# igraph Reference Manual

For using the igraph C library

Search the manual:

# Chapter 17. Graph isomorphism

## 1. The simple interface

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 3-6 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.

### 1.1. `igraph_isomorphic` — Are two graphs isomorphic?

```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 one-to-one 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 self-loops, 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:

1. If one graph is directed and the other undirected then an error is triggered.

2. If one of the graphs has multi-edges then an error is triggered.

3. If the two graphs does not have the same number of vertices and edges it returns with `false`.

4. Otherwise, if the `igraph_isoclass()` function supports both graphs (which is true for directed graphs with 3 and 4 vertices, and undirected graphs with 3-6 vertices), an O(1) algorithm is used with precomputed data.

5. 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:

 `graph1`: The first graph. `graph2`: The second graph. `iso`: Pointer to a logical variable, will be set to `true` if the two graphs are isomorphic, and `false` otherwise.

Returns:

 Error code.

Time complexity: exponential.

### 1.2. `igraph_subisomorphic` — Decide subgraph isomorphism.

```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 non-simple graphs.

Arguments:

 `graph1`: The first input graph, may be directed or undirected. This is supposed to be the bigger graph. `graph2`: The second input graph, it must have the same directedness as `graph2`, or an error is triggered. This is supposed to be the smaller graph. `iso`: Pointer to a boolean, the result is stored here.

Returns:

 Error code.

Time complexity: exponential.

## 2. The BLISS algorithm

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/10.1007/978-3-642-19754-3_16

Bliss works with both directed graphs and undirected graphs. It supports graphs with self-loops, but not graphs with multi-edges.

Bliss version 0.75 is included in igraph.

### 2.1. `igraph_bliss_sh_t` — Splitting heuristics for Bliss.

```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:

 `IGRAPH_BLISS_F`: First non-singleton cell. `IGRAPH_BLISS_FL`: First largest non-singleton cell. `IGRAPH_BLISS_FS`: First smallest non-singleton cell. `IGRAPH_BLISS_FM`: First maximally non-trivially connected non-singleton cell. `IGRAPH_BLISS_FLM`: Largest maximally non-trivially connected non-singleton cell. `IGRAPH_BLISS_FSM`: Smallest maximally non-trivially connected non-singletion cell.

### 2.2. `igraph_bliss_info_t` — Information about a Bliss run.

```typedef struct igraph_bliss_info_t {
unsigned long nof_nodes;
unsigned long nof_leaf_nodes;
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:

 `nof_nodes`: The number of nodes in the search tree. `nof_leaf_nodes`: The number of leaf nodes in the search tree. `nof_bad_nodes`: Number of bad nodes. `nof_canupdates`: Number of canrep updates. `nof_generators`: Number of generators of the automorphism group. `max_level`: Maximum level. `group_size`: The size of the automorphism group of the graph, given as a string. It should be deallocated via `igraph_free()` if not needed any more.

See https://users.aalto.fi/~tjunttil/bliss/ for details about the algorithm and these parameters.

### 2.3. `igraph_canonical_permutation` — Canonical permutation using Bliss.

```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:

 `graph`: The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. `colors`: An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. `labeling`: 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. `sh`: The splitting heuristics to be used in Bliss. See `igraph_bliss_sh_t`. `info`: If not `NULL` then information on Bliss internals is stored here. The memory used by this structure must to be freed when no longer needed, see `igraph_bliss_info_t`.

Returns:

 Error code.

Time complexity: exponential, in practice it is fast for many graphs.

### 2.4. `igraph_isomorphic_bliss` — Graph isomorphism via Bliss.

```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:

 `graph1`: The first input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. `graph2`: The second input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. `colors1`: An optional vertex color vector for the first graph. Supply a null pointer if your graph is not colored. `colors2`: An optional vertex color vector for the second graph. Supply a null pointer if your graph is not colored. `iso`: Pointer to a boolean, the result is stored here. `map12`: A vector or `NULL` pointer. If not `NULL` then an isomorphic mapping from `graph1` to `graph2` is stored here. If the input graphs are not isomorphic then this vector is cleared, i.e. it will have length zero. `map21`: Similar to `map12`, but for the mapping from `graph2` to `graph1`. `sh`: Splitting heuristics to be used for the graphs. See `igraph_bliss_sh_t`. `info1`: If not `NULL`, information about the canonization of the first input graph is stored here. Note that if the two graphs have different number of vertices or edges, then this is only partially filled. The memory used by this structure should be released when no longer needed, see `igraph_bliss_info_t` for details. `info2`: Same as `info1`, but for the second graph.

