# igraph Reference Manual

For using the igraph C library

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# Chapter 16. Cliques and independent vertex sets

These functions calculate various graph properties related to cliques and independent vertex sets.

## 1. Cliques

### 1.1. `igraph_cliques` — Finds all or some cliques in a graph.

```int igraph_cliques(const igraph_t *graph, igraph_vector_ptr_t *res,
igraph_integer_t min_size, igraph_integer_t max_size);
```

Cliques are fully connected subgraphs of a graph.

If you are only interested in the size of the largest clique in the graph, use `igraph_clique_number()` instead.

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html

Arguments:

 `graph`: The input graph. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in a clique. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: Exponential

Example 16.1.  File `examples/simple/igraph_cliques.c`

```/* -*- mode: C -*-  */
/*
IGraph library.
Copyright (C) 2006-2012  Gabor Csardi <csardi.gabor@gmail.com>
334 Harvard street, Cambridge, MA 02139 USA

This program is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc.,  51 Franklin Street, Fifth Floor, Boston, MA
02110-1301 USA

*/

#include <igraph.h>
#include <stdlib.h>

int compare_vectors(const void *p1, const void *p2) {
igraph_vector_t *v1, *v2;
long s1, s2, i;

v1 = *((igraph_vector_t **) p1);
v2 = *((igraph_vector_t **) p2);
s1 = igraph_vector_size(v1);
s2 = igraph_vector_size(v2);
if (s1 < s2) {
return -1;
}
if (s1 > s2) {
return 1;
}
for (i = 0; i < s1; ++i) {
if (VECTOR(*v1)[i] < VECTOR(*v2)[i]) {
return -1;
}
if (VECTOR(*v1)[i] > VECTOR(*v2)[i]) {
return 1;
}
}
return 0;
}

/* Takes a pointer vector of vectors. Sorts each vector, then sorts the pointer vector */
void canonicalize_list(igraph_vector_ptr_t *list) {
long i, len;
len = igraph_vector_ptr_size(list);
for (i = 0; i < len; ++i) {
igraph_vector_sort((igraph_vector_t *) VECTOR(*list)[i]);
}
qsort(&(VECTOR(*list)), len, sizeof(void *), &compare_vectors);
}

void print_vector(igraph_vector_t *v) {
long int i, n = igraph_vector_size(v);
for (i = 0; i < n; i++) {
printf(" %li", (long int) VECTOR(*v)[i]);
}
printf("\n");
}

void warning_handler_ignore(const char* reason, const char* file, int line, int e) {
}

struct userdata {
int i;
igraph_vector_ptr_t *list;
};

igraph_bool_t handler(igraph_vector_t *clique, void *arg) {
struct userdata *ud;
igraph_bool_t cont;

ud = (struct userdata *) arg;
cont = 1; /* true */

if (compare_vectors(&clique, &(VECTOR(*(ud->list))[ud->i])) != 0) {
printf("igraph_cliques() and igraph_cliques_callback() give different results.\n");
cont = 0; /* false */
}

igraph_vector_destroy(clique);
igraph_free(clique);

ud->i += 1;

return cont;
}

void test_callback(const igraph_t *graph) {
igraph_vector_ptr_t list;
struct userdata ud;

igraph_vector_ptr_init(&list, 0);
igraph_cliques(graph, &list, 0, 0);

ud.i = 0;
ud.list = &list;

igraph_cliques_callback(graph, 0, 0, &handler, (void *) &ud);

IGRAPH_VECTOR_PTR_SET_ITEM_DESTRUCTOR(&list, igraph_vector_destroy);
igraph_vector_ptr_destroy_all(&list);
}

int main() {

igraph_t g;
igraph_vector_ptr_t result;
igraph_es_t es;
igraph_integer_t omega;
long int i, j, n;
const int params[] = {4, -1, 2, 2, 0, 0, -1, -1};

igraph_set_warning_handler(warning_handler_ignore);

igraph_vector_ptr_init(&result, 0);
igraph_full(&g, 6, 0, 0);
igraph_es_pairs_small(&es, 0, 0, 1, 0, 2, 3, 5, -1);
igraph_delete_edges(&g, es);
igraph_es_destroy(&es);

for (j = 0; j < sizeof(params) / (2 * sizeof(params)); j++) {
if (params[2 * j + 1] != 0) {
igraph_cliques(&g, &result, params[2 * j], params[2 * j + 1]);
} else {
igraph_largest_cliques(&g, &result);
}
n = igraph_vector_ptr_size(&result);
printf("%ld cliques found\n", (long)n);
canonicalize_list(&result);
for (i = 0; i < n; i++) {
igraph_vector_t* v = (igraph_vector_t*) igraph_vector_ptr_e(&result, i);
print_vector(v);
igraph_vector_destroy(v);
igraph_free(v);
}
}

igraph_clique_number(&g, &omega);
printf("omega=%ld\n", (long)omega);

test_callback(&g);

igraph_destroy(&g);

igraph_tree(&g, 5, 2, IGRAPH_TREE_OUT);
igraph_cliques(&g, &result, 5, 5);
if (igraph_vector_ptr_size(&result) != 0) {
return 1;
}

igraph_destroy(&g);
igraph_vector_ptr_destroy(&result);

return 0;
}
```

