"""
Graph isomorphism functions.
"""
import networkx as nx
from networkx.exception import NetworkXError
__all__ = [
"could_be_isomorphic",
"fast_could_be_isomorphic",
"faster_could_be_isomorphic",
"is_isomorphic",
]
[docs]
@nx._dispatchable(graphs={"G1": 0, "G2": 1})
def could_be_isomorphic(G1, G2):
"""Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters
----------
G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
-----
Checks for matching degree, triangle, and number of cliques sequences.
The triangle sequence contains the number of triangles each node is part of.
The clique sequence contains for each node the number of maximal cliques
involving that node.
"""
# Check global properties
if G1.order() != G2.order():
return False
# Check local properties
d1 = G1.degree()
t1 = nx.triangles(G1)
clqs_1 = list(nx.find_cliques(G1))
c1 = {n: sum(1 for c in clqs_1 if n in c) for n in G1} # number of cliques
props1 = [[d, t1[v], c1[v]] for v, d in d1]
props1.sort()
d2 = G2.degree()
t2 = nx.triangles(G2)
clqs_2 = list(nx.find_cliques(G2))
c2 = {n: sum(1 for c in clqs_2 if n in c) for n in G2} # number of cliques
props2 = [[d, t2[v], c2[v]] for v, d in d2]
props2.sort()
if props1 != props2:
return False
# OK...
return True
graph_could_be_isomorphic = could_be_isomorphic
[docs]
@nx._dispatchable(graphs={"G1": 0, "G2": 1})
def fast_could_be_isomorphic(G1, G2):
"""Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters
----------
G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
-----
Checks for matching degree and triangle sequences. The triangle
sequence contains the number of triangles each node is part of.
"""
# Check global properties
if G1.order() != G2.order():
return False
# Check local properties
d1 = G1.degree()
t1 = nx.triangles(G1)
props1 = [[d, t1[v]] for v, d in d1]
props1.sort()
d2 = G2.degree()
t2 = nx.triangles(G2)
props2 = [[d, t2[v]] for v, d in d2]
props2.sort()
if props1 != props2:
return False
# OK...
return True
fast_graph_could_be_isomorphic = fast_could_be_isomorphic
[docs]
@nx._dispatchable(graphs={"G1": 0, "G2": 1})
def faster_could_be_isomorphic(G1, G2):
"""Returns False if graphs are definitely not isomorphic.
True does NOT guarantee isomorphism.
Parameters
----------
G1, G2 : graphs
The two graphs G1 and G2 must be the same type.
Notes
-----
Checks for matching degree sequences.
"""
# Check global properties
if G1.order() != G2.order():
return False
# Check local properties
d1 = sorted(d for n, d in G1.degree())
d2 = sorted(d for n, d in G2.degree())
if d1 != d2:
return False
# OK...
return True
faster_graph_could_be_isomorphic = faster_could_be_isomorphic
[docs]
@nx._dispatchable(
graphs={"G1": 0, "G2": 1},
preserve_edge_attrs="edge_match",
preserve_node_attrs="node_match",
)
def is_isomorphic(G1, G2, node_match=None, edge_match=None):
"""Returns True if the graphs G1 and G2 are isomorphic and False otherwise.
Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be the same type.
node_match : callable
A function that returns True if node n1 in G1 and n2 in G2 should
be considered equal during the isomorphism test.
If node_match is not specified then node attributes are not considered.
The function will be called like
node_match(G1.nodes[n1], G2.nodes[n2]).
That is, the function will receive the node attribute dictionaries
for n1 and n2 as inputs.
edge_match : callable
A function that returns True if the edge attribute dictionary
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during the isomorphism test. If edge_match is
not specified then edge attributes are not considered.
The function will be called like
edge_match(G1[u1][v1], G2[u2][v2]).
That is, the function will receive the edge attribute dictionaries
of the edges under consideration.
Notes
-----
Uses the vf2 algorithm [1]_.
Examples
--------
>>> import networkx.algorithms.isomorphism as iso
For digraphs G1 and G2, using 'weight' edge attribute (default: 1)
>>> G1 = nx.DiGraph()
>>> G2 = nx.DiGraph()
>>> nx.add_path(G1, [1, 2, 3, 4], weight=1)
>>> nx.add_path(G2, [10, 20, 30, 40], weight=2)
>>> em = iso.numerical_edge_match("weight", 1)
>>> nx.is_isomorphic(G1, G2) # no weights considered
True
>>> nx.is_isomorphic(G1, G2, edge_match=em) # match weights
False
For multidigraphs G1 and G2, using 'fill' node attribute (default: '')
>>> G1 = nx.MultiDiGraph()
>>> G2 = nx.MultiDiGraph()
>>> G1.add_nodes_from([1, 2, 3], fill="red")
>>> G2.add_nodes_from([10, 20, 30, 40], fill="red")
>>> nx.add_path(G1, [1, 2, 3, 4], weight=3, linewidth=2.5)
>>> nx.add_path(G2, [10, 20, 30, 40], weight=3)
>>> nm = iso.categorical_node_match("fill", "red")
>>> nx.is_isomorphic(G1, G2, node_match=nm)
True
For multidigraphs G1 and G2, using 'weight' edge attribute (default: 7)
>>> G1.add_edge(1, 2, weight=7)
1
>>> G2.add_edge(10, 20)
1
>>> em = iso.numerical_multiedge_match("weight", 7, rtol=1e-6)
>>> nx.is_isomorphic(G1, G2, edge_match=em)
True
For multigraphs G1 and G2, using 'weight' and 'linewidth' edge attributes
with default values 7 and 2.5. Also using 'fill' node attribute with
default value 'red'.
>>> em = iso.numerical_multiedge_match(["weight", "linewidth"], [7, 2.5])
>>> nm = iso.categorical_node_match("fill", "red")
>>> nx.is_isomorphic(G1, G2, edge_match=em, node_match=nm)
True
See Also
--------
numerical_node_match, numerical_edge_match, numerical_multiedge_match
categorical_node_match, categorical_edge_match, categorical_multiedge_match
References
----------
.. [1] L. P. Cordella, P. Foggia, C. Sansone, M. Vento,
"An Improved Algorithm for Matching Large Graphs",
3rd IAPR-TC15 Workshop on Graph-based Representations in
Pattern Recognition, Cuen, pp. 149-159, 2001.
https://www.researchgate.net/publication/200034365_An_Improved_Algorithm_for_Matching_Large_Graphs
"""
if G1.is_directed() and G2.is_directed():
GM = nx.algorithms.isomorphism.DiGraphMatcher
elif (not G1.is_directed()) and (not G2.is_directed()):
GM = nx.algorithms.isomorphism.GraphMatcher
else:
raise NetworkXError("Graphs G1 and G2 are not of the same type.")
gm = GM(G1, G2, node_match=node_match, edge_match=edge_match)
return gm.is_isomorphic()
