For using the igraph C library
igraph provides four set of functions to deal with graph isomorphism problems.
The igraph_isomorphic()
and igraph_subisomorphic()
functions make up the first set (in addition with the igraph_permute_vertices()
function). These functions choose the
algorithm which is best for the supplied input graph. (The choice is
not very sophisticated though, see their documentation for
details.)
The VF2 graph (and subgraph) isomorphism algorithm is implemented in
igraph, these functions are the second set. See igraph_isomorphic_vf2()
and igraph_subisomorphic_vf2()
for
starters.
Functions for the Bliss algorithm constitute the third set,
see igraph_isomorphic_bliss()
.
Finally, the isomorphism classes of all directed graphs with three and four vertices and all undirected graphs with 36 vertices are precomputed and stored in igraph, so for these small graphs there is a separate fast path in the code that does not use more complex, generic isomorphism algorithms.
igraph_error_t igraph_isomorphic(const igraph_t *graph1, const igraph_t *graph2, igraph_bool_t *iso);
In simple terms, two graphs are isomorphic if they become indistinguishable from each other once their vertex labels are removed (rendering the vertices within each graph indistiguishable). More precisely, two graphs are isomorphic if there is a onetoone mapping from the vertices of the first one to the vertices of the second such that it transforms the edge set of the first graph into the edge set of the second. This mapping is called an isomorphism.
Currently, this function supports simple graphs and graphs with selfloops, but does not support multigraphs.
This function decides which graph isomorphism algorithm to be used based on the input graphs. Right now it does the following:
If one graph is directed and the other undirected then an error is triggered.
If one of the graphs has multiedges then an error is triggered.
If the two graphs does not have the same number of vertices
and edges it returns with false
.
Otherwise, if the igraph_isoclass()
function supports both
graphs (which is true for directed graphs with 3 and 4 vertices, and
undirected graphs with 36 vertices), an O(1) algorithm is used with
precomputed data.
Otherwise Bliss is used, see igraph_isomorphic_bliss()
.
Please call the VF2 and Bliss functions directly if you need something more sophisticated, e.g. you need the isomorphic mapping.
Arguments:

The first graph. 

The second graph. 

Pointer to a logical variable, will be set to 
Returns:
Error code. 
See also:
Time complexity: exponential.
igraph_error_t igraph_subisomorphic(const igraph_t *graph1, const igraph_t *graph2, igraph_bool_t *iso);
Check whether graph2
is isomorphic to a subgraph of graph1
.
Currently this function just calls igraph_subisomorphic_vf2()
for all graphs.
Currently this function does not support nonsimple graphs.
Arguments:

The first input graph, may be directed or undirected. This is supposed to be the bigger graph. 

The second input graph, it must have the same
directedness as 

Pointer to a boolean, the result is stored here. 
Returns:
Error code. 
Time complexity: exponential.
igraph_bliss_sh_t
— Splitting heuristics for Bliss.igraph_bliss_info_t
— Information about a Bliss run.igraph_canonical_permutation
— Canonical permutation using Bliss.igraph_isomorphic_bliss
— Graph isomorphism via Bliss.igraph_automorphisms
— Number of automorphisms using Bliss.igraph_automorphism_group
— Automorphism group generators using Bliss.Bliss is a successor of the famous NAUTY algorithm and implementation. While using the same ideas in general, with better heuristics and data structures Bliss outperforms NAUTY on most graphs.
Bliss was developed and implemented by Tommi Junttila and Petteri Kaski at Helsinki University of Technology, Finland. For more information, see the Bliss homepage at https://users.aalto.fi/~tjunttil/bliss/ and the following publication:
Tommi Junttila and Petteri Kaski: "Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs" In ALENEX 2007, pages 135–149, 2007 https://doi.org/10.1137/1.9781611972870.13
Tommi Junttila and Petteri Kaski: "Conflict Propagation and Component Recursion for Canonical Labeling" in TAPAS 2011, pages 151–162, 2011. https://doi.org/10.1007/9783642197543_16
Bliss works with both directed graphs and undirected graphs. It supports graphs with selfloops, but not graphs with multiedges.
Bliss version 0.75 is included in igraph.
typedef enum { IGRAPH_BLISS_F = 0, IGRAPH_BLISS_FL, IGRAPH_BLISS_FS, IGRAPH_BLISS_FM, IGRAPH_BLISS_FLM, IGRAPH_BLISS_FSM } igraph_bliss_sh_t;
IGRAPH_BLISS_FL
provides good performance for many graphs, and is a reasonable
default choice. IGRAPH_BLISS_FSM
is recommended for graphs that have some
combinatorial structure, and is the default of the Bliss library's command
line tool.
Values:

First nonsingleton cell. 

First largest nonsingleton cell. 

First smallest nonsingleton cell. 

First maximally nontrivially connected nonsingleton cell. 

Largest maximally nontrivially connected nonsingleton cell. 

Smallest maximally nontrivially connected nonsingletion cell. 
typedef struct igraph_bliss_info_t { unsigned long nof_nodes; unsigned long nof_leaf_nodes; unsigned long nof_bad_nodes; unsigned long nof_canupdates; unsigned long nof_generators; unsigned long max_level; char *group_size; } igraph_bliss_info_t;
Some secondary information found by the Bliss algorithm is stored here. It is useful if you wany to study the internal working of the algorithm.
Values:

The number of nodes in the search tree. 

The number of leaf nodes in the search tree. 

Number of bad nodes. 

Number of canrep updates. 

Number of generators of the automorphism group. 

Maximum level. 

The size of the automorphism group of the graph,
given as a string. It should be deallocated via

See https://users.aalto.fi/~tjunttil/bliss/ for details about the algorithm and these parameters.
igraph_error_t igraph_canonical_permutation(const igraph_t *graph, const igraph_vector_int_t *colors, igraph_vector_int_t *labeling, igraph_bliss_sh_t sh, igraph_bliss_info_t *info);
This function computes the vertex permutation which transforms
the graph into a canonical form, using the Bliss algorithm.
Two graphs have the same canonical form if and only if they
are isomorphic. Use igraph_is_same_graph()
to compare
two canonical forms.
Arguments:

The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. 

An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. 

Pointer to a vector, the result is stored here. The permutation takes vertex 0 to the first element of the vector, vertex 1 to the second, etc. The vector will be resized as needed. 

The splitting heuristics to be used in Bliss. See 

If not 
Returns:
Error code. 
See also:
Time complexity: exponential, in practice it is fast for many graphs.
igraph_error_t igraph_isomorphic_bliss(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *colors1, const igraph_vector_int_t *colors2, igraph_bool_t *iso, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_bliss_sh_t sh, igraph_bliss_info_t *info1, igraph_bliss_info_t *info2);
This function uses the Bliss graph isomorphism algorithm, a successor of the famous NAUTY algorithm and implementation. Bliss is open source and licensed according to the GNU LGPL. See https://users.aalto.fi/~tjunttil/bliss/ for details. Currently the 0.75 version of Bliss is included in igraph.
Isomorphism testing is implemented by producing the canonical form
of both graphs using igraph_canonical_permutation()
and
comparing them.
Arguments:

The first input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. 

The second input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. 

An optional vertex color vector for the first graph. Supply a null pointer if your graph is not colored. 

An optional vertex color vector for the second graph. Supply a null pointer if your graph is not colored. 

Pointer to a boolean, the result is stored here. 

A vector or 

Similar to 

Splitting heuristics to be used for the graphs. See


If not 

Same as 
Returns:
Error code. 
Time complexity: exponential, but in practice it is quite fast.
igraph_error_t igraph_automorphisms(const igraph_t *graph, const igraph_vector_int_t *colors, igraph_bliss_sh_t sh, igraph_bliss_info_t *info);
The number of automorphisms of a graph is computed using Bliss. The
result is returned as part of the info
structure, in tag group_size
. It is returned as a string, as it can be very high even
for relatively small graphs. If the GNU MP library is used then
this number is exact, otherwise a long double is used
and it is only approximate. See also igraph_bliss_info_t
.
Arguments:

The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. 

