To remove in the future –– networkx.algorithms.isomorphism.isomorphvf2
An implementation of VF2 algorithm for graph isomorphism testing.
The simplest interface to use this module is to call networkx.is_isomorphic().
The GraphMatcher and DiGraphMatcher are responsible for matching graphs or directed graphs in a predetermined manner. This usually means a check for an isomorphism, though other checks are also possible. For example, a subgraph of one graph can be checked for isomorphism to a second graph.
Matching is done via syntactic feasibility. It is also possible to check for semantic feasibility. Feasibility, then, is defined as the logical AND of the two functions.
To include a semantic check, the (Di)GraphMatcher class should be subclassed, and the semantic_feasibility() function should be redefined. By default, the semantic feasibility function always returns True. The effect of this is that semantics are not considered in the matching of G1 and G2.
Suppose G1 and G2 are isomorphic graphs. Verification is as follows:
>>> from networkx.algorithms import isomorphism >>> G1 = nx.path_graph(4) >>> G2 = nx.path_graph(4) >>> GM = isomorphism.GraphMatcher(G1, G2) >>> GM.is_isomorphic() True
GM.mapping stores the isomorphism mapping from G1 to G2.
>>> GM.mapping
{0: 0, 1: 1, 2: 2, 3: 3}
Suppose G1 and G2 are isomorphic directed graphs. Verification is as follows:
>>> G1 = nx.path_graph(4, create_using=nx.DiGraph()) >>> G2 = nx.path_graph(4, create_using=nx.DiGraph()) >>> DiGM = isomorphism.DiGraphMatcher(G1, G2) >>> DiGM.is_isomorphic() True
DiGM.mapping stores the isomorphism mapping from G1 to G2.
>>> DiGM.mapping
{0: 0, 1: 1, 2: 2, 3: 3}
Graph theory literature can be ambiguous about the meaning of the above statement, and we seek to clarify it now.
In the VF2 literature, a mapping M is said to be a graph-subgraph isomorphism iff M is an isomorphism between G2 and a subgraph of G1. Thus, to say that G1 and G2 are graph-subgraph isomorphic is to say that a subgraph of G1 is isomorphic to G2.
Other literature uses the phrase 'subgraph isomorphic' as in 'G1 does not have a subgraph isomorphic to G2'. Another use is as an in adverb for isomorphic. Thus, to say that G1 and G2 are subgraph isomorphic is to say that a subgraph of G1 is isomorphic to G2.
Finally, the term 'subgraph' can have multiple meanings. In this context, 'subgraph' always means a 'node-induced subgraph'. Edge-induced subgraph isomorphisms are not directly supported, but one should be able to perform the check by making use of nx.line_graph(). For subgraphs which are not induced, the term 'monomorphism' is preferred over 'isomorphism'.
Let G=(N,E) be a graph with a set of nodes N and set of edges E.
If G'=(N',E') is a subgraph, then:
N' is a subset of N E' is a subset of E
If G'=(N',E') is a node-induced subgraph, then:
N' is a subset of N E' is the subset of edges in E relating nodes in N'
If G'=(N',E') is an edge-induced subgraph, then:
N' is the subset of nodes in N related by edges in E' E' is a subset of E
If G'=(N',E') is a monomorphism, then:
N' is a subset of N E' is a subset of the set of edges in E relating nodes in N'
Note that if G' is a node-induced subgraph of G, then it is always a subgraph monomorphism of G, but the opposite is not always true, as a monomorphism can have fewer edges.
[1] Luigi P. Cordella, Pasquale Foggia, Carlo Sansone, Mario Vento,
"A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 10, pp. 1367-1372, Oct., 2004. http://ieeexplore.ieee.org/iel5/34/29305/01323804.pdf
[2] 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://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.101.5342
syntactic_feasibility(), semantic_feasibility()
The implementation handles both directed and undirected graphs as well as multigraphs.
In general, the subgraph isomorphism problem is NP-complete whereas the graph isomorphism problem is most likely not NP-complete (although no polynomial-time algorithm is known to exist).
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them