Returns:

 Error code.

Time complexity: exponential, but in practice it is quite fast.

### 2.5. `igraph_automorphisms` — Number of automorphisms using Bliss.

```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:

 `graph`: The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. `colors`: An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. `sh`: The splitting heuristics to be used in Bliss. See `igraph_bliss_sh_t`. `info`: The result is stored here, in particular in the `group_size` tag of `info`. The memory used by this structure must be released when no longer needed, see `igraph_bliss_info_t`.

Returns:

 Error code.

Time complexity: exponential, in practice it is fast for many graphs.

### 2.6. `igraph_automorphism_group` — Automorphism group generators using Bliss.

```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 zero-based indexing.

Arguments:

 `graph`: The input graph. Multiple edges between the same nodes are not supported and will cause an incorrect result to be returned. `colors`: An optional vertex color vector for the graph. Supply a null pointer is the graph is not colored. `generators`: Must be an initialized pointer vector. It will contain pointers to `igraph_vector_int_t` objects representing generators of the automorphism group. `sh`: The splitting heuristics to be used in Bliss. See `igraph_bliss_sh_t`. `info`: If not `NULL` then information on Bliss internals is stored here. The memory used by this structure must to be freed when no longer needed, see `igraph_bliss_info_t`.

Returns:

 Error code.

Time complexity: exponential, in practice it is fast for many graphs.

## 3. The VF2 algorithm

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 IAPR-TC-15 International Workshop on Graph-based Representations, Italy, 2001.

VF2 supports both vertex and edge-colored graphs, as well as custom vertex or edge compatibility functions.

VF2 works with both directed and undirected graphs. Only simple graphs are supported. Self-loops or multi-edges 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.

### 3.1. `igraph_isomorphic_vf2` — Isomorphism via VF2.

```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:

 `graph1`: The first graph, may be directed or undirected. `graph2`: The second graph. It must have the same directedness as `graph1`, otherwise an error is reported. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `iso`: Pointer to a logical constant, the result of the algorithm will be placed here. `map12`: Pointer to an initialized vector or a NULL pointer. If not a NULL pointer then the mapping from `graph1` to `graph2` is stored here. If the graphs are not isomorphic then the vector is cleared (i.e. has zero elements). `map21`: Pointer to an initialized vector or a NULL pointer. If not a NULL pointer then the mapping from `graph2` to `graph1` is stored here. If the graphs are not isomorphic then the vector is cleared (i.e. has zero elements). `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

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;
}
```

### 3.2. `igraph_count_isomorphisms_vf2` — Number of isomorphisms via VF2.

```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:

 `graph1`: The first input graph, may be directed or undirected. `graph2`: The second input graph, it must have the same directedness as `graph1`, or an error will be reported. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `count`: Point to an integer, the result will be stored here. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.3. `igraph_get_isomorphisms_vf2` — Collect all isomorphic mappings of two graphs.

```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:

 `graph1`: The first input graph, may be directed or undirected. `graph2`: The second input graph, it must have the same directedness as `graph1`, or an error will be reported. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `maps`: Pointer to a list of integer vectors. On return it is empty if the input graphs are not isomorphic. Otherwise it contains pointers to `igraph_vector_int_t` objects, each vector is an isomorphic mapping of `graph2` to `graph1`. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.4. `igraph_get_isomorphisms_vf2_callback` — The generic VF2 interface

```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 IAPR-TC-15 International Workshop on Graph-based 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:

 `graph1`: The first input graph. `graph2`: The second input graph. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `map12`: Pointer to an initialized vector or `NULL`. If not `NULL` and the supplied graphs are isomorphic then the permutation taking `graph1` to `graph` is stored here. If not `NULL` and the graphs are not isomorphic then a zero-length vector is returned. `map21`: This is the same as `map12`, but for the permutation taking `graph2` to `graph1`. `isohandler_fn`: The callback function to be called if an isomorphism is found. See also `igraph_isohandler_t`. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `isohandler_fn`, `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.5. `igraph_isohandler_t` — Callback type, called when an isomorphism was found

```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:

 `map12`: The mapping from the first graph to the second. `map21`: The mapping from the second graph to the first, the inverse of `map12` basically. `arg`: This extra argument was passed to `igraph_get_isomorphisms_vf2_callback()` when it was called.