### 1.2. `igraph_clique_size_hist` — Counts cliques of each size in the graph.

```int igraph_clique_size_hist(const igraph_t *graph, igraph_vector_t *hist,
igraph_integer_t min_size, igraph_integer_t max_size);
```

Cliques are fully connected subgraphs of a graph.

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html

Arguments:

 `graph`: The input graph. `hist`: Pointer to an initialized vector. The result will be stored here. The first element will store the number of size-1 cliques, the second element the number of size-2 cliques, etc. For cliques smaller than `min_size`, zero counts will be returned. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: Exponential

### 1.3. `igraph_cliques_callback` — Calls a function for each clique in the graph.

```int igraph_cliques_callback(const igraph_t *graph,
igraph_integer_t min_size, igraph_integer_t max_size,
igraph_clique_handler_t *cliquehandler_fn, void *arg);
```

Cliques are fully connected subgraphs of a graph. This function enumerates all cliques within the given size range and calls `cliquehandler_fn` for each of them. The cliques are passed to the callback function as a pointer to an `igraph_vector_t`. Destroying and freeing this vector is left up to the user. Use `igraph_vector_destroy()` to destroy it first, then free it using `igraph_free()`.

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html

Arguments:

 `graph`: The input graph. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used. `cliquehandler_fn`: Callback function to be called for each clique. See also `igraph_clique_handler_t`. `arg`: Extra argument to supply to `cliquehandler_fn`.

Returns:

 Error code.

Time complexity: Exponential

### 1.4. `igraph_clique_handler_t` — Type of clique handler functions.

```typedef igraph_bool_t igraph_clique_handler_t(igraph_vector_t *clique, void *arg);
```

Callback type, called when a clique was found. See the details at the documentation of `igraph_cliques_callback()`.

Arguments:

 `clique`: The current clique. Destroying and freeing this vector is left to the user. Use `igraph_vector_destroy()` and `igraph_free()` to do this. `arg`: This extra argument was passed to `igraph_cliques_callback()` when it was called.

Returns:

 Boolean, whether to continue with the clique search.

### 1.5. `igraph_largest_cliques` — Finds the largest clique(s) in a graph.

```int igraph_largest_cliques(const igraph_t *graph, igraph_vector_ptr_t *res);
```

A clique is largest (quite intuitively) if there is no other clique in the graph which contains more vertices.

Note that this is not necessarily the same as a maximal clique, i.e. the largest cliques are always maximal but a maximal clique is not always largest.

The current implementation of this function searches for maximal cliques using `igraph_maximal_cliques_callback()` and drops those that are not the largest.

The implementation of this function changed between igraph 0.5 and 0.6, so the order of the cliques and the order of vertices within the cliques will almost surely be different between these two versions.

Arguments:

 `graph`: The input graph. `res`: Pointer to an initialized pointer vector, the result will be stored here. It will be resized as needed. Note that vertices of a clique may be returned in arbitrary order.

Returns:

 Error code.

Time complexity: O(3^(|V|/3)) worst case.