An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. 

The splitting heuristics to be used in Bliss. See 

The result is stored here, in particular in the 
Returns:
Error code. 
Time complexity: exponential, in practice it is fast for many graphs.
igraph_error_t igraph_automorphism_group( const igraph_t *graph, const igraph_vector_int_t *colors, igraph_vector_int_list_t *generators, igraph_bliss_sh_t sh, igraph_bliss_info_t *info);
The generators of the automorphism group of a graph are computed using Bliss. The generator set may not be minimal and may depend on the splitting heuristics. The generators are permutations represented using zerobased indexing.
Arguments:

The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. 

An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. 

Must be an initialized pointer vector. It will
contain pointers to 

The splitting heuristics to be used in Bliss. See 

If not 
Returns:
Error code. 
Time complexity: exponential, in practice it is fast for many graphs.
igraph_isomorphic_vf2
— Isomorphism via VF2.igraph_count_isomorphisms_vf2
— Number of isomorphisms via VF2.igraph_get_isomorphisms_vf2
— Collect all isomorphic mappings of two graphs.igraph_get_isomorphisms_vf2_callback
— The generic VF2 interfaceigraph_isohandler_t
— Callback type, called when an isomorphism was foundigraph_isocompat_t
— Callback type, called to check whether two vertices or edges are compatibleigraph_subisomorphic_vf2
— Decide subgraph isomorphism using VF2igraph_count_subisomorphisms_vf2
— Number of subgraph isomorphisms using VF2igraph_get_subisomorphisms_vf2
— Return all subgraph isomorphic mappings.igraph_get_subisomorphisms_vf2_callback
— Generic VF2 function for subgraph isomorphism problems.The VF2 algorithm can search for a subgraph in a larger graph, or check if two graphs are isomorphic. See P. Foggia, C. Sansone, M. Vento, An Improved algorithm for matching large graphs, Proc. of the 3rd IAPRTC15 International Workshop on Graphbased Representations, Italy, 2001.
VF2 supports both vertex and edgecolored graphs, as well as custom vertex or edge compatibility functions.
VF2 works with both directed and undirected graphs. Only simple graphs are supported. Selfloops or multiedges must not be present in the graphs. Currently, the VF2 functions do not check that the input graph is simple: it is the responsibility of the user to pass in valid input.
igraph_error_t igraph_isomorphic_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_bool_t *iso, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
This function performs the VF2 algorithm via calling igraph_get_isomorphisms_vf2_callback()
.
Note that this function cannot be used for
deciding subgraph isomorphism, use igraph_subisomorphic_vf2()
for that.
Arguments:

The first graph, may be directed or undirected. 

The second graph. It must have the same directedness
as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to a logical constant, the result of the algorithm will be placed here. 

Pointer to an initialized vector or a NULL pointer. If not
a NULL pointer then the mapping from 

