def min_crossing_path(G, u, v):
D = G.dual()
if len(G[u]) == 1:
_, x = {0} | G[u]
D.vertex_faces[u] = D.vertex_faces[x]
if len(G[v]) == 1:
_, x = {0} | G[v]
D.vertex_faces[v] = D.vertex_faces[v]
paths = [shortest_path(D, start, D.vertex_faces[v]) for start in D.vertex_faces[u]]
best = min(paths, key = len) # path of minimum length
edge_path = []
append = edge_path.append
for f1_id, f2_id in pairwise(best):
edges = tuple(D.faces_sets[f1_id - 1] & D.faces_sets[f2_id - 1]) # common edges to cross a face boundary
append(edges[0])
return edge_path
def planar_gadget(G, x = None, y = None):
V = [G.add_named_vertex("dummy" + str(G.v_id.next())) for i in range(13)]
# V = G + 13 # which notation is clearer?
for u, v in pairwise(V):
G.add_edge(u, v)
G.add_edge(V[0], V[7])
G.add_edge(V[0], V[8])
G.add_edge(V[1], V[9])
G.add_edge(V[2], V[9])
G.add_edge(V[3], V[10])
G.add_edge(V[4], V[10])
G.add_edge(V[5], V[11])
G.add_edge(V[6], V[11])
for i in (V[8], V[9], V[10]):
G.add_edge(V[12], i)
G.add_edge(V[8], V[11])
return G, V[0], V[2], V[4], V[6]
def random_connected_graph(n, m):
""" Return the random connected graph G_{n,m}.
Gives a graph picked randomly out of the set of all graphs
with n nodes and m edges.
Parameters
----------
n : int
The number of nodes.
m : int
The number of edges.
"""
G = Graph()
V = G + n # add n vertices
max_edges = int((n * (n - 1.0)) / 2.0)
m = min(m, max_edges)
# add the first connection line, (n-1) edges, assuring a connected graph
for u, v in pairwise(V):
G.add_edge(u, v)
AddEdge = G.add_edge
E_star = set_copy(combinations(G.vertices, 2))
for u, v in random.sample(E_star - G.edges, m - n + 1):
AddEdge(u, v)
return G
def octahedron():
G = Graph()
v, w, x, y, z1, z2 = (G.add_vertex() for i in xrange(6))
for i, j in pairwise((v, w, x, y, v)):
G.add_edge(i, j)
G.add_edge(i, z1)
G.add_edge(i, z2)
G.is_planar()
return G
def phi_C(G):
"""
for each vertex
N increases by 4
"""
u = random.sample(G.Vertices(), 1)
u = u[0] # u is actually a 1-element list!
w, v, y, x = (G.subdivide(u, i) for i in G.embedding[u])
for i, j in pairwise((w, v, y, x, w)):
G.add_edge(i, j)
return G