### 1.6. `igraph_maximal_cliques` — Finds all maximal cliques in a graph.

```int igraph_maximal_cliques(const igraph_t *graph,
igraph_vector_ptr_t *res,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

A maximal clique is a clique which can't be extended any more by adding a new vertex to it.

If you are only interested in the size of the largest clique in the graph, use `igraph_clique_number()` instead.

The current implementation uses a modified Bron-Kerbosch algorithm to find the maximal cliques, see: David Eppstein, Maarten Löffler, Darren Strash: Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time. Algorithms and Computation, Lecture Notes in Computer Science Volume 6506, 2010, pp 403-414.

The implementation of this function changed between igraph 0.5 and 0.6 and also between 0.6 and 0.7, so the order of the cliques and the order of vertices within the cliques will almost surely be different between these three versions.

Arguments:

 `graph`: The input graph. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in a clique. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed. Note that vertices of a clique may be returned in arbitrary order. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.

Example 16.2.  File `examples/simple/igraph_maximal_cliques.c`

```/* -*- mode: C -*-  */
/*
IGraph library.
Copyright (C) 2010-2012  Gabor Csardi <csardi.gabor@gmail.com>
334 Harvard st, Cambridge MA, 02139 USA

This program is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301 USA

*/

#include <igraph.h>

#define NODES 1000
#define CLIQUE_SIZE 10
#define NO_CLIQUES 10
#define INT(a) (igraph_rng_get_integer(igraph_rng_default(), 0, (a)))

int permutation(igraph_vector_t *vec) {
int i, r, tmp;
for (i = 0; i < CLIQUE_SIZE; i++) {
r = INT(NODES - 1);
tmp = VECTOR(*vec)[i];
VECTOR(*vec)[i] = VECTOR(*vec)[r];
VECTOR(*vec)[r] = tmp;
}
return 0;
}

int sort_cmp(const void *a, const void *b) {
const igraph_vector_t **da = (const igraph_vector_t **) a;
const igraph_vector_t **db = (const igraph_vector_t **) b;
int i, alen = igraph_vector_size(*da), blen = igraph_vector_size(*db);
if (alen != blen) {
return (alen < blen) - (alen > blen);
}
for (i = 0; i < alen; i++) {
int ea = VECTOR(**da)[i], eb = VECTOR(**db)[i];
if (ea != eb) {
return (ea > eb) - (ea < eb);
}
}
return 0;
}

void sort_cliques(igraph_vector_ptr_t *cliques) {
int i, n = igraph_vector_ptr_size(cliques);
for (i = 0; i < n; i++) {
igraph_vector_t *v = VECTOR(*cliques)[i];
igraph_vector_sort(v);
}
igraph_qsort(VECTOR(*cliques), (size_t) n,
sizeof(igraph_vector_t *), sort_cmp);
}

void print_and_destroy_cliques(igraph_vector_ptr_t *cliques) {
int i;
sort_cliques(cliques);
for (i = 0; i < igraph_vector_ptr_size(cliques); i++) {
igraph_vector_t *v = VECTOR(*cliques)[i];
igraph_vector_print(v);
igraph_vector_destroy(v);
igraph_free(v);
}
}

int main() {

igraph_t g, g2, cli;
igraph_vector_t perm;
igraph_vector_ptr_t cliques;
igraph_integer_t no;
int i;

igraph_rng_seed(igraph_rng_default(), 42);

/* Create a graph that has a random component, plus a number of
relatively small cliques */

igraph_vector_init_seq(&perm, 0, NODES - 1);
igraph_erdos_renyi_game(&g, IGRAPH_ERDOS_RENYI_GNM, NODES, NODES,
/*directed=*/ 0, /*loops=*/ 0);
igraph_full(&cli, CLIQUE_SIZE, /*directed=*/ 0, /*loops=*/ 0);

for (i = 0; i < NO_CLIQUES; i++) {
/* Permute vertices of g */
permutation(&perm);
igraph_permute_vertices(&g, &g2, &perm);
igraph_destroy(&g);
g = g2;

igraph_union(&g2, &g, &cli, /*edge_map1=*/ 0, /*edge_map2=*/ 0);
igraph_destroy(&g);
g = g2;
}
igraph_simplify(&g, /*multiple=*/ 1, /*loop=*/ 0, /*edge_comb=*/ 0);

igraph_vector_destroy(&perm);
igraph_destroy(&cli);

/* Find the maximal cliques */

igraph_vector_ptr_init(&cliques, 0);
igraph_maximal_cliques(&g, &cliques, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);
igraph_maximal_cliques_count(&g, &no, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);

if (no != igraph_vector_ptr_size(&cliques)) {
return 1;
}

/* Print and destroy them */

print_and_destroy_cliques(&cliques);

/* Clean up */

igraph_vector_ptr_destroy(&cliques);
igraph_destroy(&g);

/* Build a triangle with a loop (thanks to Emmanuel Navarro) */

igraph_small(&g, 3, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 0, 0, 0, -1);