Pointer to an initialized vector or a NULL pointer. If not
a NULL pointer then the mapping from 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
See also:
Time complexity: exponential, what did you expect?
Example 17.1. File examples/simple/igraph_isomorphic_vf2.c
#include <igraph.h> #include <stdio.h> #include <stdlib.h> int main(void) { igraph_t ring1, ring2; igraph_vector_int_t color1, color2; igraph_vector_int_t perm; igraph_bool_t iso; igraph_integer_t count; igraph_integer_t i; igraph_rng_seed(igraph_rng_default(), 12345); igraph_ring(&ring1, 100, /*directed=*/ 0, /*mutual=*/ 0, /*circular=*/1); igraph_vector_int_init_range(&perm, 0, igraph_vcount(&ring1)); igraph_vector_int_shuffle(&perm); igraph_permute_vertices(&ring1, &ring2, &perm); /* Everything has the same colors */ igraph_vector_int_init(&color1, igraph_vcount(&ring1)); igraph_vector_int_init(&color2, igraph_vcount(&ring2)); igraph_isomorphic_vf2(&ring1, &ring2, &color1, &color2, 0, 0, &iso, 0, 0, 0, 0, 0); if (!iso) { fprintf(stderr, "Single color failed.\n"); return 1; } /* Two colors, just counting */ for (i = 0; i < igraph_vector_int_size(&color1); i += 2) { VECTOR(color1)[i] = VECTOR(color2)[VECTOR(perm)[i]] = 1; } igraph_count_isomorphisms_vf2(&ring1, &ring2, &color1, &color2, 0, 0, &count, 0, 0, 0); if (count != 100) { fprintf(stderr, "Count with two colors failed, expected 100, got %" IGRAPH_PRId ".\n", count); return 2; } igraph_destroy(&ring1); igraph_destroy(&ring2); igraph_vector_int_destroy(&color2); igraph_vector_int_destroy(&perm); /* Two colors, count subisomorphisms */ igraph_ring(&ring1, 100, /*directed=*/ 0, /*mutual=*/ 0, /*circular=*/0); igraph_ring(&ring2, 80, /*directed=*/ 0, /*mutual=*/ 0, /*circular=*/0); igraph_vector_int_init(&color2, igraph_vcount(&ring2)); for (i = 0; i < igraph_vector_int_size(&color1); i += 2) { VECTOR(color1)[i] = 0; VECTOR(color1)[i + 1] = 1; } for (i = 0; i < igraph_vector_int_size(&color2); i += 2) { VECTOR(color2)[i] = 0; VECTOR(color2)[i + 1] = 1; } igraph_count_subisomorphisms_vf2(&ring1, &ring2, &color1, &color2, 0, 0, &count, 0, 0, 0); if (count != 21) { fprintf(stderr, "Count with two colors failed, expected 21, got %" IGRAPH_PRId ".\n", count); return 3; } igraph_vector_int_destroy(&color1); igraph_vector_int_destroy(&color2); igraph_destroy(&ring1); igraph_destroy(&ring2); return 0; }
igraph_error_t igraph_count_isomorphisms_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_integer_t *count, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
This function counts the number of isomorphic mappings between two
graphs. It uses the generic igraph_get_isomorphisms_vf2_callback()
function.
Arguments:

The first input graph, may be directed or undirected. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Point to an integer, the result will be stored here. 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_get_isomorphisms_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_list_t *maps, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
This function finds all the isomorphic mappings between two simple
graphs. It uses the igraph_get_isomorphisms_vf2_callback()
function. Call the function with the same graph as graph1
and graph2
to get automorphisms.
Arguments:

The first input graph, may be directed or undirected. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to a list of integer vectors. On return it is empty if
the input graphs are not isomorphic. Otherwise it contains pointers to


A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_get_isomorphisms_vf2_callback( const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isohandler_t *isohandler_fn, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg );
This function is an implementation of the VF2 isomorphism algorithm, see P. Foggia, C. Sansone, M. Vento, An Improved algorithm for matching large graphs, Proc. of the 3rd IAPRTC15 International Workshop on Graphbased Representations, Italy, 2001.
For using it you need to define a callback function of type
igraph_isohandler_t
. This function will be called whenever VF2
finds an isomorphism between the two graphs. The mapping between
the two graphs will be also provided to this function. If the
callback returns IGRAPH_SUCCESS
, then the search is continued,
otherwise it stops. IGRAPH_STOP
as a return value can be used to
indicate normal premature termination; any other return value will be
treated as an igraph error code, making the caller function return the
same error code as well. The callback function must not destroy the
mapping vectors that are passed to it.
Arguments:

The first input graph. 

The second input graph. 

