Fast compiled graphs¶

This is a Cython implementation of the base class for sparse and dense graphs in Sage. It is not intended for use on its own. Specific graph types should extend this base class and implement missing functionalities. Whenever possible, specific methods should also be overridden with implementations that suit the graph type under consideration.

Data structure¶

The class CGraph maintains the following variables:

cdef int num_verts
cdef int num_arcs
cdef int *in_degrees
cdef int *out_degrees
cdef bitset_t active_vertices

The bitset active_vertices is a list of all available vertices for use, but only the ones which are set are considered to actually be in the graph. The variables num_verts and num_arcs are self-explanatory. Note that num_verts is the number of bits set in active_vertices, not the full length of the bitset. The arrays in_degrees and out_degrees are of the same length as the bitset.

For more information about active vertices, see the documentation for the method realloc.

Classes and methods¶

class sage.graphs.base.c_graph.CGraph¶

Bases: object

Compiled sparse and dense graphs.

add_arc(u, v)¶

Add the given arc to this graph.

INPUT:

u – integer; the tail of an arc.
v – integer; the head of an arc.

OUTPUT:

Raise NotImplementedError. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

add_arc – add_arc method for sparse graphs.
add_arc – add_arc method for dense graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.add_arc(0, 1)
...
NotImplementedError

add_vertex(k=-1)¶

Adds vertex k to the graph.

INPUT:

k – nonnegative integer or -1 (default: -1). If $k = -1$ , a new vertex is added and the integer used is returned. That is, for $k = -1$ , this function will find the first available vertex that is not in self and add that vertex to this graph.

OUTPUT:

-1 – indicates that no vertex was added because the current allocation is already full or the vertex is out of range.
nonnegative integer – this vertex is now guaranteed to be in the graph.

See also

add_vertex_unsafe() – add a vertex to a graph. This method is potentially unsafe. You should instead use add_vertex().
add_vertices() – add a bunch of vertices to a graph.

EXAMPLES:

Adding vertices to a sparse graph:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: G = SparseGraph(3, extra_vertices=3)
sage: G.add_vertex(3)
3
sage: G.add_arc(2, 5)
...
RuntimeError: Vertex (5) is not a vertex of the graph.
sage: G.add_arc(1, 3)
sage: G.has_arc(1, 3)
True
sage: G.has_arc(2, 3)
False

Adding vertices to a dense graph:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(3, extra_vertices=3)
sage: G.add_vertex(3)
3
sage: G.add_arc(2,5)
...
RuntimeError: Vertex (5) is not a vertex of the graph.
sage: G.add_arc(1, 3)
sage: G.has_arc(1, 3)
True
sage: G.has_arc(2, 3)
False

Repeatedly adding a vertex using $k = -1$ will allocate more memory as required:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: G = SparseGraph(3, extra_vertices=0)
sage: G.verts()
[0, 1, 2]
sage: for i in range(10):
...       _ = G.add_vertex(-1);
...
sage: G.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(3, extra_vertices=0)
sage: G.verts()
[0, 1, 2]
sage: for i in range(12):
...       _ = G.add_vertex(-1);
...
sage: G.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

TESTS:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: G = SparseGraph(3, extra_vertices=0)
sage: G.add_vertex(6)
...
RuntimeError: Requested vertex is past twice the allocated range: use realloc.

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(3, extra_vertices=0)
sage: G.add_vertex(6)
...
RuntimeError: Requested vertex is past twice the allocated range: use realloc.

add_vertices(verts)¶

Adds vertices from the iterable verts.

INPUT:

verts – an iterable of vertices.

OUTPUT:

Same as for add_vertex().

See also

add_vertex() – add a vertex to a graph.

EXAMPLE:

Adding vertices for sparse graphs:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=4, extra_vertices=4)
sage: S.verts()
[0, 1, 2, 3]
sage: S.add_vertices([3,5,7,9])
sage: S.verts()
[0, 1, 2, 3, 5, 7, 9]
sage: S.realloc(20)
sage: S.verts()
[0, 1, 2, 3, 5, 7, 9]

Adding vertices for dense graphs:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: D = DenseGraph(nverts=4, extra_vertices=4)
sage: D.verts()
[0, 1, 2, 3]
sage: D.add_vertices([3,5,7,9])
sage: D.verts()
[0, 1, 2, 3, 5, 7, 9]
sage: D.realloc(20)
sage: D.verts()
[0, 1, 2, 3, 5, 7, 9]

all_arcs(u, v)¶

Return the labels of all arcs from u to v.