/* Find the maximal cliques */

igraph_vector_ptr_init(&cliques, 0);
igraph_maximal_cliques(&g, &cliques, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);
igraph_maximal_cliques_count(&g, &no, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);

if (no != igraph_vector_ptr_size(&cliques)) {
return 2;
}

/* Print and destroy them */

print_and_destroy_cliques(&cliques);

/* Clean up */

igraph_vector_ptr_destroy(&cliques);
igraph_destroy(&g);

return 0;
}
```

### 1.7. `igraph_maximal_cliques_count` — Count the number of maximal cliques in a graph

```int igraph_maximal_cliques_count(const igraph_t *graph,
igraph_integer_t *res,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

The current implementation uses a modified Bron-Kerbosch algorithm to find the maximal cliques, see: David Eppstein, Maarten Löffler, Darren Strash: Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time. Algorithms and Computation, Lecture Notes in Computer Science Volume 6506, 2010, pp 403-414.

Arguments:

 `graph`: The input graph. `res`: Pointer to an `igraph_integer_t`; the number of maximal cliques will be stored here. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.

Example 16.3.  File `examples/simple/igraph_maximal_cliques.c`

```/* -*- mode: C -*-  */
/*
IGraph library.
Copyright (C) 2010-2012  Gabor Csardi <csardi.gabor@gmail.com>
334 Harvard st, Cambridge MA, 02139 USA

This program is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301 USA

*/

#include <igraph.h>

#define NODES 1000
#define CLIQUE_SIZE 10
#define NO_CLIQUES 10
#define INT(a) (igraph_rng_get_integer(igraph_rng_default(), 0, (a)))

int permutation(igraph_vector_t *vec) {
int i, r, tmp;
for (i = 0; i < CLIQUE_SIZE; i++) {
r = INT(NODES - 1);
tmp = VECTOR(*vec)[i];
VECTOR(*vec)[i] = VECTOR(*vec)[r];
VECTOR(*vec)[r] = tmp;
}
return 0;
}

int sort_cmp(const void *a, const void *b) {
const igraph_vector_t **da = (const igraph_vector_t **) a;
const igraph_vector_t **db = (const igraph_vector_t **) b;
int i, alen = igraph_vector_size(*da), blen = igraph_vector_size(*db);
if (alen != blen) {
return (alen < blen) - (alen > blen);
}
for (i = 0; i < alen; i++) {
int ea = VECTOR(**da)[i], eb = VECTOR(**db)[i];
if (ea != eb) {
return (ea > eb) - (ea < eb);
}
}
return 0;
}

void sort_cliques(igraph_vector_ptr_t *cliques) {
int i, n = igraph_vector_ptr_size(cliques);
for (i = 0; i < n; i++) {
igraph_vector_t *v = VECTOR(*cliques)[i];
igraph_vector_sort(v);
}
igraph_qsort(VECTOR(*cliques), (size_t) n,
sizeof(igraph_vector_t *), sort_cmp);
}

void print_and_destroy_cliques(igraph_vector_ptr_t *cliques) {
int i;
sort_cliques(cliques);
for (i = 0; i < igraph_vector_ptr_size(cliques); i++) {
igraph_vector_t *v = VECTOR(*cliques)[i];
igraph_vector_print(v);
igraph_vector_destroy(v);
igraph_free(v);
}
}

int main() {

igraph_t g, g2, cli;
igraph_vector_t perm;
igraph_vector_ptr_t cliques;
igraph_integer_t no;
int i;

igraph_rng_seed(igraph_rng_default(), 42);

/* Create a graph that has a random component, plus a number of
relatively small cliques */

igraph_vector_init_seq(&perm, 0, NODES - 1);
igraph_erdos_renyi_game(&g, IGRAPH_ERDOS_RENYI_GNM, NODES, NODES,
/*directed=*/ 0, /*loops=*/ 0);
igraph_full(&cli, CLIQUE_SIZE, /*directed=*/ 0, /*loops=*/ 0);

for (i = 0; i < NO_CLIQUES; i++) {
/* Permute vertices of g */
permutation(&perm);
igraph_permute_vertices(&g, &g2, &perm);
igraph_destroy(&g);
g = g2;

igraph_union(&g2, &g, &cli, /*edge_map1=*/ 0, /*edge_map2=*/ 0);
igraph_destroy(&g);
g = g2;
}
igraph_simplify(&g, /*multiple=*/ 1, /*loop=*/ 0, /*edge_comb=*/ 0);

igraph_vector_destroy(&perm);
igraph_destroy(&cli);

/* Find the maximal cliques */

igraph_vector_ptr_init(&cliques, 0);
igraph_maximal_cliques(&g, &cliques, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);
igraph_maximal_cliques_count(&g, &no, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);

if (no != igraph_vector_ptr_size(&cliques)) {
return 1;
}

/* Print and destroy them */

print_and_destroy_cliques(&cliques);

/* Clean up */

igraph_vector_ptr_destroy(&cliques);
igraph_destroy(&g);

/* Build a triangle with a loop (thanks to Emmanuel Navarro) */

igraph_small(&g, 3, IGRAPH_UNDIRECTED, 0, 1, 1, 2, 2, 0, 0, 0, -1);

/* Find the maximal cliques */

igraph_vector_ptr_init(&cliques, 0);
igraph_maximal_cliques(&g, &cliques, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);
igraph_maximal_cliques_count(&g, &no, /*min_size=*/ 3,
/*max_size=*/ 0 /*no limit*/);

if (no != igraph_vector_ptr_size(&cliques)) {
return 2;
}

/* Print and destroy them */

print_and_destroy_cliques(&cliques);

/* Clean up */

igraph_vector_ptr_destroy(&cliques);
igraph_destroy(&g);

return 0;
}
```