An optional color vector for the first graph. If color vectors are given for both graphs, then the isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to an initialized vector or 

This is the same as 

The callback function to be called if an
isomorphism is found. See also 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
typedef igraph_error_t igraph_isohandler_t(const igraph_vector_int_t *map12, const igraph_vector_int_t *map21, void *arg);
See the details at the documentation of igraph_get_isomorphisms_vf2_callback()
.
Arguments:

The mapping from the first graph to the second. 

The mapping from the second graph to the first, the
inverse of 

This extra argument was passed to 
Returns:

typedef igraph_bool_t igraph_isocompat_t(const igraph_t *graph1, const igraph_t *graph2, const igraph_integer_t g1_num, const igraph_integer_t g2_num, void *arg);
VF2 (subgraph) isomorphism functions can be restricted by defining relations on the vertices and/or edges of the graphs, and then checking whether the vertices (edges) match according to these relations.
This feature is implemented by two callbacks, one for vertices, one for edges. Every time igraph tries to match a vertex (edge) of the first (sub)graph to a vertex of the second graph, the vertex (edge) compatibility callback is called. The callback returns a logical value, giving whether the two vertices match.
Both callback functions are of type igraph_isocompat_t
.
Arguments:

The first graph. 

The second graph. 

The id of a vertex or edge in the first graph. 

The id of a vertex or edge in the second graph. 

Extra argument to pass to the callback functions. 
Returns:
Logical scalar, whether vertex (or edge) 
igraph_error_t igraph_subisomorphic_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_bool_t *iso, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
Decides whether a subgraph of graph1
is isomorphic to graph2
. It uses igraph_get_subisomorphisms_vf2_callback()
.
Arguments:

The first input graph, may be directed or undirected. This is supposed to be the larger graph. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the subgraph isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to a boolean. The result of the decision problem is stored here. 

Pointer to a vector or 

Pointer to a vector ot 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_count_subisomorphisms_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_integer_t *count, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
Count the number of isomorphisms between subgraphs of graph1
and
graph2
. This function uses igraph_get_subisomorphisms_vf2_callback()
.
Arguments:

The first input graph, may be directed or undirected. This is supposed to be the larger graph. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the subgraph isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to an integer. The number of subgraph isomorphisms is stored here. 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_get_subisomorphisms_vf2(const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_list_t *maps, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg);
This function collects all isomorphic mappings of graph2
to a
subgraph of graph1
. It uses the igraph_get_subisomorphisms_vf2_callback()
function. The graphs should be simple.
Arguments:

The first input graph, may be directed or undirected. This is supposed to be the larger graph. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the subgraph isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to a list of integer vectors. On return it contains
pointers to 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_get_subisomorphisms_vf2_callback( const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isohandler_t *isohandler_fn, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg );
This function is the pair of igraph_get_isomorphisms_vf2_callback()
,
for subgraph isomorphism problems. It searches for subgraphs of graph1
which are isomorphic to graph2
. When it founds an
isomorphic mapping it calls the supplied callback isohandler_fn
.
The mapping (and its inverse) and the additional arg
argument
are supplied to the callback.
Arguments:

The first input graph, may be directed or undirected. This is supposed to be the larger graph. 

The second input graph, it must have the same
directedness as 

An optional color vector for the first graph. If color vectors are given for both graphs, then the subgraph isomorphism is calculated on the colored graphs; i.e. two vertices can match only if their color also matches. Supply a null pointer here if your graphs are not colored. 

An optional color vector for the second graph. See the previous argument for explanation. 

An optional edge color vector for the first graph. The matching edges in the two graphs must have matching colors as well. Supply a null pointer here if your graphs are not edgecolored. 

The edge color vector for the second graph. 

