For using the igraph C library
Most functions that create graphs in a deterministic manner are documented here. See also stochastic generators, spatial graph generators, bipartite graph generators, and operators that transform graphs.
igraph_error_t igraph_create(igraph_t *graph, const igraph_vector_int_t *edges,
igraph_int_t n, igraph_bool_t directed);
Arguments:
|
An uninitialized graph object. |
|
The edges to add, the first two elements are the first edge, etc. |
|
The number of vertices in the graph, if smaller or equal
to the highest vertex ID in the |
|
Boolean, whether to create a directed graph or
not. If yes, then the first edge points from the first
vertex ID in |
Returns:
Error code:
|
Time complexity: O(|V|+|E|), |V| is the number of vertices, |E| the number of edges in the graph.
Example 11.1. File examples/simple/igraph_create.c
#include <igraph.h> int main(void) { igraph_t g; igraph_vector_int_t v1, v2; /* Initialize the library. */ igraph_setup(); /* simple use */ igraph_vector_int_init(&v1, 8); VECTOR(v1)[0] = 0; VECTOR(v1)[1] = 1; VECTOR(v1)[2] = 1; VECTOR(v1)[3] = 2; VECTOR(v1)[4] = 2; VECTOR(v1)[5] = 3; VECTOR(v1)[6] = 2; VECTOR(v1)[7] = 2; igraph_create(&g, &v1, 0, 0); if (igraph_vcount(&g) != 4) { return 1; } igraph_vector_int_init(&v2, 0); igraph_get_edgelist(&g, &v2, 0); igraph_vector_int_sort(&v1); igraph_vector_int_sort(&v2); if (!igraph_vector_int_all_e(&v1, &v2)) { return 2; } igraph_destroy(&g); /* higher number of vertices */ igraph_create(&g, &v1, 10, 0); if (igraph_vcount(&g) != 10) { return 1; } igraph_get_edgelist(&g, &v2, 0); igraph_vector_int_sort(&v1); igraph_vector_int_sort(&v2); if (!igraph_vector_int_all_e(&v1, &v2)) { return 3; } igraph_destroy(&g); igraph_vector_int_destroy(&v1); igraph_vector_int_destroy(&v2); return 0; }
igraph_error_t igraph_small(igraph_t *graph, igraph_int_t n, igraph_bool_t directed,
int first, ...);
This function is handy when a relatively small graph needs to be created.
Instead of giving the edges as a vector, they are given simply as
arguments and a -1 needs to be given after the last meaningful
edge argument.
This function is intended to be used with vertex IDs that are entered as literal integers. If you use a variable instead of a literal, make sure that it is of type int, as this is the type that this function assumes for all variadic arguments. Using a different integer type is undefined behaviour and likely to cause platform-specific issues.
Arguments:
|
Pointer to an uninitialized graph object. The result will be stored here. |
||||
|
The number of vertices in the graph; a non-negative integer. |
||||
|
Boolean constant; gives whether the graph should be directed. Supported values are:
|
||||
|
The additional arguments giving the edges of the graph,
and must be of type int. Don't forget to supply an
additional |
Returns:
Error code. |
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph to create.
Example 11.2. File examples/simple/igraph_small.c
#include <igraph.h> int main(void) { igraph_t g; /* Initialize the library. */ igraph_setup(); igraph_small(&g, 0, IGRAPH_DIRECTED, 0, 1, 1, 2, 2, 3, 3, 4, 6, 1, -1); igraph_write_graph_edgelist(&g, stdout); igraph_destroy(&g); return 0; }
igraph_adjacency — Creates a graph from an adjacency matrix.igraph_weighted_adjacency — Creates a graph from a weighted adjacency matrix.igraph_sparse_adjacency — Creates a graph from a sparse adjacency matrix.igraph_sparse_weighted_adjacency — Creates a graph from a weighted sparse adjacency matrix.igraph_adjlist — Creates a graph from an adjacency list.These functions create graphs from weighted or unweighted adjacency matrices, or an adjacency list.
igraph_error_t igraph_adjacency(
igraph_t *graph, const igraph_matrix_t *adjmatrix, igraph_adjacency_t mode,
igraph_loops_t loops
);
The order of the vertices in the matrix is preserved, i.e. the vertex corresponding to the first row/column will be vertex with id 0, the next row is for vertex 1, etc. No guarantees are given about the ordering of edges.
Arguments:
|
Pointer to an uninitialized graph object. |
||||||||||||||
|
The adjacency matrix. How it is interpreted
depends on the |
||||||||||||||
|
Constant to specify how the given matrix is interpreted
as an adjacency matrix. Possible values (A(i,j) is the element in
row i and column j in the adjacency matrix
|
||||||||||||||
|
Constant of type
|
Returns:
Error code,
|
Time complexity: O(|V||V|), |V| is the number of vertices in the graph.
igraph_error_t igraph_weighted_adjacency(
igraph_t *graph, const igraph_matrix_t *adjmatrix, igraph_adjacency_t mode,
igraph_vector_t *weights, igraph_loops_t loops
);
The order of the vertices in the matrix is preserved, i.e. the vertex corresponding to the first row/column will be vertex with id 0, the next row is for vertex 1, etc. No guarantees are given for the ordering of edges.
Arguments:
|
Pointer to an uninitialized graph object. |
||||||||||||||
|
The weighted adjacency matrix. How it is interpreted
depends on the |
||||||||||||||
|
Constant to specify how the given matrix is interpreted
as an adjacency matrix. Possible values (A(i,j) is the element in row
i and column j in the adjacency matrix
|
||||||||||||||
|
Pointer to an initialized vector, the weights will be stored here. |
||||||||||||||
|
Constant to specify how the diagonal of the matrix should be
treated when creating loop edges. Ignored for modes
|
Returns:
Error code,
|
Time complexity: O(|V||V|), |V| is the number of vertices in the graph.