### 1.8. `igraph_maximal_cliques_file` — Find maximal cliques and write them to a file.

```int igraph_maximal_cliques_file(const igraph_t *graph,
FILE *outfile,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

This function enumerates all maximal cliques and writes them to file.

Edge directions are ignored.

Arguments:

 `graph`: The input graph. `outfile`: Pointer to the output file, it should be writable. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.*

### 1.9. `igraph_maximal_cliques_subset` — Maximal cliques for a subset of initial vertices

```int igraph_maximal_cliques_subset(const igraph_t *graph,
igraph_vector_int_t *subset,
igraph_vector_ptr_t *res,
igraph_integer_t *no,
FILE *outfile,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

This function enumerates all maximal cliques for a subset of initial vertices and writes them to file.

Edge directions are ignored.

Arguments:

 `graph`: The input graph. `subset`: Pointer to an `igraph_vector_int_t` containing the subset of initial vertices `res`: Pointer to an `igraph_ptr_t`; the cliques will be stored here `no`: Pointer to an `igraph_integer_t`; the number of maximal cliques will be stored here. `outfile`: Pointer to an output file or `NULL`. When not `NULL`, the file should be writable. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.

### 1.10. `igraph_maximal_cliques_hist` — Counts the number of maximal cliques of each size in a graph.

```int igraph_maximal_cliques_hist(const igraph_t *graph,
igraph_vector_t *hist,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

This function counts how many maximal cliques of each size are present in the graph. Size-1 maximal cliques are simply isolated vertices.

Edge directions are ignored.

Arguments:

 `graph`: The input graph. `hist`: Pointer to an initialized vector. The result will be stored here. The first element will store the number of size-1 maximal cliques, the second element the number of size-2 maximal cliques, etc. For cliques smaller than `min_size`, zero counts will be returned. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.

### 1.11. `igraph_maximal_cliques_callback` — Finds maximal cliques in a graph and calls a function for each one.

```int igraph_maximal_cliques_callback(const igraph_t *graph,
igraph_clique_handler_t *cliquehandler_fn, void *arg,
igraph_integer_t min_size, igraph_integer_t max_size);
```

This function enumerates all maximal cliques within the given size range and calls `cliquehandler_fn` for each of them. The cliques are passed to the callback function as a pointer to an `igraph_vector_t`. Destroying and freeing this vector is left up to the user. Use `igraph_vector_destroy()` to destroy it first, then free it using `igraph_free()`.

Edge directions are ignored.

Arguments:

 `graph`: The input graph. `cliquehandler_fn`: Callback function to be called for each clique. See also `igraph_clique_handler_t`. `arg`: Extra argument to supply to `cliquehandler_fn`. `min_size`: Integer giving the minimum size of the cliques to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the cliques to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: O(d(n-d)3^(d/3)) worst case, d is the degeneracy of the graph, this is typically small for sparse graphs.

### 1.12. `igraph_clique_number` — Finds the clique number of the graph.