INPUT:

u – integer; the tail of an arc.
v – integer; the head of an arc.

OUTPUT:

Raise NotImplementedError. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

all_arcs – all_arcs method for sparse graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.all_arcs(0, 1)
...
NotImplementedError

check_vertex(n)¶

Checks that n is a vertex of self.

This method is different from has_vertex(). The current method raises an error if n is not a vertex of this graph. On the other hand, has_vertex() returns a boolean to signify whether or not n is a vertex of this graph.

INPUT:

n – a nonnegative integer representing a vertex.

OUTPUT:

Raise an error if n is not a vertex of this graph.

See also

has_vertex() – determine whether this graph has a specific vertex.

EXAMPLES:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=10, expected_degree=3, extra_vertices=10)
sage: S.check_vertex(4)
sage: S.check_vertex(12)
...
RuntimeError: Vertex (12) is not a vertex of the graph.
sage: S.check_vertex(24)
...
RuntimeError: Vertex (24) is not a vertex of the graph.
sage: S.check_vertex(-19)
...
RuntimeError: Vertex (-19) is not a vertex of the graph.

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: D = DenseGraph(nverts=10, extra_vertices=10)
sage: D.check_vertex(4)
sage: D.check_vertex(12)
...
RuntimeError: Vertex (12) is not a vertex of the graph.
sage: D.check_vertex(24)
...
RuntimeError: Vertex (24) is not a vertex of the graph.
sage: D.check_vertex(-19)
...
RuntimeError: Vertex (-19) is not a vertex of the graph.

current_allocation()¶

Report the number of vertices allocated.

INPUT:

None.

OUTPUT:

The number of vertices allocated. This number is usually different from the order of a graph. We may have allocated enough memory for a graph to hold $n > 0$ vertices, but the order (actual number of vertices) of the graph could be less than $n$ .

EXAMPLES:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=4, extra_vertices=4)
sage: S.current_allocation()
8
sage: S.add_vertex(6)
6
sage: S.current_allocation()
8
sage: S.add_vertex(10)
10
sage: S.current_allocation()
16
sage: S.add_vertex(40)
...
RuntimeError: Requested vertex is past twice the allocated range: use realloc.
sage: S.realloc(50)
sage: S.add_vertex(40)
40
sage: S.current_allocation()
50
sage: S.realloc(30)
-1
sage: S.current_allocation()
50
sage: S.del_vertex(40)
sage: S.realloc(30)
sage: S.current_allocation()
30

The actual number of vertices in a graph might be less than the number of vertices allocated for the graph:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(nverts=3, extra_vertices=2)
sage: order = len(G.verts())
sage: order
3
sage: G.current_allocation()
5
sage: order < G.current_allocation()
True

del_all_arcs(u, v)¶

Delete all arcs from u to v.

INPUT:

u – integer; the tail of an arc.
v – integer; the head of an arc.

OUTPUT:

Raise NotImplementedError. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

del_all_arcs – del_all_arcs method for sparse graphs.
del_all_arcs – del_all_arcs method for dense graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.del_all_arcs(0,1)
...
NotImplementedError

del_vertex(v)¶

Deletes the vertex v, along with all edges incident to it. If v is not in self, fails silently.

INPUT:

v – a nonnegative integer representing a vertex.

OUTPUT:

None.

See also

del_vertex_unsafe() – delete a vertex from a graph. This method is potentially unsafe. Use del_vertex() instead.