Example 11.3. File examples/simple/igraph_weighted_adjacency.c
#include <igraph.h> int main(void) { igraph_t graph; igraph_real_t data[4][4] = { { 0, 1.2, 2.3, 0 }, { 2.0, 0, 0, 1.0 }, { 0, 0, 1.5, 0 }, { 0, 1.0, 0, 0 } }; /* C arrays use row-major storage, while igraph's matrix uses column-major. * The matrix 'mat' will be the transpose of 'data'. */ const igraph_matrix_t mat = igraph_matrix_view(*data, sizeof(data[0]) / sizeof(data[0][0]), sizeof(data) / sizeof(data[0])); igraph_vector_t weights; igraph_vector_int_t edges; igraph_int_t n; /* Initialize the library. */ igraph_setup(); /* Initialize vector into which weights will be written. */ igraph_vector_init(&weights, 0); igraph_weighted_adjacency(&graph, &mat, IGRAPH_ADJ_DIRECTED, &weights, IGRAPH_LOOPS_ONCE); /* When igraph_weighted_adjacency() returns, 'weights' will typically have * more capacity allocated than what it uses. We may optionally free any * unused capacity to save memory, although in most applications this * is not necessary. */ igraph_vector_resize_min(&weights); /* Get the edge list of the graph and output it, along with the weights. */ igraph_vector_int_init(&edges, 0); igraph_get_edgelist(&graph, &edges, 0); n = igraph_ecount(&graph); for (igraph_int_t i = 0; i < n; i++) { printf("%" IGRAPH_PRId " --> %" IGRAPH_PRId ": %g\n", VECTOR(edges)[2*i], VECTOR(edges)[2*i + 1], VECTOR(weights)[i]); } /* Free all allocated storage. */ igraph_vector_int_destroy(&edges); igraph_destroy(&graph); igraph_vector_destroy(&weights); return 0; }
igraph_error_t igraph_sparse_adjacency(igraph_t *graph, igraph_sparsemat_t *adjmatrix,
igraph_adjacency_t mode, igraph_loops_t loops);
This has the same functionality as igraph_adjacency(), but uses
a column-compressed adjacency matrix.
Time complexity: O(|E|), where |E| is the number of edges in the graph.
igraph_error_t igraph_sparse_weighted_adjacency(
igraph_t *graph, igraph_sparsemat_t *adjmatrix, igraph_adjacency_t mode,
igraph_vector_t *weights, igraph_loops_t loops
);
This has the same functionality as igraph_weighted_adjacency(), but uses
a column-compressed adjacency matrix.
Time complexity: O(|E|), where |E| is the number of edges in the graph.
igraph_error_t igraph_adjlist(igraph_t *graph, const igraph_adjlist_t *adjlist,
igraph_neimode_t mode, igraph_bool_t duplicate);
An adjacency list is a list of vectors, containing the neighbors
of all vertices. For operations that involve many changes to the
graph structure, it is recommended that you convert the graph into
an adjacency list via igraph_adjlist_init(), perform the
modifications (these are cheap for an adjacency list) and then
recreate the igraph graph via this function.
Arguments:
|
Pointer to an uninitialized graph object. |
|
The adjacency list. |
|
Whether or not to create a directed graph. |
|
Boolean constant. For undirected graphs this specifies
whether each edge is included twice, in the vectors of
both adjacent vertices. If this is |
Returns:
Error code. |
See also:
|
Time complexity: O(|V|+|E|).
igraph_star — Creates a star graph, every vertex connects only to the center.igraph_wheel — Creates a wheel graph, a union of a star and a cycle graph.igraph_hypercube — The n-dimensional hypercube graph.igraph_square_lattice — Arbitrary dimensional square lattices.igraph_triangular_lattice — A triangular lattice with the given shape.igraph_hexagonal_lattice — A hexagonal lattice with the given shape.igraph_ring — Creates a cycle graph or a path graph.igraph_path_graph — A path graph P_n.igraph_cycle_graph — A cycle graph C_n.igraph_lcf — Creates a graph from LCF notation.igraph_lcf_small — Shorthand to create a graph from LCF notation, giving shifts as the arguments.igraph_circulant — Creates a circulant graph.igraph_extended_chordal_ring — Create an extended chordal ring.These functions produce various basic regular graph structures, such as paths, cycles or lattices.
igraph_error_t igraph_star(igraph_t *graph, igraph_int_t n, igraph_star_mode_t mode,
igraph_int_t center);
Arguments:
|
Pointer to an uninitialized graph object, this will be the result. |
||||||||
|
Integer constant, the number of vertices in the graph. |
||||||||
|
Constant, gives the type of the star graph to create. Possible values:
|
||||||||
|
Id of the vertex which will be the center of the graph. |
Returns:
|
Error code:
|
Time complexity: O(|V|), the number of vertices in the graph.
See also:
|
Example 11.4. File examples/simple/igraph_star.c
#include <igraph.h> #include <stdio.h> int main(void) { igraph_t graph; /* Initialize the library. */ igraph_setup(); /* Create an undirected 6-star, with the 0th node as the centre. */ igraph_star(&graph, 7, IGRAPH_STAR_UNDIRECTED, 0); /* Output the edge list of the graph. */ igraph_write_graph_edgelist(&graph, stdout); /* Destroy the graph when we are done using it. */ igraph_destroy(&graph); return 0; }
igraph_error_t igraph_wheel(igraph_t *graph, igraph_int_t n, igraph_wheel_mode_t mode,
igraph_int_t center);
A wheel graph on n vertices can be thought of as a wheel with
n - 1 spokes. The cycle graph part makes up the rim,
while the star graph part adds the spokes.
Note that the two and three-vertex wheel graphs are non-simple: The two-vertex wheel graph contains a self-loop, while the three-vertex wheel graph contains parallel edges (a 1-cycle and a 2-cycle, respectively).
Arguments:
|
Pointer to an uninitialized graph object, this will be the result. |
||||||||
|
Integer constant, the number of vertices in the graph. |
||||||||
|
Constant, gives the type of the star graph to create. Possible values:
|
||||||||
|
Id of the vertex which will be the center of the graph. |
Returns:
|
Error code:
|
Time complexity: O(|V|), the number of vertices in the graph.
See also:
|
igraph_error_t igraph_hypercube(igraph_t *graph,
igraph_int_t n, igraph_bool_t directed);
The hypercube graph Q_n has 2^n vertices and
2^(n-1) n edges. Two vertices are connected when the binary
representations of their zero-based vertex IDs differs in precisely one bit.
Arguments:
|
An uninitialized graph object. |
|
The dimension of the hypercube graph. |
|
Whether the graph should be directed. Edges will point from lower index vertices towards higher index ones. |
Returns:
Error code. |
See also:
Time complexity: O(2^n)
igraph_error_t igraph_square_lattice(
igraph_t *graph, const igraph_vector_int_t *dimvector, igraph_int_t nei,
igraph_bool_t directed, igraph_bool_t mutual, const igraph_vector_bool_t *periodic
);
Creates d-dimensional square lattices of the given size. Optionally, the lattice can be made periodic, and the neighbors within a given graph distance can be connected.
