), [[1, 0, 1, 1], [0, 1, 1, 2]]),
....:             regular=True)
sage: N.is_valid()
True
sage: N.is_isomorphic(M)
False