def spanning_stars(M):
r"""
Returns the edges of a connected subgraph that is a union of
all edges incident some subset of vertices.
INPUT:
- ``M`` -- a matrix defining a bipartite graph G. The vertices are the
rows and columns, if `M[i,j]` is non-zero, then there is an edge
between row i and column 0.
OUTPUT:
A list of tuples `(row,column)` in a spanning forest of the bipartite graph defined by ``M``
EXAMPLES::
sage: edges = sage.matroids.utilities.spanning_stars(matrix([[1,1,1],[1,1,1],[1,1,1]]))
sage: Graph([(x+3, y) for x,y in edges]).is_connected()
True
"""
G = Graph()
m = M.ncols()
for (x,y) in M.dict():
G.add_edge(x+m,y)
delta = (M.nrows()+m)**0.5
# remove low degree vertices
H = []
# candidate vertices
V_0 = set([])
d = 0
while G.order()>0:
(x,d) = min(G.degree_iterator(labels=True),key=itemgetter(1))
if d < delta:
V_0.add(x)
H.extend(G.edges_incident(x,False))
G.delete_vertex(x)
else:
break
# min degree is at least sqrt(n)
# greedily remove vertices
G2 = G.copy()
# set of picked vertices
V_1 = set([])
while G2.order()>0:
# choose vertex with maximum degree in G2
(x,d) = max(G2.degree_iterator(labels=True),key=itemgetter(1))
V_1.add(x)
G2.delete_vertices(G2.neighbors(x))
G2.delete_vertex(x)
# G2 is a graph of all edges incident to V_1
G2 = Graph()
for v in V_1:
for u in G.neighbors(v):
G2.add_edge(u,v)
V = V_0 | V_1
# compute a spanning tree
T = spanning_forest(M)
for (x,y) in T:
if not x in V and not y in V:
V.add(v)
for v in V:
if G.has_vertex(v): # some vertices are not in G
H.extend(G.edges_incident(v,False))
# T contain all edges in some spanning tree
T = []
for (x,y) in H:
if x < m:
t = x
x = y
y = t
T.append((x-m,y))
return T
def spanning_stars(M):
r"""
Returns the edges of a connected subgraph that is a union of
all edges incident some subset of vertices.
INPUT:
- ``M`` -- a matrix defining a bipartite graph G. The vertices are the
rows and columns, if `M[i,j]` is non-zero, then there is an edge
between row i and column 0.
OUTPUT:
A list of tuples `(row,column)` in a spanning forest of the bipartite graph defined by ``M``
EXAMPLES::
sage: edges = sage.matroids.utilities.spanning_stars(matrix([[1,1,1],[1,1,1],[1,1,1]]))
sage: Graph([(x+3, y) for x,y in edges]).is_connected()
True
"""
G = Graph()
m = M.ncols()
for (x, y) in M.dict():
G.add_edge(x + m, y)
delta = (M.nrows() + m)**0.5
# remove low degree vertices
H = []
# candidate vertices
V_0 = set([])
d = 0
while G.order() > 0:
(x, d) = min(G.degree_iterator(labels=True), key=itemgetter(1))
if d < delta:
V_0.add(x)
H.extend(G.edges_incident(x, False))
G.delete_vertex(x)
else:
break
# min degree is at least sqrt(n)
# greedily remove vertices
G2 = G.copy()
# set of picked vertices
V_1 = set([])
while G2.order() > 0:
# choose vertex with maximum degree in G2
(x, d) = max(G2.degree_iterator(labels=True), key=itemgetter(1))
V_1.add(x)
G2.delete_vertices(G2.neighbors(x))
G2.delete_vertex(x)
# G2 is a graph of all edges incident to V_1
G2 = Graph()
for v in V_1:
for u in G.neighbors(v):
G2.add_edge(u, v)
V = V_0 | V_1
# compute a spanning tree
T = spanning_forest(M)
for (x, y) in T:
if not x in V and not y in V:
V.add(v)
for v in V:
if G.has_vertex(v): # some vertices are not in G
H.extend(G.edges_incident(v, False))
# T contain all edges in some spanning tree
T = []
for (x, y) in H:
if x < m:
t = x
x = y
y = t
T.append((x - m, y))
return T
class Isogeny_graph:
# Class for construction isogeny graphs with 1 or more degrees
# ls is a list of primes which define the defining degrees
def __init__(self, E, ls, special=False):
self._ls = ls
j = E.j_invariant()
self.field = E.base_field()
self._graph = Graph(multiedges=True, loops=True)
self._special = False
self._graph.add_vertex(j)
queue = [j]
R = rainbow(len(ls) + 1)
self._rainbow = R
self._edge_colors = {R[i]: [] for i in range(len(ls))}
while queue:
color = 0
s = queue.pop(0)
if s == 0 or s == 1728:
self._special = True
for l in ls:
neighb = isogenous_curves(s, l)
for i in neighb:
if not self._graph.has_vertex(i[0]):
queue.append(i[0])
self._graph.add_vertex(i[0])
if not ((s, i[0], l) in self._edge_colors[R[color]] or
(i[0], s, l) in self._edge_colors[R[color]]):
for _ in range(i[1]):
self._graph.add_edge((s, i[0], l))
self._edge_colors[R[color]].append((s, i[0], l))
color += 1
if self._special and special:
print("Curve with j_invariant 0 or 1728 found, may malfunction.")
def __repr__(self):
return "Isogeny graph of degrees %r" % (self._ls)
# Returns degrees of isogenies
def degrees(self):
return self._ls
# Returns list of all edges
def edges(self):
return self._graph.edges()
# Returns list of all vertices
def vertices(self):
return self._graph.vertices()
# Plots the graph
# Optional arguments figsize, vertex_size, vertex_labels and layout which are the same as in igraph
def plot(self,
figsize=None,
edge_labels=False,
vertex_size=None,
layout=None):
if vertex_size == None:
return self._graph.plot(edge_colors=self._edge_colors,
figsize=figsize,
edge_labels=edge_labels,
layout=layout)
else:
return self._graph.plot(edge_colors=self._edge_colors,
figsize=figsize,
edge_labels=edge_labels,
vertex_size=vertex_size,
layout=layout)