EXAMPLES:

Deleting vertices of sparse graphs:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: G = SparseGraph(3)
sage: G.add_arc(0, 1)
sage: G.add_arc(0, 2)
sage: G.add_arc(1, 2)
sage: G.add_arc(2, 0)
sage: G.del_vertex(2)
sage: for i in range(2):
...       for j in range(2):
...           if G.has_arc(i, j):
...               print i, j
0 1
sage: G = SparseGraph(3)
sage: G.add_arc(0, 1)
sage: G.add_arc(0, 2)
sage: G.add_arc(1, 2)
sage: G.add_arc(2, 0)
sage: G.del_vertex(1)
sage: for i in xrange(3):
...       for j in xrange(3):
...           if G.has_arc(i, j):
...               print i, j
0 2
2 0

Deleting vertices of dense graphs:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(4)
sage: G.add_arc(0, 1); G.add_arc(0, 2)
sage: G.add_arc(3, 1); G.add_arc(3, 2)
sage: G.add_arc(1, 2)
sage: G.verts()
[0, 1, 2, 3]
sage: G.del_vertex(3); G.verts()
[0, 1, 2]
sage: for i in range(3):
...       for j in range(3):
...           if G.has_arc(i, j):
...               print i, j
...
0 1
0 2
1 2

If the vertex to be deleted is not in this graph, then fail silently:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: G = SparseGraph(3)
sage: G.verts()
[0, 1, 2]
sage: G.has_vertex(3)
False
sage: G.del_vertex(3)
sage: G.verts()
[0, 1, 2]

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.verts()
[0, 1, 2, 3, 4]
sage: G.has_vertex(6)
False
sage: G.del_vertex(6)
sage: G.verts()
[0, 1, 2, 3, 4]

has_arc(u, v)¶

Determine whether or not the given arc is in this graph.

INPUT:

u – integer; the tail of an arc.
v – integer; the head of an arc.

OUTPUT:

Print a Not Implemented! message. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

has_arc – has_arc method for sparse graphs.
has_arc – has_arc method for dense graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.has_arc(0, 1)
Not Implemented!
False

has_vertex(n)¶

Determine whether the vertex n is in self.

This method is different from check_vertex(). The current method returns a boolean to signify whether or not n is a vertex of this graph. On the other hand, check_vertex() raises an error if n is not a vertex of this graph.

INPUT:

n – a nonnegative integer representing a vertex.

OUTPUT:

True if n is a vertex of this graph; False otherwise.

See also

check_vertex() – raise an error if this graph does not contain a specific vertex.

EXAMPLES:

Upon initialization, a SparseGraph or DenseGraph has the first nverts vertices:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=10, expected_degree=3, extra_vertices=10)
sage: S.has_vertex(6)
True
sage: S.has_vertex(12)
False
sage: S.has_vertex(24)
False
sage: S.has_vertex(-19)
False

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: D = DenseGraph(nverts=10, extra_vertices=10)
sage: D.has_vertex(6)
True
sage: D.has_vertex(12)
False
sage: D.has_vertex(24)
False
sage: D.has_vertex(-19)
False

in_neighbors(v)¶

Gives the in-neighbors of the vertex v.

INPUT:

v – integer representing a vertex of this graph.

OUTPUT:

Raise NotImplementedError. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

in_neighbors – in_neighbors method for sparse graphs.
in_neighbors – in_neighbors method for dense graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.in_neighbors(0)
...
NotImplementedError

out_neighbors(u)¶

Gives the out-neighbors of the vertex u.

INPUT:

u – integer representing a vertex of this graph.

OUTPUT:

Raise NotImplementedError. This method is not implemented at the CGraph level. A child class should provide a suitable implementation.

See also

out_neighbors – out_neighbors implementation for sparse graphs.
out_neighbors – out_neighbors implementation for dense graphs.

EXAMPLE:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.out_neighbors(0)
...
NotImplementedError

realloc(total)¶

Reallocate the number of vertices to use, without actually adding any.

INPUT:

total – integer; the total size to make the array of vertices.

OUTPUT:

Raise a NotImplementedError. This method is not implemented in this base class. A child class should provide a suitable implementation.

See also

realloc – a realloc implementation for sparse graphs.
realloc – a realloc implementation for dense graphs.

