Circuit closures matroids¶
Matroids are characterized by a list of all tuples \((C, k)\), where \(C\) is the closure of a circuit, and \(k\) the rank of \(C\). The CircuitClosuresMatroid class implements matroids using this information as data.
Construction¶
A CircuitClosuresMatroid can be created from another matroid or from a
list of circuit-closures. For a full description of allowed inputs, see
below.
It is recommended to use the
Matroid() function for a more
flexible construction of a CircuitClosuresMatroid. For direct access to
the CircuitClosuresMatroid constructor, run:
sage: from sage.matroids.advanced import *
>>> from sage.all import *
>>> from sage.matroids.advanced import *
See also sage.matroids.advanced.
EXAMPLES:
sage: from sage.matroids.advanced import *
sage: M1 = CircuitClosuresMatroid(groundset='abcdef',
....:                 circuit_closures={2: ['abc', 'ade'], 3: ['abcdef']})
sage: M2 = Matroid(circuit_closures={2: ['abc', 'ade'], 3: ['abcdef']})
sage: M3 = Matroid(circuit_closures=[(2, 'abc'),
....:                                (3, 'abcdef'), (2, 'ade')])
sage: M1 == M2
True
sage: M1 == M3
True
>>> from sage.all import *
>>> from sage.matroids.advanced import *
>>> M1 = CircuitClosuresMatroid(groundset='abcdef',
...                 circuit_closures={Integer(2): ['abc', 'ade'], Integer(3): ['abcdef']})
>>> M2 = Matroid(circuit_closures={Integer(2): ['abc', 'ade'], Integer(3): ['abcdef']})
>>> M3 = Matroid(circuit_closures=[(Integer(2), 'abc'),
...                                (Integer(3), 'abcdef'), (Integer(2), 'ade')])
>>> M1 == M2
True
>>> M1 == M3
True
Note that the class does not implement custom minor and dual operations:
sage: from sage.matroids.advanced import *
sage: M = CircuitClosuresMatroid(groundset='abcdef',
....:                 circuit_closures={2: ['abc', 'ade'], 3: ['abcdef']})
sage: isinstance(M.contract('a'), MinorMatroid)
True
sage: isinstance(M.dual(), DualMatroid)
True
>>> from sage.all import *
>>> from sage.matroids.advanced import *
>>> M = CircuitClosuresMatroid(groundset='abcdef',
...                 circuit_closures={Integer(2): ['abc', 'ade'], Integer(3): ['abcdef']})
>>> isinstance(M.contract('a'), MinorMatroid)
True
>>> isinstance(M.dual(), DualMatroid)
True
AUTHORS:
- Rudi Pendavingh, Stefan van Zwam (2013-04-01): initial version 
- class sage.matroids.circuit_closures_matroid.CircuitClosuresMatroid[source]¶
- Bases: - Matroid- A general matroid \(M\) is characterized by its rank \(r(M)\) and the set of pairs - \((k, \{\) closure \((C) : C ` circuit of ` M, r(C)=k\})\) for \(k=0, .., r(M)-1\) - As each independent set of size \(k\) is in at most one closure(\(C\)) of rank \(k\), and each closure(\(C\)) of rank \(k\) contains at least \(k + 1\) independent sets of size \(k\), there are at most \(\binom{n}{k}/(k + 1)\) such closures-of-circuits of rank \(k\). Each closure(\(C\)) takes \(O(n)\) bits to store, giving an upper bound of \(O(2^n)\) on the space complexity of the entire matroid. - A subset \(X\) of the groundset is independent if and only if - \(| X \cap ` closure `(C) | \leq k\) for all circuits \(C\) of \(M\) with \(r(C)=k\). - So determining whether a set is independent takes time proportional to the space complexity of the matroid. - INPUT: - M– matroid (default:- None)
- groundset– groundset of a matroid (default:- None)
- circuit_closures– dictionary (default:- None); the collection of circuit closures of a matroid presented as a dictionary whose keys are ranks, and whose values are sets of circuit closures of the specified rank
 - OUTPUT: - If the input is a matroid - M, return a- CircuitClosuresMatroidinstance representing- M.
- Otherwise, return a - CircuitClosuresMatroidinstance based on- groundsetand- circuit_closures.
