def make_regular_matroid_from_matroid(matroid):
r"""
Attempt to construct a regular representation of a matroid.
INPUT:
- ``matroid`` -- a matroid.
OUTPUT:
Return a `(0, 1, -1)`-matrix over the integers such that, if the input is
a regular matroid, then the output is a totally unimodular matrix
representing that matroid.
EXAMPLES::
sage: from sage.matroids.utilities import make_regular_matroid_from_matroid
sage: make_regular_matroid_from_matroid(
....: matroids.CompleteGraphic(6)).is_isomorphic(
....: matroids.CompleteGraphic(6))
True
"""
import sage.matroids.linear_matroid
M = matroid
if isinstance(M, sage.matroids.linear_matroid.RegularMatroid):
return M
rk = M.full_rank()
# First create a reduced 0-1 matrix
B = list(M.basis())
NB = list(M.groundset().difference(B))
dB = {}
i = 0
for e in B:
dB[e] = i
i += 1
dNB = {}
i = 0
for e in NB:
dNB[e] = i
i += 1
A = Matrix(ZZ, len(B), len(NB), 0)
G = BipartiteGraph(
A.transpose()
) # Sage's BipartiteGraph uses the column set as first color class. This is an edgeless graph.
for e in NB:
C = M.circuit(B + [e])
for f in C.difference([e]):
A[dB[f], dNB[e]] = 1
# Change some entries from -1 to 1
entries = BipartiteGraph(A.transpose()).edges(labels=False)
while len(entries) > 0:
L = [G.shortest_path(u, v) for u, v in entries]
mindex, minval = min(enumerate(L), key=lambda x: len(x[1]))
# if minval = 0, there is an edge not spanned by the current subgraph. Its entry is free to be scaled any way.
if len(minval) > 0:
# Check the subdeterminant
S = frozenset(L[mindex])
rows = []
cols = []
for i in S:
if i < rk:
rows.append(i)
else:
cols.append(i - rk)
if A[rows, cols].det() != 0:
A[entries[mindex][0], entries[mindex][1] - rk] = -1
G.add_edge(entries[mindex][0], entries[mindex][1])
entries.pop(mindex)
return sage.matroids.linear_matroid.RegularMatroid(groundset=B + NB,
reduced_matrix=A)
def make_regular_matroid_from_matroid(matroid):
r"""
Attempt to construct a regular representation of a matroid.
INPUT:
- ``matroid`` -- a matroid.
OUTPUT:
Return a `(0, 1, -1)`-matrix over the integers such that, if the input is
a regular matroid, then the output is a totally unimodular matrix
representing that matroid.
EXAMPLES::
sage: from sage.matroids.utilities import make_regular_matroid_from_matroid
sage: make_regular_matroid_from_matroid(
....: matroids.CompleteGraphic(6)).is_isomorphic(
....: matroids.CompleteGraphic(6))
True
"""
import sage.matroids.linear_matroid
M = matroid
if isinstance(M, sage.matroids.linear_matroid.RegularMatroid):
return M
rk = M.full_rank()
# First create a reduced 0-1 matrix
B = list(M.basis())
NB = list(M.groundset().difference(B))
dB = {}
i = 0
for e in B:
dB[e] = i
i += 1
dNB = {}
i = 0
for e in NB:
dNB[e] = i
i += 1
A = Matrix(ZZ, len(B), len(NB), 0)
G = BipartiteGraph(A.transpose()) # Sage's BipartiteGraph uses the column set as first color class. This is an edgeless graph.
for e in NB:
C = M.circuit(B + [e])
for f in C.difference([e]):
A[dB[f], dNB[e]] = 1
# Change some entries from -1 to 1
entries = BipartiteGraph(A.transpose()).edges(labels=False)
while len(entries) > 0:
L = [G.shortest_path(u, v) for u, v in entries]
mindex, minval = min(enumerate(L), key=lambda x: len(x[1]))
# if minval = 0, there is an edge not spanned by the current subgraph. Its entry is free to be scaled any way.
if len(minval) > 0:
# Check the subdeterminant
S = frozenset(L[mindex])
rows = []
cols = []
for i in S:
if i < rk:
rows.append(i)
else:
cols.append(i - rk)
if A[rows, cols].det() != 0:
A[entries[mindex][0], entries[mindex][1] - rk] = -1
G.add_edge(entries[mindex][0], entries[mindex][1])
entries.pop(mindex)
return sage.matroids.linear_matroid.RegularMatroid(groundset=B + NB, reduced_matrix=A)