def _is_valid_digraph_edge_set( edges, frozen=0 ):
"""
Returns True if the input data is the edge set of a digraph for a quiver (no loops, no 2-cycles, edge-labels of the specified format), and returns False otherwise.
INPUT:
- ``frozen`` -- (integer; default:0) The number of frozen vertices.
EXAMPLES::
sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _is_valid_digraph_edge_set
sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)]] )
The given digraph has edge labels which are not integral or integral 2-tuples.
False
sage: _is_valid_digraph_edge_set( [[0,1],[2,3,(1,-1)]] )
True
sage: _is_valid_digraph_edge_set( [[0,1,'a'],[2,3,(1,-1)],[3,2,(1,-1)]] )
The given digraph or edge list contains oriented 2-cycles.
False
"""
try:
dg = DiGraph()
dg.allow_multiple_edges(True)
dg.add_edges( edges )
# checks if the digraph contains loops
if dg.has_loops():
print "The given digraph or edge list contains loops."
return False
# checks if the digraph contains oriented 2-cycles
if _has_two_cycles( dg ):
print "The given digraph or edge list contains oriented 2-cycles."
return False
# checks if all edge labels are 'None', positive integers or tuples of positive integers
if not all( i == None or ( i in ZZ and i > 0 ) or ( type(i) == tuple and len(i) == 2 and i[0] in ZZ and i[1] in ZZ ) for i in dg.edge_labels() ):
print "The given digraph has edge labels which are not integral or integral 2-tuples."
return False
# checks if all edge labels for multiple edges are 'None' or positive integers
if dg.has_multiple_edges():
for e in set( dg.multiple_edges(labels=False) ):
if not all( i == None or ( i in ZZ and i > 0 ) for i in dg.edge_label( e[0], e[1] ) ):
print "The given digraph or edge list contains multiple edges with non-integral labels."
return False
n = dg.order() - frozen
if n < 0:
print "The number of frozen variables is larger than the number of vertices."
return False
if [ e for e in dg.edges(labels=False) if e[0] >= n] <> []:
print "The given digraph or edge list contains edges within the frozen vertices."
return False
return True
except StandardError:
print "Could not even build a digraph from the input data."
return False
def _digraph_mutate( dg, k, n, m ):
"""
Returns a digraph obtained from dg with n+m vertices by mutating at vertex k.
INPUT:
- ``dg`` -- a digraph with integral edge labels with ``n+m`` vertices
- ``k`` -- the vertex at which ``dg`` is mutated
EXAMPLES::
sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _digraph_mutate
sage: dg = ClusterQuiver(['A',4]).digraph()
sage: dg.edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]
sage: _digraph_mutate(dg,2,4,0).edges()
[(0, 1, (1, -1)), (1, 2, (1, -1)), (3, 2, (1, -1))]
"""
edges = dict( ((v1,v2),label) for v1,v2,label in dg._backend.iterator_in_edges(dg,True) )
in_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v2 == k ]
out_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v1 == k ]
in_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in in_edges ]
out_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in out_edges ]
diag_edges_new = []
diag_edges_del = []
for (v1,v2,label1) in in_edges:
for (w1,w2,label2) in out_edges:
l11,l12 = label1
l21,l22 = label2
if (v1,w2) in edges:
diag_edges_del.append( (v1,w2,edges[(v1,w2)]) )
a,b = edges[(v1,w2)]
a,b = a+l11*l21, b-l12*l22
diag_edges_new.append( (v1,w2,(a,b)) )
elif (w2,v1) in edges:
diag_edges_del.append( (w2,v1,edges[(w2,v1)]) )
a,b = edges[(w2,v1)]
a,b = b+l11*l21, a-l12*l22
if a<0:
diag_edges_new.append( (w2,v1,(b,a)) )
elif a>0:
diag_edges_new.append( (v1,w2,(a,b)) )
else:
a,b = l11*l21,-l12*l22
diag_edges_new.append( (v1,w2,(a,b)) )
del_edges = in_edges + out_edges + diag_edges_del
new_edges = in_edges_new + out_edges_new + diag_edges_new
new_edges += [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if not (v1,v2,edges[(v1,v2)]) in del_edges ]
dg_new = DiGraph()
for v1,v2,label in new_edges:
dg_new._backend.add_edge(v1,v2,label,True)
if dg_new.order() < n+m:
dg_new_vertices = [ v for v in dg_new ]
for i in [ v for v in dg if v not in dg_new_vertices ]:
dg_new.add_vertex(i)
return dg_new
def _matrix_to_digraph( M ):
"""
Returns the digraph obtained from the matrix ``M``.
In order to generate a quiver, we assume that ``M`` is skew-symmetrizable.
EXAMPLES::
sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _matrix_to_digraph
sage: _matrix_to_digraph(matrix(3,[0,1,0,-1,0,-1,0,1,0]))
Digraph on 3 vertices
"""
n = M.ncols()
dg = DiGraph(sparse=True)
for i,j in M.nonzero_positions():
if i >= n: a,b = M[i,j],-M[i,j]
else: a,b = M[i,j],M[j,i]
if a > 0:
dg._backend.add_edge(i,j,(a,b),True)
elif i >= n:
dg._backend.add_edge(j,i,(-a,-b),True)
if dg.order() < M.nrows():
for i in [ index for index in xrange(M.nrows()) if index not in dg ]:
dg.add_vertex(i)
return dg