 - Note - For a more flexible means of input, use the - Matroid()function.- EXAMPLES: - sage: from sage.matroids.advanced import * sage: M = CircuitClosuresMatroid(matroids.catalog.Fano()) sage: M Matroid of rank 3 on 7 elements with circuit-closures {2: {{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'}, {'b', 'c', 'd'}, {'b', 'e', 'g'}, {'c', 'f', 'g'}, {'d', 'e', 'f'}}, 3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g'}}} sage: M = CircuitClosuresMatroid(groundset='abcdefgh', ....: circuit_closures={3: ['edfg', 'acdg', 'bcfg', 'cefh', ....: 'afgh', 'abce', 'abdf', 'begh', 'bcdh', 'adeh'], ....: 4: ['abcdefgh']}) sage: M.equals(matroids.catalog.P8()) True - >>> from sage.all import * >>> from sage.matroids.advanced import * >>> M = CircuitClosuresMatroid(matroids.catalog.Fano()) >>> M Matroid of rank 3 on 7 elements with circuit-closures {2: {{'a', 'b', 'f'}, {'a', 'c', 'e'}, {'a', 'd', 'g'}, {'b', 'c', 'd'}, {'b', 'e', 'g'}, {'c', 'f', 'g'}, {'d', 'e', 'f'}}, 3: {{'a', 'b', 'c', 'd', 'e', 'f', 'g'}}} >>> M = CircuitClosuresMatroid(groundset='abcdefgh', ... circuit_closures={Integer(3): ['edfg', 'acdg', 'bcfg', 'cefh', ... 'afgh', 'abce', 'abdf', 'begh', 'bcdh', 'adeh'], ... Integer(4): ['abcdefgh']}) >>> M.equals(matroids.catalog.P8()) True - circuit_closures()[source]¶
- Return the closures of circuits of the matroid. - A circuit closure is a closed set containing a circuit. - OUTPUT: dictionary containing the circuit closures of the matroid, indexed by their ranks - See also - EXAMPLES: - sage: from sage.matroids.advanced import * sage: M = CircuitClosuresMatroid(matroids.catalog.Fano()) sage: CC = M.circuit_closures() sage: len(CC[2]) 7 sage: len(CC[3]) 1 sage: len(CC[1]) Traceback (most recent call last): ... KeyError: 1 sage: [sorted(X) for X in CC[3]] [['a', 'b', 'c', 'd', 'e', 'f', 'g']] - >>> from sage.all import * >>> from sage.matroids.advanced import * >>> M = CircuitClosuresMatroid(matroids.catalog.Fano()) >>> CC = M.circuit_closures() >>> len(CC[Integer(2)]) 7 >>> len(CC[Integer(3)]) 1 >>> len(CC[Integer(1)]) Traceback (most recent call last): ... KeyError: 1 >>> [sorted(X) for X in CC[Integer(3)]] [['a', 'b', 'c', 'd', 'e', 'f', 'g']] 
 - full_rank()[source]¶
- Return the rank of the matroid. - The rank of the matroid is the size of the largest independent subset of the groundset. - OUTPUT: integer - EXAMPLES: - sage: M = matroids.catalog.Vamos() sage: M.full_rank() 4 sage: M.dual().full_rank() 4 - >>> from sage.all import * >>> M = matroids.catalog.Vamos() >>> M.full_rank() 4 >>> M.dual().full_rank() 4 
 - groundset()[source]¶
- Return the groundset of the matroid. - The groundset is the set of elements that comprise the matroid. - OUTPUT: frozenset - EXAMPLES: - sage: M = matroids.catalog.Pappus() sage: sorted(M.groundset()) ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'] - >>> from sage.all import * >>> M = matroids.catalog.Pappus() >>> sorted(M.groundset()) ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'] 
 - relabel(mapping)[source]¶
- Return an isomorphic matroid with relabeled groundset. - The output is obtained by relabeling each element \(e\) by - mapping[e], where- mappingis a given injective map. If- mapping[e]is not defined, then the identity map is assumed.- INPUT: - mapping– a Python object such that- mapping[e]is the new label of \(e\)
 - OUTPUT: matroid - EXAMPLES: - sage: from sage.matroids.circuit_closures_matroid import CircuitClosuresMatroid sage: M = CircuitClosuresMatroid(matroids.catalog.RelaxedNonFano()) sage: sorted(M.groundset()) [0, 1, 2, 3, 4, 5, 6] sage: N = M.relabel({'g': 'x', 0: 'z'}) # 'g': 'x' is ignored sage: from sage.matroids.utilities import cmp_elements_key sage: sorted(N.groundset(), key=cmp_elements_key) [1, 2, 3, 4, 5, 6, 'z'] sage: M.is_isomorphic(N) True - >>> from sage.all import * >>> from sage.matroids.circuit_closures_matroid import CircuitClosuresMatroid >>> M = CircuitClosuresMatroid(matroids.catalog.RelaxedNonFano()) >>> sorted(M.groundset()) [0, 1, 2, 3, 4, 5, 6] >>> N = M.relabel({'g': 'x', Integer(0): 'z'}) # 'g': 'x' is ignored >>> from sage.matroids.utilities import cmp_elements_key >>> sorted(N.groundset(), key=cmp_elements_key) [1, 2, 3, 4, 5, 6, 'z'] >>> M.is_isomorphic(N) True