EXAMPLES:

First, note that realloc() is implemented for SparseGraph and DenseGraph differently, and is not implemented at the CGraph level:

sage: from sage.graphs.base.c_graph import CGraph
sage: G = CGraph()
sage: G.realloc(20)
...
NotImplementedError

The realloc implementation for sparse graphs:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=4, extra_vertices=4)
sage: S.current_allocation()
8
sage: S.add_vertex(6)
6
sage: S.current_allocation()
8
sage: S.add_vertex(10)
10
sage: S.current_allocation()
16
sage: S.add_vertex(40)
...
RuntimeError: Requested vertex is past twice the allocated range: use realloc.
sage: S.realloc(50)
sage: S.add_vertex(40)
40
sage: S.current_allocation()
50
sage: S.realloc(30)
-1
sage: S.current_allocation()
50
sage: S.del_vertex(40)
sage: S.realloc(30)
sage: S.current_allocation()
30

The realloc implementation for dense graphs:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: D = DenseGraph(nverts=4, extra_vertices=4)
sage: D.current_allocation()
8
sage: D.add_vertex(6)
6
sage: D.current_allocation()
8
sage: D.add_vertex(10)
10
sage: D.current_allocation()
16
sage: D.add_vertex(40)
...
RuntimeError: Requested vertex is past twice the allocated range: use realloc.
sage: D.realloc(50)
sage: D.add_vertex(40)
40
sage: D.current_allocation()
50
sage: D.realloc(30)
-1
sage: D.current_allocation()
50
sage: D.del_vertex(40)
sage: D.realloc(30)
sage: D.current_allocation()
30

verts()¶

Returns a list of the vertices in self.

INPUT:

None.

OUTPUT:

A list of all vertices in this graph.

EXAMPLE:

sage: from sage.graphs.base.sparse_graph import SparseGraph
sage: S = SparseGraph(nverts=4, extra_vertices=4)
sage: S.verts()
[0, 1, 2, 3]
sage: S.add_vertices([3,5,7,9])
sage: S.verts()
[0, 1, 2, 3, 5, 7, 9]
sage: S.realloc(20)
sage: S.verts()
[0, 1, 2, 3, 5, 7, 9]

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(3, extra_vertices=2)
sage: G.verts()
[0, 1, 2]
sage: G.del_vertex(0)
sage: G.verts()
[1, 2]

class sage.graphs.base.c_graph.CGraphBackend¶

Bases: sage.graphs.base.graph_backends.GenericGraphBackend

Base class for sparse and dense graph backends.

sage: from sage.graphs.base.c_graph import CGraphBackend

This class is extended by SparseGraphBackend and DenseGraphBackend, which are fully functional backends. This class is mainly just for vertex functions, which are the same for both. A CGraphBackend will not work on its own:

sage: from sage.graphs.base.c_graph import CGraphBackend
sage: CGB = CGraphBackend()
sage: CGB.degree(0, True)
...
AttributeError: 'CGraphBackend' object has no attribute 'vertex_ints'

The appropriate way to use these backends is via Sage graphs:

sage: G = Graph(30, implementation="c_graph")
sage: G.add_edges([(0,1), (0,3), (4,5), (9, 23)])
sage: G.edges(labels=False)
[(0, 1), (0, 3), (4, 5), (9, 23)]

See also

SparseGraphBackend – backend for sparse graphs.
DenseGraphBackend – backend for dense graphs.

add_vertex(name)¶

Add a vertex to self.

INPUT:

name – the vertex to be added (must be hashable).

OUTPUT:

None.

See also

add_vertices() – add a bunch of vertices of this graph.
has_vertex() – returns whether or not this graph has a specific vertex.

EXAMPLE:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_vertex(10)
sage: D.add_vertex([])
...
TypeError: unhashable type: 'list'

sage: S = sage.graphs.base.sparse_graph.SparseGraphBackend(9)
sage: S.add_vertex(10)
sage: S.add_vertex([])
...
TypeError: unhashable type: 'list'

add_vertices(vertices)¶

Add vertices to self.

INPUT:

vertices – iterator of vertex labels.

OUTPUT:

None.

See also

add_vertex() – add a vertex to this graph.