In the zero-dimensional case, the singleton graph is returned.
The vertices of the resulting graph are ordered such that the
index of the vertex at position (i_1, i_2, i_3, ..., i_d)
in a lattice of size (n_1, n_2, ..., n_d) will be
i_1 + n_1 * i_2 + n_1 * n_2 * i_3 + ....
Arguments:
|
An uninitialized graph object. |
|
Vector giving the sizes of the lattice in each of its dimensions. The dimension of the lattice will be the same as the length of this vector. |
|
Integer value giving the distance (number of steps) within which two vertices will be connected. |
|
Boolean, whether to create a directed graph.
If the |
|
Boolean, if the graph is directed this gives whether to create all connections as mutual. |
|
Boolean vector, defines whether the generated lattice is
periodic along each dimension. The length of this vector must match
the length of |
Returns:
Error code:
|
See also:
|
Time complexity: If nei is less than two then it is O(|V|+|E|) (as
far as I remember), |V| and |E| are the number of vertices
and edges in the generated graph. Otherwise it is O(|V|*d^k+|E|), d
is the average degree of the graph, k is the nei argument.
igraph_error_t igraph_triangular_lattice(
igraph_t *graph, const igraph_vector_int_t *dims,
igraph_bool_t directed, igraph_bool_t mutual);
Creates a triangular lattice whose vertices have the form (i, j) for non-negative
integers i and j and (i, j) is generally connected with (i + 1, j), (i, j + 1),
and (i - 1, j + 1). The function constructs a planar dual of the graph
constructed by igraph_hexagonal_lattice(). In particular, there a one-to-one
correspondence between the vertices in the constructed graph and the cycles of
length 6 in the graph constructed by igraph_hexagonal_lattice()
with the same dims parameter.
The vertices of the resulting graph are ordered lexicographically with the 2nd coordinate being more significant, e.g., (i, j) < (i + 1, j) and (i + 1, j) < (i, j + 1)
Arguments:
|
An uninitialized graph object. |
|
Integer vector, defines the shape of the lattice.
If |
|
Boolean, whether to create a directed graph.
If the |
|
Boolean, if the graph is directed this gives whether to create all connections as mutual. |
Returns:
Error code:
|
See also:
|
Time complexity: O(|V|), where |V| is the number of vertices in the generated graph.
igraph_error_t igraph_hexagonal_lattice(
igraph_t *graph, const igraph_vector_int_t *dims,
igraph_bool_t directed, igraph_bool_t mutual);
Creates a hexagonal lattice whose vertices have the form (i, j) for non-negative
integers i and j and (i, j) is generally connected with (i + 1, j), and if i is
odd also with (i - 1, j + 1). The function constructs a planar dual of the graph
constructed by igraph_triangular_lattice(). In particular, there a one-to-one
correspondence between the cycles of length 6 in the constructed graph and the
vertices of the graph constructed by igraph_triangular_lattice() function
with the same dims parameter.
The vertices of the resulting graph are ordered lexicographically with the 2nd coordinate being more significant, e.g., (i, j) < (i + 1, j) and (i + 1, j) < (i, j + 1)
Arguments:
|
An uninitialized graph object. |
|
Integer vector, defines the shape of the lattice.
If |
|
Boolean, whether to create a directed graph.
If the |
|
Boolean, if the graph is directed this gives whether to create all connections as mutual. |
Returns:
Error code:
|
See also:
|
Time complexity: O(|V|), where |V| is the number of vertices in the generated graph.
igraph_error_t igraph_ring(igraph_t *graph, igraph_int_t n, igraph_bool_t directed,
igraph_bool_t mutual, igraph_bool_t circular);
A circular ring on n vertices is commonly known in graph
theory as the cycle graph, and often denoted by C_n.
Removing a single edge from the cycle graph C_n results
in the path graph P_n. This function can generate both.
When n is 1 or 2, the result may not be a simple graph:
the one-cycle contains a self-loop and the undirected or reciprocally
connected directed two-cycle contains parallel edges.
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
Whether to create a directed graph. All edges will be oriented in the same direction along the cycle or path. |
|
Whether to create mutual edges in directed graphs. It is ignored for undirected graphs. |
|
Whether to create a closed ring (a cycle) or an open path. |
Returns:
Error code:
|
Time complexity: O(|V|), the number of vertices in the graph.
See also:
|
Example 11.5. File examples/simple/igraph_ring.c
#include <igraph.h> #include <stdio.h> int main(void) { igraph_t graph; /* Initialize the library. */ igraph_setup(); /* Create a directed path graph on 10 vertices. */ igraph_ring(&graph, 10, IGRAPH_DIRECTED, /* mutual= */ 0, /* circular= */ 0); /* Output the edge list of the graph. */ printf("10-path graph:\n"); igraph_write_graph_edgelist(&graph, stdout); /* Destroy the graph. */ igraph_destroy(&graph); /* Create a 4-cycle graph. */ igraph_ring(&graph, 4, IGRAPH_UNDIRECTED, /* mutual= */ 0, /* circular= */ 1); /* Output the edge list of the graph. */ printf("\n4-cycle graph:\n"); igraph_write_graph_edgelist(&graph, stdout); /* Destroy the graph. */ igraph_destroy(&graph); return 0; }
igraph_error_t igraph_path_graph(
igraph_t *graph, igraph_int_t n,
igraph_bool_t directed, igraph_bool_t mutual);
Creates the path graph P_n on n vertices.
This is a convenience wrapper to igraph_ring().
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
Whether to create a directed graph. |
|
Whether to create mutual edges in directed graphs. It is ignored for undirected graphs. |
Returns:
Error code. |
Time complexity: O(|V|), the number of vertices in the graph.
igraph_error_t igraph_cycle_graph(
igraph_t *graph, igraph_int_t n,
igraph_bool_t directed, igraph_bool_t mutual);
Creates the cycle graph C_n on n vertices.
When n is 1 or 2, the result may not be a simple graph:
the one-cycle contains a self-loop and the undirected or reciprocally
connected directed two-cycle contains parallel edges.
This is a convenience wrapper to igraph_ring().