Returns:

 `IGRAPH_SUCCESS` to continue the search, `IGRAPH_STOP` to terminate the search. Any other return value is interpreted as an igraph error code, which will then abort the search and return the same error code from the caller function.

### 3.6. `igraph_isocompat_t` — Callback type, called to check whether two vertices or edges are compatible

```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:

 `graph1`: The first graph. `graph2`: The second graph. `g1_num`: The id of a vertex or edge in the first graph. `g2_num`: The id of a vertex or edge in the second graph. `arg`: Extra argument to pass to the callback functions.

Returns:

 Logical scalar, whether vertex (or edge) `g1_num` in `graph1` is compatible with vertex (or edge) `g2_num` in `graph2`.

### 3.7. `igraph_subisomorphic_vf2` — Decide subgraph isomorphism using VF2

```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:

 `graph1`: The first input graph, may be directed or undirected. This is supposed to be the larger graph. `graph2`: The second input graph, it must have the same directedness as `graph1`. This is supposed to be the smaller graph. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `iso`: Pointer to a boolean. The result of the decision problem is stored here. `map12`: Pointer to a vector or `NULL`. If not `NULL`, then an isomorphic mapping from `graph1` to `graph2` is stored here. `map21`: Pointer to a vector ot `NULL`. If not `NULL`, then an isomorphic mapping from `graph2` to `graph1` is stored here. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.8. `igraph_count_subisomorphisms_vf2` — Number of subgraph isomorphisms using VF2

```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:

 `graph1`: The first input graph, may be directed or undirected. This is supposed to be the larger graph. `graph2`: The second input graph, it must have the same directedness as `graph1`. This is supposed to be the smaller graph. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `count`: Pointer to an integer. The number of subgraph isomorphisms is stored here. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.9. `igraph_get_subisomorphisms_vf2` — Return all subgraph isomorphic mappings.

```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:

 `graph1`: The first input graph, may be directed or undirected. This is supposed to be the larger graph. `graph2`: The second input graph, it must have the same directedness as `graph1`. This is supposed to be the smaller graph. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `maps`: Pointer to a list of integer vectors. On return it contains pointers to `igraph_vector_int_t` objects, each vector is an isomorphic mapping of `graph2` to a subgraph of `graph1`. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.10. `igraph_get_subisomorphisms_vf2_callback` — Generic VF2 function for subgraph isomorphism problems.

```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:

 `graph1`: The first input graph, may be directed or undirected. This is supposed to be the larger graph. `graph2`: The second input graph, it must have the same directedness as `graph1`. This is supposed to be the smaller graph. `vertex_color1`: 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. `vertex_color2`: An optional color vector for the second graph. See the previous argument for explanation. `edge_color1`: 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 edge-colored. `edge_color2`: The edge color vector for the second graph. `map12`: Pointer to a vector or `NULL`. If not `NULL`, then an isomorphic mapping from `graph1` to `graph2` is stored here. `map21`: Pointer to a vector ot `NULL`. If not `NULL`, then an isomorphic mapping from `graph2` to `graph1` is stored here. `isohandler_fn`: A pointer to a function of type `igraph_isohandler_t`. This will be called whenever a subgraph isomorphism is found. If the function returns `IGRAPH_SUCCESS`, then the search is continued. If the function returns `IGRAPH_STOP`, the search is terminated normally. Any other value is treated as an igraph error code. `node_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two nodes are compatible. `edge_compat_fn`: A pointer to a function of type `igraph_isocompat_t`. This function will be called by the algorithm to determine whether two edges are compatible. `arg`: Extra argument to supply to functions `isohandler_fn`, `node_compat_fn` and `edge_compat_fn`.

Returns:

 Error code.

Time complexity: exponential.

### 3.11. Deprecated aliases

#### 3.11.1. `igraph_isomorphic_function_vf2` — The generic VF2 interface (deprecated alias).

```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
);
```

### Warning

Deprecated since version 0.10.0. Please do not use this function in new code; use `igraph_get_isomorphisms_vf2_callback()` instead.

#### 3.11.2. `igraph_subisomorphic_function_vf2` — Generic VF2 function for subgraph isomorphism problems (deprecated alias).