EXAMPLE:

sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(1)
sage: D.add_vertices([1,2,3])

sage: G = sage.graphs.base.sparse_graph.SparseGraphBackend(0)
sage: G.add_vertices([0,1])
sage: list(G.iterator_verts(None))
[0, 1]
sage: list(G.iterator_edges([0,1], True))
[]

sage: import sage.graphs.base.dense_graph
sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_vertices([10,11,12])

bidirectional_dijkstra(x, y)¶

Returns the shortest path between x and y using a bidirectional version of Dijkstra’s algorithm.

INPUT:

x – the starting vertex in the shortest path from x to y.
y – the end vertex in the shortest path from x to y.

OUTPUT:

A list of vertices in the shortest path from x to y.

EXAMPLE:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: for (u,v) in G.edges(labels=None):
...      G.set_edge_label(u,v,1)
sage: G.shortest_path(0, 1, by_weight=True)
[0, 1]

TEST:

Bugfix from #7673

sage: G = Graph(implementation="networkx")
sage: G.add_edges([(0,1,9),(0,2,8),(1,2,7)])
sage: Gc = G.copy(implementation='c_graph')
sage: sp = G.shortest_path_length(0,1,by_weight=True)
sage: spc = Gc.shortest_path_length(0,1,by_weight=True)
sage: sp == spc
True

breadth_first_search(v, reverse=False, ignore_direction=False)¶

Returns a breadth-first search from vertex v.

INPUT:

v – a vertex from which to start the breadth-first search.
reverse – boolean (default: False). This is only relevant to digraphs. If this is a digraph, consider the reversed graph in which the out-neighbors become the in-neighbors and vice versa.
ignore_direction – boolean (default: False). This is only relevant to digraphs. If this is a digraph, ignore all orientations and consider the graph as undirected.

ALGORITHM:

Below is a general template for breadth-first search.

Input: A directed or undirected graph $G = (V, E)$ of order $n > 0$ . A vertex $s$ from which to start the search. The vertices are numbered from 1 to $n = |V|$ , i.e. $V = \{1, 2, \dots, n\}$ .
Output: A list $D$ of distances of all vertices from $s$ . A tree $T$ rooted at $s$ .

$Q \leftarrow [s]$ # a queue of nodes to visit
$D \leftarrow [\infty, \infty, \dots, \infty]$ # $n$ copies of $\infty$
$D[s] \leftarrow 0$
$T \leftarrow [\,]$
while do
1. $v \leftarrow \text{dequeue}(Q)$
2. for each do # for digraphs, use out-neighbor set
  1. if then
    1. $D[w] \leftarrow D[v] + 1$
    2. $\text{enqueue}(Q, w)$
    3. $\text{append}(T, vw)$
return $(D, T)$

See also

breadth_first_search – breadth-first search for generic graphs.
depth_first_search – depth-first search for generic graphs.
depth_first_search() – depth-first search for fast compiled graphs.

EXAMPLES:

Breadth-first search of the Petersen graph starting at vertex 0:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: list(G.breadth_first_search(0))
[0, 1, 4, 5, 2, 6, 3, 9, 7, 8]

Visiting German cities using breadth-first search:

sage: G = Graph({"Mannheim": ["Frankfurt","Karlsruhe"],
...   "Frankfurt": ["Mannheim","Wurzburg","Kassel"],
...   "Kassel": ["Frankfurt","Munchen"],
...   "Munchen": ["Kassel","Nurnberg","Augsburg"],
...   "Augsburg": ["Munchen","Karlsruhe"],
...   "Karlsruhe": ["Mannheim","Augsburg"],
...   "Wurzburg": ["Frankfurt","Erfurt","Nurnberg"],
...   "Nurnberg": ["Wurzburg","Stuttgart","Munchen"],
...   "Stuttgart": ["Nurnberg"],
...   "Erfurt": ["Wurzburg"]}, implementation="c_graph")
sage: list(G.breadth_first_search("Frankfurt"))
['Frankfurt', 'Mannheim', 'Kassel', 'Wurzburg', 'Karlsruhe', 'Munchen', 'Erfurt', 'Nurnberg', 'Augsburg', 'Stuttgart']

degree(v, directed)¶

Return the degree of the vertex v.

INPUT:

v – a vertex of the graph.
directed – boolean; whether to take into account the orientation of this graph in counting the degree of v.