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of vertices in the graph. |
|
Whether to create a directed graph. |
|
Whether to create mutual edges in directed graphs. It is ignored for undirected graphs. |
Returns:
Error code. |
Time complexity: O(|V|), the number of vertices in the graph.
igraph_error_t igraph_lcf(igraph_t *graph, igraph_int_t n,
const igraph_vector_int_t *shifts,
igraph_int_t repeats);
LCF notation (named after Lederberg, Coxeter and Frucht) is a concise notation for 3-regular Hamiltonian graphs. It consists of three parameters: the number of vertices in the graph, a list of shifts giving additional edges to a cycle backbone, and another integer giving how many times the shifts should be performed. See https://mathworld.wolfram.com/LCFNotation.html for details.
Arguments:
|
Pointer to an uninitialized graph object. |
|
Integer constant giving the number of vertices. This is normally set to the number of shifts multiplied by the number of repeats. |
|
An integer vector giving the shifts. |
|
The number of repeats for the shifts. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), linear in the number of vertices plus the number of edges.
igraph_error_t igraph_lcf_small(igraph_t *graph, igraph_int_t n, ...);
This function provides a shorthand to give the shifts of the LCF notation
directly as function arguments. See igraph_lcf() for an explanation
of LCF notation.
Arguments:
|
Pointer to an uninitialized graph object. |
|
Integer, the number of vertices in the graph. |
|
The shifts and the number of repeats for the shifts, plus an additional 0 to mark the end of the arguments. |
Returns:
Error code. |
See also:
See |
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
Example 11.6. File examples/simple/igraph_lcf.c
#include <igraph.h> int main(void) { igraph_t g1, g2; igraph_vector_int_t edges; igraph_bool_t iso; /* Initialize the library. */ igraph_setup(); // Heawood graph through LCF notation: [5, -5]^7 // The number of vertices is normally the number of shifts // multiplied by the number of repeats, in this case 2*7 = 14. igraph_lcf_small(&g1, /* n */ 14, /* shifts */ 5, -5, /* repeats */ 7, 0); printf("edges:\n"); igraph_vector_int_init(&edges, 0); igraph_get_edgelist(&g1, &edges, false); igraph_vector_int_print(&edges); igraph_vector_int_destroy(&edges); // Built-in Heawood graph: igraph_famous(&g2, "Heawood"); igraph_isomorphic(&g1, &g2, &iso); printf("isomorphic: %s\n", iso ? "true" : "false"); igraph_destroy(&g2); igraph_destroy(&g1); return 0; }
igraph_error_t igraph_circulant(igraph_t *graph, igraph_int_t n, const igraph_vector_int_t *shifts, igraph_bool_t directed);
A circulant graph G(n, shifts) consists of n vertices v_0, ...,
v_(n-1) such that for each s_i in the list of offsets shifts, v_j is
connected to v_((j + s_i) mod n) for all j.
The function can generate either directed or undirected graphs. It does not generate multi-edges or self-loops.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, the number of vertices in the circulant graph. |
|
Integer vector, a list of the offsets within the circulant graph. |
|
Boolean, whether to create a directed graph. |
Returns:
Error code. |
See also:
Time complexity: O(|V| |shifts|), the number of vertices in the graph times the number of shifts.
igraph_error_t igraph_extended_chordal_ring(
igraph_t *graph, igraph_int_t nodes, const igraph_matrix_int_t *W,
igraph_bool_t directed);
An extended chordal ring is a cycle graph with additional chords
connecting its vertices.
Each row L of the matrix W specifies a set of chords to be
inserted, in the following way: vertex i will connect to a vertex
L[(i mod p)] steps ahead of it along the cycle, where
p is the length of L.
In other words, vertex i will be connected to vertex
(i + L[(i mod p)]) mod nodes. If multiple edges are
defined in this way, this will output a non-simple graph. The result
can be simplified using igraph_simplify().
See also Kotsis, G: Interconnection Topologies for Parallel Processing Systems, PARS Mitteilungen 11, 1-6, 1993. The igraph extended chordal rings are not identical to the ones in the paper. In igraph the matrix specifies which edges to add. In the paper, a condition is specified which should simultaneously hold between two endpoints and the reverse endpoints.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer constant, the number of vertices in the graph. It must be at least 3. |
|
The matrix specifying the extra edges. The number of columns should divide the number of total vertices. The elements are allowed to be negative. |
|
Whether the graph should be directed. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
igraph_kary_tree — Creates a k-ary tree in which almost all vertices have k children.igraph_symmetric_tree — Creates a symmetric tree with the specified number of branches at each level.igraph_regular_tree — Creates a regular tree.igraph_tree_from_parent_vector — Constructs a tree or forest from a vector encoding the parent of each vertex.igraph_from_prufer — Generates a tree from a Prüfer sequence.These functions generate tree graphs.
igraph_error_t igraph_kary_tree(igraph_t *graph, igraph_int_t n, igraph_int_t children,
igraph_tree_mode_t type);
To obtain a completely symmetric tree with l layers, where each
vertex has precisely children descendants, use
n = (children^(l+1) - 1) / (children - 1).
Such trees are often called k-ary trees, where k refers
to the number of children.
Note that for n=0, the null graph is returned,
which is not considered to be a tree by igraph_is_tree().
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
Integer, the number of vertices in the graph. |
||||||
|
Integer, the number of children of a vertex in the tree. |
||||||
|
Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
|
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
See also:
|
Example 11.7. File examples/simple/igraph_kary_tree.c
#include <igraph.h> int main(void) { igraph_t graph; igraph_bool_t res; /* Initialize the library. */ igraph_setup(); /* Create a directed binary tree on 15 nodes, with edges pointing towards the root. */ igraph_kary_tree(&graph, 15, 2, IGRAPH_TREE_IN); igraph_is_tree(&graph, &res, NULL, IGRAPH_IN); printf("Is it an in-tree? %s\n", res ? "Yes" : "No"); igraph_is_tree(&graph, &res, NULL, IGRAPH_OUT); printf("Is it an out-tree? %s\n", res ? "Yes" : "No"); igraph_destroy(&graph); return 0; }
igraph_error_t igraph_symmetric_tree(igraph_t *graph, const igraph_vector_int_t *branches,
igraph_tree_mode_t type);
This function creates a tree in which all vertices at distance d from the
root have branching_counts[d] children.