```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
);
```

### Warning

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: AllDifferent-based Filtering for Subgraph Isomorphism. Artificial Intelligence, 174(12-13):850-864, 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 multi-edges are not supported.

### 4.1. `igraph_subisomorphic_lad` — Check subgraph isomorphism with the LAD algorithm

```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: AllDifferent-based Filtering for Subgraph Isomorphism. Artificial Intelligence, 174(12-13):850-864, 2010. https://doi.org/10.1016/j.artint.2010.05.002

Arguments:

 `pattern`: The smaller graph, it can be directed or undirected. `target`: The bigger graph, it can be directed or undirected. `domains`: A pointer vector, or a null pointer. If a pointer vector, then it must contain pointers to `igraph_vector_int_t` objects and the length of the vector must match the number of vertices in the `pattern` graph. For each vertex, the IDs of the compatible vertices in the target graph are listed. `iso`: Pointer to a boolean, or a null pointer. If not a null pointer, then the boolean is set to `true` if a subgraph isomorphism is found, and to `false` otherwise. `map`: 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. `maps`: 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 `igraph_vector_int_t` objects. `induced`: Boolean, whether to search for induced matching subgraphs. `time_limit`: 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

 `igraph_subisomorphic_vf2()` for the VF2 algorithm.

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;
}
```

## 5. Functions for small graphs

### 5.1. `igraph_isoclass` — Determine the isomorphism class of small graphs.

```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 4-vertex directed graphs and 3 to 6 vertex undirected graphs are supported.

Multi-edges and self-loops are ignored by this function.

Arguments:

 `graph`: The graph object. `isoclass`: Pointer to an integer, the isomorphism class will be stored here.

Returns:

 Error code.

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.

### 5.2. `igraph_isoclass_subgraph` — The isomorphism class of a subgraph of a 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 4-vertex directed graphs and 3 to 6 vertex undirected graphs are supported.

Multi-edges and self-loops are ignored by this function.

Arguments:

 `graph`: The graph object. `vids`: A vector containing the vertex IDs to be considered as a subgraph. Each vertex ID should be included at most once. `isoclass`: Pointer to an integer, this will be set to the isomorphism class.

Returns:

 Error code.

Time complexity: O((d+n)*n), d is the average degree in the network, and n is the number of vertices in `vids`.

### 5.3. `igraph_isoclass_create` — Creates a graph from the given isomorphism class.

```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. non-isomorphic) 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 4-vertex directed graphs and 3 to 6 vertex undirected graphs are supported.

Arguments:

 `graph`: Pointer to an uninitialized graph object. `size`: The number of vertices to add to the graph. `number`: The isomorphism class. `directed`: Logical constant, whether to create a directed graph.

Returns:

 Error code.

Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph to create.

### 5.4. `igraph_graph_count` — The number of unlabelled graphs on the given number of vertices.

```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:

 `n`: The number of vertices. `directed`: Boolean, whether to consider directed graphs. `count`: Pointer to an integer, the result will be stored here.

Returns:

 Error code.

Time complexity: O(1).

## 6. Utility functions

### 6.1. `igraph_permute_vertices` — Permute the vertices.

```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:

 `graph`: The input graph. `res`: Pointer to an uninitialized graph object. The new graph is created here. `permutation`: 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.

### 6.2. `igraph_simplify_and_colorize` — Simplify the graph and compute self-loop and edge multiplicities.

```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 self-loops 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:

 `graph`: The graph object, typically having self-loops or multi-edges. `res`: An uninitialized graph object. The result will be stored here `vertex_color`: Computed vertex colors corresponding to self-loop multiplicities. `edge_color`: Computed edge colors corresponding to edge multiplicities

Returns:

 Error code.

## 7. Deprecated functions

### 7.1. `igraph_isomorphic_34` — Graph isomorphism for 3-4 vertices (deprecated alias).

```igraph_error_t igraph_isomorphic_34(
const igraph_t *graph1, const igraph_t *graph2, igraph_bool_t *iso
);
```

### Warning

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 3-6 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 self-loops.

Arguments:

 `graph1`: The first input graph. `graph2`: The second input graph. Must have the same directedness as `graph1`. `iso`: Pointer to a boolean, the result is stored here.

Returns:

 Error code.

Time complexity: O(1).