OUTPUT:

The degree of vertex v.

EXAMPLE:

sage: from sage.graphs.base.sparse_graph import SparseGraphBackend
sage: B = SparseGraphBackend(7)
sage: B.degree(3, False)
0

del_vertex(v)¶

Delete a vertex in self, failing silently if the vertex is not in the graph.

INPUT:

v – vertex to be deleted.

OUTPUT:

None.

See also

del_vertices() – delete a bunch of vertices from this graph.

EXAMPLE:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.del_vertex(0)
sage: D.has_vertex(0)
False

sage: S = sage.graphs.base.sparse_graph.SparseGraphBackend(9)
sage: S.del_vertex(0)
sage: S.has_vertex(0)
False

del_vertices(vertices)¶

Delete vertices from an iterable container.

INPUT:

vertices – iterator of vertex labels.

OUTPUT:

Same as for del_vertex().

See also

del_vertex() – delete a vertex of this graph.

EXAMPLE:

sage: import sage.graphs.base.dense_graph
sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.del_vertices([7,8])
sage: D.has_vertex(7)
False
sage: D.has_vertex(6)
True

sage: D = sage.graphs.base.sparse_graph.SparseGraphBackend(9)
sage: D.del_vertices([1,2,3])
sage: D.has_vertex(1)
False
sage: D.has_vertex(0)
True

depth_first_search(v, reverse=False, ignore_direction=False)¶

Returns a depth-first search from vertex v.

INPUT:

v – a vertex from which to start the depth-first search.
reverse – boolean (default: False). This is only relevant to digraphs. If this is a digraph, consider the reversed graph in which the out-neighbors become the in-neighbors and vice versa.
ignore_direction – boolean (default: False). This is only relevant to digraphs. If this is a digraph, ignore all orientations and consider the graph as undirected.

ALGORITHM:

Below is a general template for depth-first search.

Input: A directed or undirected graph $G = (V, E)$ of order $n > 0$ . A vertex $s$ from which to start the search. The vertices are numbered from 1 to $n = |V|$ , i.e. $V = \{1, 2, \dots, n\}$ .
Output: A list $D$ of distances of all vertices from $s$ . A tree $T$ rooted at $s$ .

$S \leftarrow [s]$ # a stack of nodes to visit
$D \leftarrow [\infty, \infty, \dots, \infty]$ # $n$ copies of $\infty$
$D[s] \leftarrow 0$
$T \leftarrow [\,]$
while do
1. $v \leftarrow \text{pop}(S)$
2. for each do # for digraphs, use out-neighbor set
  1. if then
    1. $D[w] \leftarrow D[v] + 1$
    2. $\text{push}(S, w)$
    3. $\text{append}(T, vw)$
return $(D, T)$

See also

breadth_first_search() – breadth-first search for fast compiled graphs.
breadth_first_search – breadth-first search for generic graphs.
depth_first_search – depth-first search for generic graphs.

EXAMPLES:

Traversing the Petersen graph using depth-first search:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: list(G.depth_first_search(0))
[0, 5, 8, 6, 9, 7, 2, 3, 4, 1]

Visiting German cities using depth-first search:

sage: G = Graph({"Mannheim": ["Frankfurt","Karlsruhe"],
...   "Frankfurt": ["Mannheim","Wurzburg","Kassel"],
...   "Kassel": ["Frankfurt","Munchen"],
...   "Munchen": ["Kassel","Nurnberg","Augsburg"],
...   "Augsburg": ["Munchen","Karlsruhe"],
...   "Karlsruhe": ["Mannheim","Augsburg"],
...   "Wurzburg": ["Frankfurt","Erfurt","Nurnberg"],
...   "Nurnberg": ["Wurzburg","Stuttgart","Munchen"],
...   "Stuttgart": ["Nurnberg"],
...   "Erfurt": ["Wurzburg"]}, implementation="c_graph")
sage: list(G.depth_first_search("Frankfurt"))
['Frankfurt', 'Wurzburg', 'Nurnberg', 'Munchen', 'Kassel', 'Augsburg', 'Karlsruhe', 'Mannheim', 'Stuttgart', 'Erfurt']

has_vertex(v)¶

Returns whether v is a vertex of self.