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
Vector detailing the number of branches at each level. |
||||||
|
Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
|
Returns:
Error code:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
See also:
|
Example 11.8. File examples/simple/igraph_symmetric_tree.c
#include <igraph.h> int main(void) { igraph_t graph; igraph_bool_t res; igraph_vector_int_t v; igraph_vector_int_init_int(&v, 3, 3, 4, 5); /* Initialize the library. */ igraph_setup(); /* Create a directed symmetric tree with 2 levels - 3 children in first and 4 children in second level, 5 children in third level with edges pointing towards the root. */ igraph_symmetric_tree(&graph, &v, IGRAPH_TREE_IN); igraph_is_tree(&graph, &res, NULL, IGRAPH_IN); printf("Is it an in-tree? %s\n", res ? "Yes" : "No"); igraph_is_tree(&graph, &res, NULL, IGRAPH_OUT); printf("Is it an out-tree? %s\n", res ? "Yes" : "No"); igraph_destroy(&graph); igraph_vector_int_destroy(&v); return 0; }
igraph_error_t igraph_regular_tree(igraph_t *graph, igraph_int_t h, igraph_int_t k, igraph_tree_mode_t type);
All vertices of a regular tree, except its leaves, have the same total degree k.
This is different from a k-ary tree (igraph_kary_tree()), where all
vertices have the same number of children, thus the degre of the root is
one less than the degree of the other internal vertices. Regular trees
are also referred to as Bethe lattices.
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
The height of the tree, i.e. the distance between the root and the leaves. |
||||||
|
The degree of the regular tree. |
||||||
|
Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
|
Returns:
Error code. |
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
See also:
|
Example 11.9. File examples/simple/igraph_regular_tree.c
#include <igraph.h> int main(void) { igraph_t tree; igraph_vector_t eccentricity; igraph_bool_t is_tree; /* Initialize the library. */ igraph_setup(); /* Create a Bethe lattice with 5 levels, i.e. height 4. */ igraph_regular_tree(&tree, 4, 3, IGRAPH_TREE_UNDIRECTED); /* Bethe lattices are trees. */ igraph_is_tree(&tree, &is_tree, NULL, IGRAPH_ALL); printf("Is it a tree? %s\n", is_tree ? "Yes." : "No."); /* Compute and print eccentricities. The root is the most central. */ igraph_vector_init(&eccentricity, 0); igraph_eccentricity(&tree, NULL, &eccentricity, igraph_vss_all(), IGRAPH_ALL); printf("Vertex eccentricities:\n"); igraph_vector_print(&eccentricity); igraph_vector_destroy(&eccentricity); /* Clean up. */ igraph_destroy(&tree); return 0; }
igraph_error_t igraph_tree_from_parent_vector(
igraph_t *graph,
const igraph_vector_int_t *parents,
igraph_tree_mode_t type);
Rooted trees and forests are conveniently represented using a parents
vector where the ID of the parent of vertex v is stored in parents[v].
This function serves to construct an igraph graph from a parent vector representation.
The result is guaranteed to be a forest or a tree. If the parents vector
is found to encode a cycle or a self-loop, an error is raised.
Several igraph functions produce such vectors, such as graph traversal
functions (igraph_bfs() and igraph_dfs()), shortest path functions
that construct a shortest path tree, as well as some other specialized
functions like igraph_dominator_tree() or igraph_cohesive_blocks().
Vertices which do not have parents (i.e. roots) get a negative entry in the
parents vector.
Use igraph_bfs() or igraph_dfs() to convert a forest into a parent
vector representation. For trees, i.e. forests with a single root, it is
more convenient to use igraph_bfs_simple().
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
The parent vector. |
||||||
|
Constant, gives whether to create a directed tree, and if this is the case, also its orientation. Possible values:
|
Returns:
Error code. |
See also:
|
Time complexity: O(n) where n is the length of parents.
igraph_error_t igraph_from_prufer(igraph_t *graph, const igraph_vector_int_t *prufer);
A Prüfer sequence is a unique sequence of integers associated
with a labelled tree. A tree on n vertices can be represented
by a sequence of n-2 integers, each between 0 and
n-1 (inclusive).
The algorithm used by this function is based on
Paulius Micikevičius, Saverio Caminiti, Narsingh Deo:
Linear-time Algorithms for Encoding Trees as Sequences of Node Labels
Arguments:
|
Pointer to an uninitialized graph object. |
|
The Prüfer sequence |
Returns:
|
Error code:
|
See also:
Time complexity: O(|V|), where |V| is the number of vertices in the tree.
These functions generate graphs with the specified degrees.
igraph_error_t igraph_realize_degree_sequence(
igraph_t *graph,
const igraph_vector_int_t *outdeg, const igraph_vector_int_t *indeg,
igraph_edge_type_sw_t allowed_edge_types,
igraph_realize_degseq_t method);
This function generates an undirected graph that realizes a given degree sequence, or a directed graph that realizes a given pair of out- and in-degree sequences.
Simple undirected graphs are constructed using the Havel-Hakimi algorithm (undirected case), or the analogous Kleitman-Wang algorithm (directed case). These algorithms work by choosing an arbitrary vertex and connecting all its stubs to other vertices of highest degree. In the directed case, the "highest" (in, out) degree pairs are determined based on lexicographic ordering. This step is repeated until all degrees have been connected up.
Loopless multigraphs are generated using an analogous algorithm: an arbitrary vertex is chosen, and it is connected with a single connection to a highest remaining degee vertex. If self-loops are also allowed, the same algorithm is used, but if a non-zero vertex remains at the end of the procedure, the graph is completed by adding self-loops to it. Thus, the result will contain at most one vertex with self-loops.
The method parameter controls the order in which the vertices to be
connected are chosen. In the undirected case, IGRAPH_REALIZE_DEGSEQ_SMALLEST
produces a connected graph when one exists. This makes this method suitable
for constructing trees with a given degree sequence.
For a undirected simple graph, the time complexity is O(V + alpha(V) * E). For an undirected multi graph, the time complexity is O(V * E + V log V). For a directed graph, the time complexity is O(E + V^2 log V).
