VF2 Algorithm¶

Introduction¶

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.

Examples

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 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}

Subgraph Isomorphism¶

Graph theory literature can be ambiguious 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’. Currently, it is not possible to check for monomorphisms.

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 subgrpah, then:

N’ is the subset of nodes in N related by edges in E’ E’ is a subset of E

References

[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. http://amalfi.dis.unina.it/graph/db/papers/vf-algorithm.pdf

Notes

Modified to handle undirected graphs. Modified to handle multiple edges.

In general, this problem is NP-Complete.