def braid(self, n, K=QQ, names=None):
r"""
The braid arrangement.
INPUT:
- ``n`` -- integer
- ``K`` -- field (default: ``QQ``)
- ``names`` -- tuple of strings or ``None`` (default); the
variable names for the ambient space
OUTPUT:
The hyperplane arrangement consisting of the `n(n-1)/2`
hyperplanes `\{ x_i - x_j = 0 : 1 \leq i \leq j \leq n \}`.
EXAMPLES::
sage: hyperplane_arrangements.braid(4)
Arrangement of 6 hyperplanes of dimension 4 and rank 3
"""
x = polygen(QQ, 'x')
A = self.graphical(graphs.CompleteGraph(n), K, names=names)
charpoly = prod(x - i for i in range(n))
A.characteristic_polynomial.set_cache(charpoly)
return A
def complete_graph_fundamental_cycles(draw):
n = draw(integers(3, 12))
G = EdgeLabelledGraph(graphs.CompleteGraph(n))
u = draw(sampled_from(G))
found = {u}
tree_edges = set()
while len(found) < len(G):
v = draw(sampled_from(G.neighbors(u)))
if not v in found:
found.add(v)
e = G.edge_label(u, v)
tree_edges.add((u, v, e))
u = v
tree_edge_labels = {e for _, _, e in tree_edges}
def find_cycle(e):
return [
label for _, _, label in G.subgraph(
edges=tree_edges
| {e}).is_forest(certificate=True, output='edge')[1]
]
cycles = [find_cycle(e) for e in G.edges() if e[2] not in tree_edge_labels]
cycles = draw(permutations(cycles))
return G, cycles
class TestSPQR(unittest.TestCase):
@settings(suppress_health_check=[HealthCheck.too_slow])
@given(complete_graph_fundamental_cycles())
@example((EdgeLabelledGraph(graphs.CompleteGraph(6)), [[10, 1, 3],
[12, 2, 3],
[11, 10, 3, 4],
[10, 5, 0, 3],
[12, 6, 0, 3],
[7, 0, 3],
[10, 11, 8, 0, 3],
[9, 10, 12],
[13, 11, 10, 12],
[11, 10, 14]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(6)), [[9, 5, 0, 2],
[9, 1, 2],
[7, 5, 9, 2, 3],
[5, 6, 9],
[8, 5, 9, 2, 4],
[7, 5, 10],
[11, 9, 2, 4],
[7, 5, 9, 12],
[13, 2, 4],
[14, 7, 5, 9, 2,
4]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(5)), [[9, 8, 1, 2],
[8, 1, 3],
[4, 0, 1],
[8, 9, 5, 0, 1],
[8, 6, 0, 1],
[9, 7, 8]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(5)), [[4, 0, 1],
[9, 2, 3],
[5, 4, 1, 2],
[9, 6, 4, 1, 2],
[7, 1, 2],
[9, 8, 1, 2]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(5)), [[7, 1, 2],
[8, 7, 2, 3],
[7, 4, 0, 2],
[5, 0, 2],
[7, 8, 6, 0, 2],
[8, 7, 9]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(5)), [[7, 1, 2],
[9, 7, 1, 3],
[4, 0, 1],
[7, 5, 0, 1],
[7, 9, 6, 0, 1],
[9, 7, 8]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(4)), [[5, 4, 0, 1],
[5, 1, 2],
[3, 0, 1]]))
@example((EdgeLabelledGraph(graphs.CompleteGraph(4)), [[5, 1, 2],
[3, 0, 1],
[5, 4, 0, 1]]))
def test_complete_graph(self, G_cycles):
G, cycles = G_cycles
reserved_labels = 1000
if len(G.edge_labels()) > 0:
reserved_labels = max(reserved_labels, max(G.edge_labels()))
decomposition = SPQRTree(reserved_labels=reserved_labels)
for cycle in cycles:
self.assertTrue(decomposition.add_circuit(cycle))
decomposition.validate()
self.assertEqual(len(decomposition), 1)
H = decomposition.graph()
expected_edge_neighborhoods = \
{ frozenset(frozenset([G.edge_label(u,v)]) for v in G.neighbors(u))
for u in
G }
actual_edge_neighborhoods = \
{ frozenset(frozenset(H.edge_label(u,v)) for v in H.neighbors(u))
for u in H }
self.assertEqual(actual_edge_neighborhoods,
expected_edge_neighborhoods)
@given(integers(0, 10))
@example(6)
def test_complete_graph_star(self, n):
decomposition = SPQRTree(reserved_labels=1000)
k = n - 1
for i in range(0, n - 1):
for j in range(i + 1, n - 1):
self.assertTrue(decomposition.add_circuit([i, j, k]))
k += 1
decomposition.validate()
@given(integers(0, 10))
def test_complete_graph_path(self, n):
decomposition = SPQRTree(reserved_labels=1000)
k = n - 1
for i in range(0, n - 1):
for j in range(i + 1, n - 1):
circuit = list(range(i, j + 1))
circuit.append(k)
self.assertTrue(decomposition.add_circuit(circuit))
k += 1
decomposition.validate()
def test_k5(self):
decomposition = SPQRTree(reserved_labels=1000)
self.assertTrue(decomposition.add_circuit([0, 1, 4]))
self.assertTrue(decomposition.add_circuit([0, 2, 5]))
self.assertTrue(decomposition.add_circuit([0, 3, 6]))
self.assertTrue(decomposition.add_circuit([1, 3, 8]))
self.assertTrue(decomposition.add_circuit([2, 3, 9]))
self.assertTrue(decomposition.add_circuit([1, 2, 7]))
decomposition.validate()