References:
V. Havel: Poznámka o existenci konečných grafů (A remark on the existence of finite graphs), Časopis pro pěstování matematiky 80, 477-480 (1955). http://eudml.org/doc/19050
S. L. Hakimi: On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph, Journal of the SIAM 10, 3 (1962). https://www.jstor.org/stable/2098770
D. J. Kleitman and D. L. Wang: Algorithms for Constructing Graphs and Digraphs with Given Valences and Factors, Discrete Mathematics 6, 1 (1973). https://doi.org/10.1016/0012-365X%2873%2990037-X P. L. Erdős, I. Miklós, Z. Toroczkai: A simple Havel-Hakimi type algorithm to realize graphical degree sequences of directed graphs, The Electronic Journal of Combinatorics 17.1 (2010). http://eudml.org/doc/227072
Sz. Horvát and C. D. Modes: Connectedness matters: construction and exact random sampling of connected networks (2021). https://doi.org/10.1088/2632-072X/abced5
Arguments:
|
Pointer to an uninitialized graph object. |
||||||||
|
The degree sequence of an undirected graph (if |
||||||||
|
The in-degree sequence of a directed graph. Pass |
||||||||
|
The types of edges to allow in the graph. See
|
||||||||
|
The method to generate the graph. Possible values:
|
Returns:
|
Error code:
|
See also:
|
Example 11.10. File examples/simple/igraph_realize_degree_sequence.c
#include <igraph.h> #include <stdio.h> int main(void){ igraph_t g1, g2, g3; igraph_int_t nodes = 500, A = 0, power = 1, m = 1; igraph_real_t assortativity; /* Initialize the library. */ igraph_setup(); igraph_rng_seed(igraph_rng_default(), 42); printf("Demonstration of difference in assortativities of graphs with the same degree sequence but different linkages:\n\nInitial graph based on the Barabasi-Albert model with %" IGRAPH_PRId " nodes.\n", nodes); /* Graph 1 generated by a randomized graph generator */ igraph_barabasi_game(&g1, nodes, power, m, NULL, /* outpref */ 0, A, IGRAPH_UNDIRECTED, IGRAPH_BARABASI_PSUMTREE, /* start from */ NULL); igraph_vector_int_t degree; igraph_vector_int_init(°ree, nodes); igraph_degree(&g1, °ree, igraph_vss_all(), IGRAPH_ALL, IGRAPH_NO_LOOPS); /* Measuring assortativity of the first graph */ igraph_assortativity_degree(&g1, &assortativity, IGRAPH_UNDIRECTED); printf("Assortativity of initial graph = %g\n\n", assortativity); igraph_destroy(&g1); /* Graph 2 (with the same degree sequence) generated by selecting vertices with the smallest degree first */ igraph_realize_degree_sequence(&g2, °ree, NULL, IGRAPH_SIMPLE_SW, IGRAPH_REALIZE_DEGSEQ_SMALLEST); igraph_assortativity_degree(&g2, &assortativity, IGRAPH_UNDIRECTED); printf("Assortativity after choosing vertices with the smallest degrees first = %g\n\n", assortativity); igraph_destroy(&g2); /* Graph 3 (with the same degree sequence) generated by selecting vertices with the largest degree first */ igraph_realize_degree_sequence(&g3, °ree, NULL, IGRAPH_SIMPLE_SW, IGRAPH_REALIZE_DEGSEQ_LARGEST); igraph_assortativity_degree(&g3, &assortativity, IGRAPH_UNDIRECTED); printf("Assortativity after choosing vertices with the largest degrees first = %g\n", assortativity); igraph_destroy(&g3); igraph_vector_int_destroy(°ree); return 0; }
igraph_error_t igraph_realize_bipartite_degree_sequence(
igraph_t *graph,
const igraph_vector_int_t *degrees1, const igraph_vector_int_t *degrees2,
const igraph_edge_type_sw_t allowed_edge_types, const igraph_realize_degseq_t method
);
This function generates a bipartite graph with the given bidegree sequence,
using a Havel-Hakimi-like construction algorithm. The order in which vertices
are connected up is controlled by the method parameter. When using the
IGRAPH_REALIZE_DEGSEQ_SMALLEST method, it is ensured that the graph will be
connected if and only if the given bidegree sequence is potentially connected.
The vertices of the graph will be ordered so that those having degrees1
come first, followed by degrees2.
Arguments:
|
Pointer to an uninitialized graph object. |
||||||
|
The degree sequence of the first partition. |
||||||
|
The degree sequence of the second partition. |
||||||
|
The types of edges to allow in the graph.
|
||||||
|
Controls the order in which vertices are selected for connection. Possible values:
|
Returns:
Error code. |
See also:
|
These functions produce single and multipartite complete graphs, as well as related graphs.
igraph_error_t igraph_full(igraph_t *graph, igraph_int_t n, igraph_bool_t directed,
igraph_bool_t loops);
In a full graph every possible edge is present: every vertex is connected to every other vertex. igraph generalizes the usual concept of complete graphs in graph theory to graphs with self-loops as well as to directed graphs.
Arguments:
|
Pointer to an uninitialized graph object. |
|
Integer, the number of vertices in the graph. |
|
Whether to create a directed graph. |
|
Whether to include self-loops. |
Returns:
Error code:
|
Time complexity: O(|V|^2) = O(|E|), where |V| is the number of vertices and |E| is the number of edges.
See also:
|
Example 11.11. File examples/simple/igraph_full.c
#include <igraph.h> #include <stdio.h> int main(void) { igraph_t graph; igraph_int_t n_vertices = 10; /* Initialize the library. */ igraph_setup(); /* Create an undirected complete graph. */ /* Use IGRAPH_UNDIRECTED and IGRAPH_NO_LOOPS instead of true and false for better readability. */ igraph_full(&graph, n_vertices, IGRAPH_UNDIRECTED, IGRAPH_NO_LOOPS); printf("The undirected complete graph on %" IGRAPH_PRId " vertices has %" IGRAPH_PRId " edges.\n", igraph_vcount(&graph), igraph_ecount(&graph)); /* Remember to destroy the object at the end. */ igraph_destroy(&graph); /* Create a directed complete graph. */ igraph_full(&graph, n_vertices, IGRAPH_DIRECTED, IGRAPH_NO_LOOPS); printf("The directed complete graph on %" IGRAPH_PRId " vertices has %" IGRAPH_PRId " edges.\n", igraph_vcount(&graph), igraph_ecount(&graph)); igraph_destroy(&graph); /* Create an undirected complete graph with self-loops. */ igraph_full(&graph, n_vertices, IGRAPH_UNDIRECTED, IGRAPH_LOOPS); printf("The undirected complete graph on %" IGRAPH_PRId " vertices with self-loops has %" IGRAPH_PRId " edges.\n", igraph_vcount(&graph), igraph_ecount(&graph)); igraph_destroy(&graph); /* Create a directed graph with self-loops. */ igraph_full(&graph, n_vertices, IGRAPH_DIRECTED, IGRAPH_LOOPS); printf("The directed complete graph on %" IGRAPH_PRId " vertices with self-loops has %" IGRAPH_PRId " edges.\n", igraph_vcount(&graph), igraph_ecount(&graph)); igraph_destroy(&graph); return 0; }
igraph_error_t igraph_full_citation(igraph_t *graph, igraph_int_t n,
igraph_bool_t directed);
This is a directed graph, where every i->j edge is
present if and only if j<i.
