def graph_isomorphism(g: 'Graph', h: 'Graph'):
# Combine the two graphs to one graph with two components
combined_graph, division = add_graphs(g, h)
# Do an initial colouring based on degree and colourize
colouring_dict, partitions, max_colour = colourize(combined_graph, True)
# Update the graph labels in the combined graph
for vertex in combined_graph:
vertex.label = colouring_dict[vertex]
# If the graph is not bijective but balanced, we can branch
while not bijective(partitions) and balanced(partitions, division):
# Find the first non bijective partition
# TODO optimize this by doing it in bijective() function, already iterating over it there
old_colouring_dict = colouring_dict
non_bijective_colour = ""
for colour in partitions:
if len(partitions[colour]) != 2:
non_bijective_colour = colour
break
# Set a variable used for breaking out of two loops at once
doublebreak = False
# Iterate over all the vertices in the first found non bijective partition
for vertex_a in partitions[non_bijective_colour]:
# If the vertex is in the component of the new graph that previously (before combining) was one graph (g)
if vertex_a in division[0]:
g_vertex = vertex_a
# Iterate over all the vertices in the first found non bijective partition again
for vertex_b in partitions[non_bijective_colour]:
# If the vertex is now in the other graph (h)
if vertex_b in division[1]:
h_vertex = vertex_b
# Change the labels of the g_vertex and h_vertex
h_vertex.label, g_vertex.label = max_colour, max_colour
# Colourize the new graph
new_colouring_dict, new_partitions, new_max_colour = colourize(
combined_graph, False)
# Update the labels
for vertex_c in combined_graph:
vertex_c.label = new_colouring_dict[vertex_c]
# Check if the new graph is balanced
if balanced(new_partitions, division):
# If balanced, update variables and break out of for-loops
partitions = new_partitions
max_colour = new_max_colour
colouring_dict = new_colouring_dict
doublebreak = True
break
else:
# If not balanced, revert the changes
for vertex_d in combined_graph:
vertex_d.label = old_colouring_dict[vertex_d]
if doublebreak:
break
# The graphs are isomorphic if the partitions are balanced and bijective at the end of the loop
result = bijective(partitions) and balanced(partitions, division)
return combined_graph, result
def recursion(colouring_dict, combined_graph, division, max_colour,
partitions):
if bijective(partitions) and balanced(partitions, division):
return True, 1
else:
old_colouring_dict = colouring_dict
non_bijective_colour = ""
for colour in partitions:
if len(partitions[colour]) != 2:
non_bijective_colour = colour
break
partition_automorphisms = 0
result = False
for vertex_a in partitions[non_bijective_colour]:
if vertex_a in division[0]:
g_vertex = vertex_a
for vertex_b in partitions[non_bijective_colour]:
if vertex_b in division[1]:
h_vertex = vertex_b
h_vertex.label, g_vertex.label = max_colour, max_colour
new_colouring_dict, new_partitions, new_max_colour = colourize(
combined_graph, False)
for vertex_c in combined_graph:
vertex_c.label = new_colouring_dict[vertex_c]
if balanced(new_partitions, division):
rec_result, automorphisms = recursion(
new_colouring_dict, combined_graph, division,
new_max_colour, new_partitions)
partition_automorphisms += automorphisms
result = result or rec_result
for vertex_d in combined_graph: # revert changes
vertex_d.label = old_colouring_dict[vertex_d]
# The graphs are isomorphic if the partitions are balanced and bijective at the end of the loop
return result, partition_automorphisms
def graph_isomorphism(g: 'Graph', h: 'Graph'):
# Combine the two graphs to one graph with two components
combined_graph, division = add_graphs(g, h)
# Do an initial colouring based on degree and colourize
colouring_dict, partitions, max_colour = colourize(combined_graph, True)
# Update the graph labels in the combined graph
for vertex in combined_graph:
vertex.label = colouring_dict[vertex]
result, total_automorphisms = recursion(colouring_dict, combined_graph,
division, max_colour, partitions)
return combined_graph, result, int(math.sqrt(total_automorphisms))
def graph_isomorphism(g: 'Graph', h: 'Graph'):
total_automorphisms = 1
# Combine the two graphs to one graph with two components
combined_graph, division = add_graphs(g, h)
# Do an initial colouring based on degree and colourize
colouring_dict, partitions, max_colour = colourize(combined_graph, True)
# Update the graph labels in the combined graph
for vertex in combined_graph:
vertex.label = colouring_dict[vertex]
# If the graph is not bijective but balanced, we can branch
while not bijective(partitions) and balanced(partitions, division):
# Find the first non bijective partition
old_colouring_dict = colouring_dict
non_bijective_colour = ""
for colour in partitions:
if len(partitions[colour]) != 2:
non_bijective_colour = colour
break
last_stable_partitions = None
last_stable_max_colour = None
last_stable_colouring_dict = None
partition_automorphisms = 0
# Pick a vertex in the non bijective partition of graph g
g_vertex = partitions[non_bijective_colour][0]
nr = 1
while g_vertex not in division[0]:
g_vertex = partitions[non_bijective_colour][nr]
nr += 1
# For every vertex in the non bijective partition of graph h
for vertex_b in partitions[non_bijective_colour]:
if vertex_b in division[1]:
h_vertex = vertex_b
# Change the labels of the g_vertex and h_vertex
h_vertex.label, g_vertex.label = max_colour, max_colour
# Colourize the new graph
new_colouring_dict, new_partitions, new_max_colour = colourize(
combined_graph, False)
# Update the labels
for vertex_c in combined_graph:
vertex_c.label = new_colouring_dict[vertex_c]
# Check if the new graph is balanced
if balanced(new_partitions, division):
# If balanced, update variables and break out of for-loops
last_stable_partitions = new_partitions
last_stable_max_colour = new_max_colour
last_stable_colouring_dict = new_colouring_dict
partition_automorphisms += 1
# Revert the changes so we can look for more stable options
for vertex_d in combined_graph:
vertex_d.label = old_colouring_dict[vertex_d]
# If a stable option was found
if last_stable_partitions is not None:
partitions = last_stable_partitions
max_colour = last_stable_max_colour
colouring_dict = last_stable_colouring_dict
# Update the labels
for vertex_e in combined_graph:
vertex_e.label = colouring_dict[vertex_e]
# Update the automorphisms counter
if partition_automorphisms > 0:
total_automorphisms = total_automorphisms * partition_automorphisms
else:
return combined_graph, False, 0
# The graphs are isomorphic if the partitions are balanced and bijective at the end of the loop
result = bijective(partitions) and balanced(partitions, division)
if not result:
total_automorphisms = 0
return combined_graph, result, total_automorphisms