Pointer to a vector or 

Pointer to a vector ot 

A pointer to a function of type 

A pointer to a function of type 

A pointer to a function of type 

Extra argument to supply to functions 
Returns:
Error code. 
Time complexity: exponential.
igraph_error_t igraph_isomorphic_function_vf2( const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isohandler_t *isohandler_fn, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg );
Deprecated since version 0.10.0. Please do not use this function in new
code; use igraph_get_isomorphisms_vf2_callback()
instead.
igraph_error_t igraph_subisomorphic_function_vf2( const igraph_t *graph1, const igraph_t *graph2, const igraph_vector_int_t *vertex_color1, const igraph_vector_int_t *vertex_color2, const igraph_vector_int_t *edge_color1, const igraph_vector_int_t *edge_color2, igraph_vector_int_t *map12, igraph_vector_int_t *map21, igraph_isohandler_t *isohandler_fn, igraph_isocompat_t *node_compat_fn, igraph_isocompat_t *edge_compat_fn, void *arg );
Deprecated since version 0.10.0. Please do not use this function in new
code; use igraph_get_subisomorphisms_vf2_callback()
instead.
The LAD algorithm can search for a subgraph in a larger graph, or check if two graphs are isomorphic. See Christine Solnon: AllDifferentbased Filtering for Subgraph Isomorphism. Artificial Intelligence, 174(1213):850864, 2010. https://doi.org/10.1016/j.artint.2010.05.002 as well as the homepage of the LAD library at http://liris.cnrs.fr/csolnon/LAD.html The implementation in igraph is based on LADv1, but it is modified to use igraph's own memory allocation and error handling.
LAD uses the concept of domains to indicate vertex compatibility when matching the pattern graph. Domains can be used to implement matching of colored vertices.
LAD works with both directed and undirected graphs. Graphs with multiedges are not supported.
igraph_error_t igraph_subisomorphic_lad(const igraph_t *pattern, const igraph_t *target, const igraph_vector_int_list_t *domains, igraph_bool_t *iso, igraph_vector_int_t *map, igraph_vector_int_list_t *maps, igraph_bool_t induced, igraph_integer_t time_limit);
Check whether pattern
is isomorphic to a subgraph os target
.
The original LAD implementation by Christine Solnon was used as the
basis of this code.
See more about LAD at http://liris.cnrs.fr/csolnon/LAD.html and in Christine Solnon: AllDifferentbased Filtering for Subgraph Isomorphism. Artificial Intelligence, 174(1213):850864, 2010. https://doi.org/10.1016/j.artint.2010.05.002
Arguments:

The smaller graph, it can be directed or undirected. 

The bigger graph, it can be directed or undirected. 

A pointer vector, or a null pointer. If a pointer
vector, then it must contain pointers to 

Pointer to a boolean, or a null pointer. If not a null
pointer, then the boolean is set to 

Pointer to a vector or a null pointer. If not a null pointer and a subgraph isomorphism is found, the matching vertices from the target graph are listed here, for each vertex (in vertex ID order) from the pattern graph. 

Pointer to a list of integer vectors or a null pointer. If not
a null pointer, then all subgraph isomorphisms are stored in the
vector list, in 

Boolean, whether to search for induced matching subgraphs. 

Processor time limit in seconds. Supply zero here for no limit. If the time limit is over, then the function signals an error. 
Returns:
Error code 
See also:

Time complexity: exponential.
Example 17.2. File examples/simple/igraph_subisomorphic_lad.c
#include <igraph.h> void print_maps(igraph_vector_int_t *map, igraph_vector_int_list_t *maps) { igraph_integer_t n, i; igraph_vector_int_print(map); n = igraph_vector_int_list_size(maps); for (i = 0; i < n; i++) { igraph_vector_int_t *v = igraph_vector_int_list_get_ptr(maps, i); igraph_vector_int_print(v); } igraph_vector_int_list_clear(maps); } int main(void) { igraph_t target, pattern; igraph_bool_t iso; igraph_vector_int_t map; igraph_vector_int_list_t maps; igraph_integer_t i; int domainsvec[] = { 0, 2, 8, 1, 4, 5, 6, 7, 1, 1, 3, 5, 6, 7, 8, 1, 0, 2, 8, 1, 1, 3, 7, 8, 1, 2 }; igraph_vector_int_list_t domains; igraph_vector_int_t v; igraph_small(&target, 9, IGRAPH_UNDIRECTED, 0, 1, 0, 4, 0, 6, 1, 4, 1, 2, 2, 3, 3, 4, 3, 5, 3, 7, 3, 8, 4, 5, 4, 6, 5, 6, 5, 8, 7, 8, 1); igraph_small(&pattern, 5, IGRAPH_UNDIRECTED, 0, 1, 0, 4, 1, 4, 1, 2, 2, 3, 3, 4, 1); igraph_vector_int_init(&map, 0); igraph_vector_int_list_init(&maps, 0); igraph_subisomorphic_lad(&pattern, &target, /*domains=*/ NULL, &iso, &map, &maps, /*induced=*/ false, /*time_limit=*/ 0); if (!iso) { return 1; } print_maps(&map, &maps); printf("\n"); igraph_subisomorphic_lad(&pattern, &target, /*domains=*/ NULL, &iso, &map, &maps, /*induced=*/ true, /*time_limit=*/ 0); if (!iso) { return 2; } print_maps(&map, &maps); printf("\n"); igraph_vector_int_list_init(&domains, 0); i = 0; igraph_vector_int_init(&v, 0); while (1) { if (domainsvec[i] == 2) { break; } else if (domainsvec[i] == 1) { igraph_vector_int_list_push_back_copy(&domains, &v); igraph_vector_int_clear(&v); } else { igraph_vector_int_push_back(&v, domainsvec[i]); } i++; } igraph_vector_int_destroy(&v); igraph_subisomorphic_lad(&pattern, &target, &domains, &iso, &map, &maps, /*induced=*/ false, /*time_limit=*/ 0); if (!iso) { return 3; } print_maps(&map, &maps); igraph_vector_int_list_destroy(&domains); igraph_vector_int_destroy(&map); igraph_vector_int_list_destroy(&maps); igraph_destroy(&pattern); igraph_destroy(&target); return 0; }
igraph_isoclass
— Determine the isomorphism class of small graphs.igraph_isoclass_subgraph
— The isomorphism class of a subgraph of a graph.igraph_isoclass_create
— Creates a graph from the given isomorphism class.igraph_graph_count
— The number of unlabelled graphs on the given number of vertices.
igraph_error_t igraph_isoclass(const igraph_t *graph, igraph_integer_t *isoclass);
All graphs with a given number of vertices belong to a number of isomorphism classes, with every graph in a given class being isomorphic to each other.
This function gives the isomorphism class (a number) of a graph. Two graphs have the same isomorphism class if and only if they are isomorphic.
The first isomorphism class is numbered zero and it contains the edgeless
graph. The last isomorphism class contains the full graph. The number of
isomorphism classes for directed graphs with three vertices is 16
(between 0 and 15), for undirected graph it is only 4. For graphs
with four vertices it is 218 (directed) and 11 (undirected).
For 5 and 6 vertex undirected graphs, it is 34 and 156, respectively.
These values can also be retrieved using igraph_graph_count()
.
For more information, see https://oeis.org/A000273 and https://oeis.org/A000088.
At the moment, 3 and 4vertex directed graphs and 3 to 6 vertex undirected graphs are supported.
Multiedges and selfloops are ignored by this function.
Arguments:

The graph object. 

Pointer to an integer, the isomorphism class will be stored here. 
Returns:
Error code. 
See also:
Because of some limitations this function works only for graphs with three of four vertices.
Time complexity: O(E), the number of edges in the graph.
igraph_error_t igraph_isoclass_subgraph(const igraph_t *graph, const igraph_vector_int_t *vids, igraph_integer_t *isoclass);
This function identifies the isomorphism class of the subgraph
induced the vertices specified in vids
.
At the moment, 3 and 4vertex directed graphs and 3 to 6 vertex undirected graphs are supported.
Multiedges and selfloops are ignored by this function.
Arguments:

The graph object. 

A vector containing the vertex IDs to be considered as a subgraph. Each vertex ID should be included at most once. 