If the directed argument is false then an undirected graph is
created, and it is just a complete graph.
Arguments:
|
Pointer to an uninitialized graph object, the result is stored here. |
|
The number of vertices. |
|
Whether to created a directed graph. If false an undirected graph is created. |
Returns:
Error code. |
See also:
Time complexity: O(|V|^2) = O(|E|), where |V| is the number of vertices and |E| is the number of edges.
igraph_error_t igraph_full_multipartite(igraph_t *graph,
igraph_vector_int_t *types,
const igraph_vector_int_t *n,
igraph_bool_t directed,
igraph_neimode_t mode);
A multipartite graph contains two or more types of vertices and connections are only possible between two vertices of different types. This function creates a complete multipartite graph.
Arguments:
|
Pointer to an uninitialized graph object, the graph will be created here. |
|
Pointer to an integer vector. If not a null pointer, the type of each vertex will be stored here. |
|
Pointer to an integer vector, the number of vertices of each type. |
|
Boolean, whether to create a directed graph. |
|
A constant that gives the type of connections for
directed graphs. If |
Returns:
Error code. |
Time complexity: O(|V|+|E|), linear in the number of vertices and edges.
See also:
|
igraph_error_t igraph_turan(igraph_t *graph,
igraph_vector_int_t *types,
igraph_int_t n,
igraph_int_t r);
Turán graphs are complete multipartite graphs with the property that the sizes of the partitions are as close to equal as possible.
The Turán graph with n vertices and r partitions is the densest
graph on n vertices that does not contain a clique of size
r+1.
This function generates undirected graphs. The null graph is returned when the number of vertices is zero. A complete graph is returned if the number of partitions is greater than the number of vertices.
Arguments:
|
Pointer to an igraph_t object, the graph will be created here. |
|
Pointer to an integer vector. If not a null pointer, the type (partition index) of each vertex will be stored here. |
|
Integer, the number of vertices in the graph. |
|
Integer, the number of partitions of the graph, must be positive. |
Returns:
Error code. |
Time complexity: O(|V|+|E|), linear in the number of vertices and edges.
See also:
|
These functions return graphs from various graph collections.
igraph_error_t igraph_famous(igraph_t *graph, const char *name);
The name of the graph can be simply supplied as a string.
Note that this function creates graphs which don't take any parameters,
there are separate functions for graphs with parameters, e.g. igraph_full() for creating a full graph.
The following graphs are supported:
|
The bull graph, 5 vertices, 5 edges, resembles the head of a bull if drawn properly. |
|
This is the smallest triangle-free graph that is both 4-chromatic and 4-regular. According to the Grunbaum conjecture there exists an m-regular, m-chromatic graph with n vertices for every m>1 and n>2. The Chvatal graph is an example for m=4 and n=12. It has 24 edges. |
|
A non-Hamiltonian cubic symmetric graph with 28 vertices and 42 edges. |
|
The Platonic graph of the cube. A convex regular polyhedron with 8 vertices and 12 edges. |
|
A graph with 4 vertices and 5 edges, resembles a schematic diamond if drawn properly. |
|
Another Platonic solid with 20 vertices and 30 edges. |
|
The semisymmetric graph with minimum number of vertices, 20 and 40 edges. A semisymmetric graph is regular, edge transitive and not vertex transitive. |
|
This is a graph whose embedding to the Klein bottle can be colored with six colors, it is a counterexample to the necessity of the Heawood conjecture on a Klein bottle. It has 12 vertices and 18 edges. |
|
The Frucht Graph is the smallest cubical graph whose automorphism group consists only of the identity element. It has 12 vertices and 18 edges. |
|
The Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, and chromatic number 4. It is named after German mathematician Herbert Grötzsch, and its existence demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable. |
|
The Heawood graph is an undirected graph with 14 vertices and 21 edges. The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6. |
|
The Herschel graph is the smallest nonhamiltonian polyhedral graph. It is the unique such graph on 11 nodes, and has 18 edges. |
|
The house graph is a 5-vertex, 6-edge graph, the schematic draw of a house if drawn properly, basically a triangle on top of a square. |
|
The same as the house graph with an X in the square. 5 vertices and 8 edges. |
|
A Platonic solid with 12 vertices and 30 edges. |
|
A social network with 10 vertices and 18 edges. Krackhardt, D. Assessing the Political Landscape: Structure, Cognition, and Power in Organizations. Admin. Sci. Quart. 35, 342-369, 1990. |
|
The graph is a 4-arc transitive cubic graph, it has 30 vertices and 45 edges. |
|
The McGee graph is the unique 3-regular 7-cage graph, it has 24 vertices and 36 edges. |
|
The Meredith graph is a quartic graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian. |
|
A connected graph with 16 vertices and 27 edges containing no perfect matching. A matching in a graph is a set of pairwise non-incident edges; that is, no two edges share a common vertex. A perfect matching is a matching which covers all vertices of the graph. |
|
A graph whose connected components are the 9 graphs whose presence as a vertex-induced subgraph in a graph makes a nonline graph. It has 50 vertices and 72 edges. |
|
Platonic solid with 6 vertices and 12 edges. |
|
A 3-regular graph with 10 vertices and 15 edges. It is the smallest hypohamiltonian graph, i.e. it is non-hamiltonian but removing any single vertex from it makes it Hamiltonian. |
|
The unique (4,5)-cage graph, i.e. a 4-regular graph of girth 5. It has 19 vertices and 38 edges. |
|
A smallest nontrivial graph whose automorphism group is cyclic. It has 9 vertices and 15 edges. |
|
Platonic solid with 4 vertices and 6 edges. |
|
The smallest hypotraceable graph, on 34 vertices and 52 edges. A hypotracable graph does not contain a Hamiltonian path but after removing any single vertex from it the remainder always contains a Hamiltonian path. A graph containing a Hamiltonian path is called traceable. |
|
Tait's Hamiltonian graph conjecture states that every 3-connected 3-regular planar graph is Hamiltonian. This graph is a counterexample. It has 46 vertices and 69 edges. |
|
Returns a 12-vertex, triangle-free graph with chromatic number 3 that is uniquely 3-colorable. |
|
An identity graph with 25 vertices and 31 edges. An identity graph has a single graph automorphism, the trivial one. |
|
Social network of friendships between 34 members of a karate club at a US university in the 1970s. See W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research 33, 452-473 (1977). |
Arguments:
|
Pointer to an uninitialized graph object. |
|
Character constant, the name of the graph to be created, it is case insensitive. |
Returns:
Error code, |
See also:
Other functions for creating graph structures:
|
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges in the graph.