```int igraph_clique_number(const igraph_t *graph, igraph_integer_t *no);
```

The clique number of a graph is the size of the largest clique.

The current implementation of this function searches for maximal cliques using `igraph_maximal_cliques_callback()` and keeps track of the size of the largest clique that was found.

Arguments:

 `graph`: The input graph. `no`: The clique number will be returned to the `igraph_integer_t` pointed by this variable.

Returns:

 Error code.

Time complexity: O(3^(|V|/3)) worst case.

## 2. Weighted cliques

### 2.1. `igraph_weighted_cliques` — Finds all cliques in a given weight range in a vertex weighted graph.

```int igraph_weighted_cliques(const igraph_t *graph,
const igraph_vector_t *vertex_weights, igraph_vector_ptr_t *res,
igraph_real_t min_weight, igraph_real_t max_weight, igraph_bool_t maximal);
```

Cliques are fully connected subgraphs of a graph. The weight of a clique is the sum of the weights of individual vertices within the clique.

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html Only positive integer vertex weights are supported.

Arguments:

 `graph`: The input graph. `vertex_weights`: A vector of vertex weights. The current implementation will truncate all weights to their integer parts. You may pass `NULL` here to make each vertex have a weight of 1. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in a clique. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed. `min_weight`: Integer giving the minimum weight of the cliques to be returned. If negative or zero, no lower bound will be used. `max_weight`: Integer giving the maximum weight of the cliques to be returned. If negative or zero, no upper bound will be used. `maximal`: If true, only maximal cliques will be returned

Returns:

 Error code.

Time complexity: Exponential

### 2.2. `igraph_largest_weighted_cliques` — Finds the largest weight clique(s) in a graph.

```int igraph_largest_weighted_cliques(const igraph_t *graph,
const igraph_vector_t *vertex_weights, igraph_vector_ptr_t *res);
```

Finds the clique(s) having the largest weight in the graph.

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html Only positive integer vertex weights are supported.

Arguments:

 `graph`: The input graph. `vertex_weights`: A vector of vertex weights. The current implementation will truncate all weights to their integer parts. You may pass `NULL` here to make each vertex have a weight of 1. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in a clique. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed.

Returns:

 Error code.

Time complexity: TODO

### 2.3. `igraph_weighted_clique_number` — Finds the weight of the largest weight clique in the graph.

```int igraph_weighted_clique_number(const igraph_t *graph,
const igraph_vector_t *vertex_weights, igraph_real_t *res);
```

The current implementation of this function uses version 1.21 of the Cliquer library by Sampo Niskanen and Patric R. J. Östergård, http://users.aalto.fi/~pat/cliquer.html Only positive integer vertex weights are supported.

Arguments:

 `graph`: The input graph. `vertex_weights`: A vector of vertex weights. The current implementation will truncate all weights to their integer parts. You may pass `NULL` here to make each vertex have a weight of 1. `res`: The largest weight will be returned to the `igraph_real_t` pointed to by this variable.

Returns:

 Error code.

Time complexity: TODO

## 3. Independent vertex sets

### 3.1. `igraph_independent_vertex_sets` — Finds all independent vertex sets in a graph.

```int igraph_independent_vertex_sets(const igraph_t *graph,
igraph_vector_ptr_t *res,
igraph_integer_t min_size,
igraph_integer_t max_size);
```

A vertex set is considered independent if there are no edges between them.

If you are interested in the size of the largest independent vertex set, use `igraph_independence_number()` instead.

The current implementation was ported to igraph from the Very Nauty Graph Library by Keith Briggs and uses the algorithm from the paper S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505--517, 1977.

Arguments:

 `graph`: The input graph. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in an independent vertex set. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed. `min_size`: Integer giving the minimum size of the sets to be returned. If negative or zero, no lower bound will be used. `max_size`: Integer giving the maximum size of the sets to be returned. If negative or zero, no upper bound will be used.

Returns:

 Error code.

Time complexity: TODO

Example 16.4.  File `examples/simple/igraph_independent_sets.c`

```/* -*- mode: C -*-  */
/*
IGraph library.
Copyright (C) 2006-2012  Gabor Csardi <csardi.gabor@gmail.com>
334 Harvard st, Cambridge MA, 02139 USA

This program is free software; you can redistribute it and/or modify
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc.,  51 Franklin Street, Fifth Floor, Boston, MA
02110-1301 USA

