Rank function matroids¶
The easiest way to define arbitrary matroids in Sage might be through the
class RankMatroid. All that is required is a groundset and a function that
computes the rank for each given subset.
Of course, since the rank function is used as black box, matroids so defined cannot take advantage of any extra structure your class might have, and rely on default implementations. Besides this, matroids in this class can’t be saved.
Constructions¶
Any function can be used, but no checks are performed, so be careful.
EXAMPLES:
sage: def f(X):
....: return min(len(X), 3)
sage: M = Matroid(groundset=range(6), rank_function=f)
sage: M.is_valid()
True
sage: M.is_isomorphic(matroids.Uniform(3, 6))
True
sage: def g(X):
....: if len(X) >= 3:
....: return 1
....: else:
....: return 0
sage: N = Matroid(groundset='abc', rank_function=g)
sage: N.is_valid()
False
>>> from sage.all import *
>>> def f(X):
... return min(len(X), Integer(3))
>>> M = Matroid(groundset=range(Integer(6)), rank_function=f)
>>> M.is_valid()
True
>>> M.is_isomorphic(matroids.Uniform(Integer(3), Integer(6)))
True
>>> def g(X):
... if len(X) >= Integer(3):
... return Integer(1)
... else:
... return Integer(0)
>>> N = Matroid(groundset='abc', rank_function=g)
>>> N.is_valid()
False
See below for more. It is
recommended to use the Matroid()
function for easy construction of a RankMatroid. For direct access to the
RankMatroid constructor, run:
sage: from sage.matroids.advanced import *
>>> from sage.all import *
>>> from sage.matroids.advanced import *
AUTHORS:
Rudi Pendavingh, Stefan van Zwam (2013-04-01): initial version
Methods¶
- class sage.matroids.rank_matroid.RankMatroid(groundset, rank_function)[source]¶
Bases:
MatroidMatroid specified by its rank function.
INPUT:
groundset– the groundset of a matroidrank_function– a function mapping subsets ofgroundsetto nonnegative integers
OUTPUT: a matroid on
groundsetwhose rank function equalsrank_functionEXAMPLES:
sage: from sage.matroids.advanced import * sage: def f(X): ....: return min(len(X), 3) sage: M = RankMatroid(groundset=range(6), rank_function=f) sage: M.is_valid() True sage: M.is_isomorphic(matroids.Uniform(3, 6)) True
>>> from sage.all import * >>> from sage.matroids.advanced import * >>> def f(X): ... return min(len(X), Integer(3)) >>> M = RankMatroid(groundset=range(Integer(6)), rank_function=f) >>> M.is_valid() True >>> M.is_isomorphic(matroids.Uniform(Integer(3), Integer(6))) True
- groundset()[source]¶
Return the groundset of
self.EXAMPLES:
sage: from sage.matroids.advanced import * sage: M = RankMatroid(range(6), ....: rank_function=matroids.Uniform(3, 6).rank) sage: sorted(M.groundset()) [0, 1, 2, 3, 4, 5]
>>> from sage.all import * >>> from sage.matroids.advanced import * >>> M = RankMatroid(range(Integer(6)), ... rank_function=matroids.Uniform(Integer(3), Integer(6)).rank) >>> sorted(M.groundset()) [0, 1, 2, 3, 4, 5]