Pointer to an integer, this will be set to the isomorphism class. 
Returns:
Error code. 
See also:
Time complexity: O((d+n)*n), d is the average degree in the network,
and n is the number of vertices in vids
.
igraph_error_t igraph_isoclass_create(igraph_t *graph, igraph_integer_t size, igraph_integer_t number, igraph_bool_t directed);
This function creates the canonical representative graph of the given isomorphism class.
The isomorphism class is an integer between 0 and the number of
unique unlabeled (i.e. nonisomorphic) graphs on the given number
of vertices and give directedness. See https://oeis.org/A000273
and https://oeis.org/A000088 for the number of directed and
undirected graphs on size
nodes.
At the moment, 3 and 4vertex directed graphs and 3 to 6 vertex undirected graphs are supported.
Arguments:

Pointer to an uninitialized graph object. 

The number of vertices to add to the graph. 

The isomorphism class. 

Logical constant, whether to create a directed graph. 
Returns:
Error code. 
See also:
Time complexity: O(V+E), the number of vertices plus the number of edges in the graph to create.
igraph_error_t igraph_graph_count(igraph_integer_t n, igraph_bool_t directed, igraph_integer_t *count);
Gives the number of unlabelled simple graphs on the specified number of vertices. The "isoclass" of a graph of this size is at most one less than this value.
This function is meant to be used in conjunction with isoclass and motif finder
functions. It will only work for small n
values for which the result is
represetable in an igraph_integer_t. For larger n
values, an overflow
error is raised.
Arguments:

The number of vertices. 

Boolean, whether to consider directed graphs. 

Pointer to an integer, the result will be stored here. 
Returns:
Error code. 
See also:
Time complexity: O(1).
igraph_error_t igraph_permute_vertices(const igraph_t *graph, igraph_t *res, const igraph_vector_int_t *permutation);
This function creates a new graph from the input graph by permuting
its vertices according to the specified mapping. Call this function
with the output of igraph_canonical_permutation()
to create
the canonical form of a graph.
Arguments:

The input graph. 

Pointer to an uninitialized graph object. The new graph is created here. 

The permutation to apply. Vertex 0 is mapped to the first element of the vector, vertex 1 to the second, etc. 
Returns:
Error code. 
Time complexity: O(V+E), linear in terms of the number of vertices and edges.
igraph_error_t igraph_simplify_and_colorize( const igraph_t *graph, igraph_t *res, igraph_vector_int_t *vertex_color, igraph_vector_int_t *edge_color);
This function creates a vertex and edge colored simple graph from the input graph. The vertex colors are computed as the number of incident selfloops to each vertex in the input graph. The edge colors are computed as the number of parallel edges in the input graph that were merged to create each edge in the simple graph.
The resulting colored simple graph is suitable for use by isomorphism checking algorithms such as VF2, which only support simple graphs, but can consider vertex and edge colors.
Arguments:

The graph object, typically having selfloops or multiedges. 

An uninitialized graph object. The result will be stored here 

Computed vertex colors corresponding to selfloop multiplicities. 

Computed edge colors corresponding to edge multiplicities 
Returns:
Error code. 
See also:
igraph_error_t igraph_isomorphic_34( const igraph_t *graph1, const igraph_t *graph2, igraph_bool_t *iso );
Deprecated since version 0.10.0. Please do not use this function in new
code; use igraph_isomorphic()
instead.
If you really care about performance and you know for sure that your
input graphs are simple and have either 3 or 4 vertices for directed graphs,
or 36 vertices for undirected graphs, you can compare their isomorphism
classes obtained from igraph_isoclass()
directly instead of calling
igraph_isomorphic()
; this saves the cost of checking whether the graphs
do not contain multiple edges or selfloops.
Arguments:

The first input graph. 

The second input graph. Must have the same
directedness as 

Pointer to a boolean, the result is stored here. 
Returns:
Error code. 
Time complexity: O(1).
← Chapter 16. Cliques and independent vertex sets  Chapter 18. Graph coloring → 