*/

#include <igraph.h>
#include <stdlib.h>

void print_vector(igraph_vector_t *v) {
long int i, n = igraph_vector_size(v);
for (i = 0; i < n; i++) {
printf(" %li", (long int) VECTOR(*v)[i]);
}
printf("\n");
}

void warning_handler_ignore(const char* reason, const char* file, int line, int e) {
}

int main() {

igraph_t g;
igraph_vector_ptr_t result;
long int i, j, n;
igraph_integer_t alpha;
const int params[] = {4, -1, 2, 2, 0, 0, -1, -1};

igraph_set_warning_handler(warning_handler_ignore);
igraph_vector_ptr_init(&result, 0);

igraph_tree(&g, 5, 2, IGRAPH_TREE_OUT);
for (j = 0; j < sizeof(params) / (2 * sizeof(params)); j++) {
if (params[2 * j + 1] != 0) {
igraph_independent_vertex_sets(&g, &result, params[2 * j], params[2 * j + 1]);
} else {
igraph_largest_independent_vertex_sets(&g, &result);
}
n = igraph_vector_ptr_size(&result);
printf("%ld independent sets found\n", (long)n);
for (i = 0; i < n; i++) {
igraph_vector_t* v;
v = igraph_vector_ptr_e(&result, i);
print_vector((igraph_vector_t*)v);
igraph_vector_destroy(v);
igraph_free(v);
}
}
igraph_destroy(&g);

igraph_tree(&g, 10, 2, IGRAPH_TREE_OUT);
igraph_maximal_independent_vertex_sets(&g, &result);
n = igraph_vector_ptr_size(&result);
printf("%ld maximal independent sets found\n", (long)n);
for (i = 0; i < n; i++) {
igraph_vector_t* v;
v = igraph_vector_ptr_e(&result, i);
print_vector((igraph_vector_t*)v);
igraph_vector_destroy(v);
igraph_free(v);
}
igraph_vector_ptr_destroy(&result);

igraph_independence_number(&g, &alpha);
printf("alpha=%ld\n", (long)alpha);

igraph_destroy(&g);

return 0;
}
```

### 3.2. `igraph_largest_independent_vertex_sets` — Finds the largest independent vertex set(s) in a graph.

```int igraph_largest_independent_vertex_sets(const igraph_t *graph,
igraph_vector_ptr_t *res);
```

An independent vertex set is largest if there is no other independent vertex set with more vertices in the graph.

The current implementation was ported to igraph from the Very Nauty Graph Library by Keith Briggs and uses the algorithm from the paper S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505--517, 1977.

Arguments:

 `graph`: The input graph. `res`: Pointer to a pointer vector, the result will be stored here. It will be resized as needed.

Returns:

 Error code.

Time complexity: TODO

### 3.3. `igraph_maximal_independent_vertex_sets` — Finds all maximal independent vertex sets of a graph.

```int igraph_maximal_independent_vertex_sets(const igraph_t *graph,
igraph_vector_ptr_t *res);
```

A maximal independent vertex set is an independent vertex set which can't be extended any more by adding a new vertex to it.

The algorithm used here is based on the following paper: S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505--517, 1977.

The implementation was originally written by Kevin O'Neill and modified by K M Briggs in the Very Nauty Graph Library. I simply re-wrote it to use igraph's data structures.

If you are interested in the size of the largest independent vertex set, use `igraph_independence_number()` instead.

Arguments:

 `graph`: The input graph. `res`: Pointer to a pointer vector, the result will be stored here, i.e. `res` will contain pointers to `igraph_vector_t` objects which contain the indices of vertices involved in an independent vertex set. The pointer vector will be resized if needed but note that the objects in the pointer vector will not be freed.

Returns:

 Error code.

Time complexity: TODO.

### 3.4. `igraph_independence_number` — Finds the independence number of the graph.

```int igraph_independence_number(const igraph_t *graph, igraph_integer_t *no);
```

The independence number of a graph is the cardinality of the largest independent vertex set.

The current implementation was ported to igraph from the Very Nauty Graph Library by Keith Briggs and uses the algorithm from the paper S. Tsukiyama, M. Ide, H. Ariyoshi and I. Shirawaka. A new algorithm for generating all the maximal independent sets. SIAM J Computing, 6:505--517, 1977.

Arguments:

 `graph`: The input graph. `no`: The independence number will be returned to the `igraph_integer_t` pointed by this variable.

Returns:

 Error code.