""" 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", ] @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 @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 @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 @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()