igraph_error_t igraph_atlas(igraph_t *graph, igraph_int_t number);
The graph atlas contains all simple undirected unlabeled graphs on between 0 and 7 vertices. The number of the graph is given as a parameter. The graphs are listed:
in increasing order of number of vertices;
for a fixed number of vertices, in increasing order of the number of edges;
for fixed numbers of vertices and edges, in lexicographically increasing order of the degree sequence, for example 111223 < 112222;
for fixed degree sequence, in increasing number of automorphisms.
The data was converted from the NetworkX software package, see https://networkx.org/.
See An Atlas of Graphs by Ronald C. Read and Robin J. Wilson, Oxford University Press, 1998.
Arguments:
|
Pointer to an uninitialized graph object. |
|
The number of the graph to generate. Must be between 0 and 1252 (inclusive). Graphs on 0-7 vertices start at numbers 0, 1, 2, 4, 8, 19, 53, and 209, respectively. |
Returns:
Error code. |
Added in version 0.2.
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
Example 11.12. File examples/simple/igraph_atlas.c
#include <igraph.h> int main(void) { igraph_t g; /* Initialize the library. */ igraph_setup(); igraph_atlas(&g, 45); igraph_write_graph_edgelist(&g, stdout); printf("\n"); igraph_destroy(&g); igraph_atlas(&g, 0); igraph_write_graph_edgelist(&g, stdout); printf("\n"); igraph_destroy(&g); igraph_atlas(&g, 1252); igraph_write_graph_edgelist(&g, stdout); printf("\n"); igraph_destroy(&g); return 0; }
igraph_error_t igraph_de_bruijn(igraph_t *graph, igraph_int_t m, igraph_int_t n);
A de Bruijn graph represents relationships between strings. An alphabet
of m letters are used and strings of length n are considered.
A vertex corresponds to every possible string and there is a directed edge
from vertex v to vertex w if the string of v can be transformed into
the string of w by removing its first letter and appending a letter to it.
Please note that the graph will have m to the power n vertices and
even more edges, so probably you don't want to supply too big numbers for
m and n.
De Bruijn graphs have some interesting properties, please see another source, e.g. Wikipedia for details.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, the number of letters in the alphabet. |
|
Integer, the length of the strings. |
Returns:
Error code. |
See also:
Time complexity: O(|V|+|E|), the number of vertices plus the number of edges.
igraph_error_t igraph_kautz(igraph_t *graph, igraph_int_t m, igraph_int_t n);
A Kautz graph is a labeled graph, vertices are labeled by strings
of length n+1 above an alphabet with m+1 letters, with
the restriction that every two consecutive letters in the string
must be different. There is a directed edge from a vertex v to
another vertex w if it is possible to transform the string of
v into the string of w by removing the first letter and
appending a letter to it. For string length 1 the new letter
cannot equal the old letter, so there are no loops.
Kautz graphs have some interesting properties, see e.g. Wikipedia for details.
Vincent Matossian wrote the first version of this function in R, thanks.
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, |
|
Integer, |
Returns:
Error code. |
See also:
Time complexity: O(|V|* [(m+1)/m]^n +|E|), in practice it is more
like O(|V|+|E|). |V| is the number of vertices, |E| is the number
of edges and m and n are the corresponding arguments.
igraph_error_t igraph_generalized_petersen(igraph_t *graph, igraph_int_t n, igraph_int_t k);
The generalized Petersen graph G(n, k) consists of n vertices
v_0, ..., v_n forming an "outer" cycle graph, and n additional vertices
u_0, ..., u_n forming an "inner" circulant graph where u_i
is connected to u_(i + k mod n). Additionally, all v_i are
connected to u_i.
G(n, k) has 2n vertices and 3n edges. The Petersen graph
itself is G(5, 2).
Reference:
M. E. Watkins, A Theorem on Tait Colorings with an Application to the Generalized Petersen Graphs, Journal of Combinatorial Theory 6, 152-164 (1969). https://doi.org/10.1016%2FS0021-9800%2869%2980116-X
Arguments:
|
Pointer to an uninitialized graph object, the result will be stored here. |
|
Integer, |
|
Integer, |
Returns:
Error code. |
See also:
|
Time complexity: O(|V|), the number of vertices in the graph.
igraph_error_t igraph_mycielski_graph(igraph_t *graph, igraph_int_t k);
The Mycielski graph of order k, denoted M_k, is a triangle-free graph on
k vertices with chromatic number k. It is defined through the Mycielski
construction described in the documentation of igraph_mycielskian().
Some authors define Mycielski graphs only for k > 1.
igraph extends this to all k >= 0.
The first few Mycielski graphs are:
M_0: Null graph
M_1: Single vertex
M_2: Path graph with 2 vertices
M_3: Cycle graph with 5 vertices
M_4: Grötzsch graph (a triangle-free graph with chromatic number 4)
The vertex count of M_k is
n_k = 3 * 2^(k-2) - 1 for k > 1 and k otherwise.
The edge count is
m_k = (7 * 3^(k-2) + 1) / 2 - 3 * 2^(k - 2) for k > 1
and 0 otherwise.
Arguments:
|
Pointer to an uninitialized graph object. The generated Mycielski graph will be stored here. |
|
Integer, the order of the Mycielski graph (must be non-negative). |
Returns:
Error code. |
See also:
Time complexity: O(3^k), i.e. exponential in k.
| ← Chapter 10. Graph, vertex and edge attributes | Chapter 12. Stochastic graph generators ("games") → |