# VF2 Algorithm#

An implementation of VF2 algorithm for graph isomorphism testing.

The simplest interface to use this module is to call the `is_isomorphic` function.

## 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. 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 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 `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` and `E'` is a subset of `E`.

If `G' = (N', E')` is a node-induced subgraph, then:

`N'` is a subset of `N` and `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'` and `E'` is a subset of `E`.

If `G' = (N', E')` is a monomorphism, then:

`N'` is a subset of `N` and `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.

## 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. https://www.researchgate.net/publication/200034365_An_Improved_Algorithm_for_Matching_Large_Graphs

## Notes#

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).

## Graph Matcher#

 `GraphMatcher.__init__`(G1, G2[, node_match, ...]) Initialize graph matcher. Reinitializes the state of the algorithm. Returns True if G1 and G2 are isomorphic graphs. Returns True if a subgraph of G1 is isomorphic to G2. Returns True if a subgraph of G1 is monomorphic to G2. Generator over isomorphisms between G1 and G2. Generator over isomorphisms between a subgraph of G1 and G2. Generator over monomorphisms between a subgraph of G1 and G2. Iterator over candidate pairs of nodes in G1 and G2. Extends the isomorphism mapping. `GraphMatcher.semantic_feasibility`(G1_node, ...) Returns True if mapping G1_node to G2_node is semantically feasible. `GraphMatcher.syntactic_feasibility`(G1_node, ...) Returns True if adding (G1_node, G2_node) is syntactically feasible.

## DiGraph Matcher#

 `DiGraphMatcher.__init__`(G1, G2[, ...]) Initialize graph matcher. Reinitializes the state of the algorithm. Returns True if G1 and G2 are isomorphic graphs. Returns True if a subgraph of G1 is isomorphic to G2. Returns True if a subgraph of G1 is monomorphic to G2. Generator over isomorphisms between G1 and G2. Generator over isomorphisms between a subgraph of G1 and G2. Generator over monomorphisms between a subgraph of G1 and G2. Iterator over candidate pairs of nodes in G1 and G2. Extends the isomorphism mapping. `DiGraphMatcher.semantic_feasibility`(G1_node, ...) Returns True if mapping G1_node to G2_node is semantically feasible. Returns True if adding (G1_node, G2_node) is syntactically feasible.

## Match helpers#

 `categorical_node_match`(attr, default) Returns a comparison function for a categorical node attribute. `categorical_edge_match`(attr, default) Returns a comparison function for a categorical edge attribute. `categorical_multiedge_match`(attr, default) Returns a comparison function for a categorical edge attribute. `numerical_node_match`(attr, default[, rtol, atol]) Returns a comparison function for a numerical node attribute. `numerical_edge_match`(attr, default[, rtol, atol]) Returns a comparison function for a numerical edge attribute. `numerical_multiedge_match`(attr, default[, ...]) Returns a comparison function for a numerical edge attribute. `generic_node_match`(attr, default, op) Returns a comparison function for a generic attribute. `generic_edge_match`(attr, default, op) Returns a comparison function for a generic attribute. `generic_multiedge_match`(attr, default, op) Returns a comparison function for a generic attribute.