INPUT:

v – any object.

OUTPUT:

True if v is a vertex of this graph; False otherwise.

EXAMPLE:

sage: from sage.graphs.base.sparse_graph import SparseGraphBackend
sage: B = SparseGraphBackend(7)
sage: B.has_vertex(6)
True
sage: B.has_vertex(7)
False

is_connected()¶

Returns whether the graph is connected.

INPUT:

None.

OUTPUT:

True if this graph is connected; False otherwise.

EXAMPLES:

Petersen’s graph is connected:

sage: DiGraph(graphs.PetersenGraph(),implementation="c_graph").is_connected()
True

While the disjoint union of two of them is not:

sage: DiGraph(2*graphs.PetersenGraph(),implementation="c_graph").is_connected()
False

A graph with non-integer vertex labels::: sage: Graph(graphs.CubeGraph(3), implementation=’c_graph’).is_connected() True

is_strongly_connected()¶

Returns whether the graph is strongly connected.

INPUT:

None.

OUTPUT:

True if this graph is strongly connected; False otherwise.

EXAMPLES:

The circuit on 3 vertices is obviously strongly connected:

sage: g = DiGraph({0: [1], 1: [2], 2: [0]}, implementation="c_graph")
sage: g.is_strongly_connected()
True

But a transitive triangle is not:

sage: g = DiGraph({0: [1,2], 1: [2]}, implementation="c_graph")
sage: g.is_strongly_connected()
False

iterator_in_nbrs(v)¶

Returns an iterator over the incoming neighbors of v.

INPUT:

v – a vertex of this graph.

OUTPUT:

An iterator over the in-neighbors of the vertex v.

See also

iterator_nbrs() – returns an iterator over the neighbors of a vertex.
iterator_out_nbrs() – returns an iterator over the out-neighbors of a vertex.

EXAMPLE:

sage: P = DiGraph(graphs.PetersenGraph().to_directed(), implementation="c_graph")
sage: list(P._backend.iterator_in_nbrs(0))
[1, 4, 5]

iterator_nbrs(v)¶

Returns an iterator over the neighbors of v.

INPUT:

v – a vertex of this graph.

OUTPUT:

An iterator over the neighbors the vertex v.

See also

iterator_in_nbrs() – returns an iterator over the in-neighbors of a vertex.
iterator_out_nbrs() – returns an iterator over the out-neighbors of a vertex.
iterator_verts() – returns an iterator over a given set of vertices.

EXAMPLE:

sage: P = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: list(P._backend.iterator_nbrs(0))
[1, 4, 5]

iterator_out_nbrs(v)¶

Returns an iterator over the outgoing neighbors of v.

INPUT:

v – a vertex of this graph.

OUTPUT:

An iterator over the out-neighbors of the vertex v.

See also

iterator_nbrs() – returns an iterator over the neighbors of a vertex.
iterator_in_nbrs() – returns an iterator over the in-neighbors of a vertex.

EXAMPLE:

sage: P = DiGraph(graphs.PetersenGraph().to_directed(), implementation="c_graph")
sage: list(P._backend.iterator_out_nbrs(0))
[1, 4, 5]

iterator_verts(verts=None)¶

Returns an iterator over the vertices of self intersected with verts.

INPUT:

verts – an iterable container of objects (default: None).

OUTPUT:

If verts=None, return an iterator over all vertices of this graph.
If verts is an iterable container of vertices, find the intersection of verts with the vertex set of this graph and return an iterator over the resulting intersection.

See also

iterator_nbrs() – returns an iterator over the neighbors of a vertex.

EXAMPLE:

sage: P = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: list(P._backend.iterator_verts(P))
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: list(P._backend.iterator_verts())
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: list(P._backend.iterator_verts([1, 2, 3]))
[1, 2, 3]
sage: list(P._backend.iterator_verts([1, 2, 10]))
[1, 2]

loops(new=None)¶

Returns whether loops are allowed in this graph.

INPUT:

new – (default: None); boolean (to set) or None (to get).

OUTPUT:

If new=None, return True if this graph allows self-loops or False if self-loops are not allowed.
If new is a boolean, set the self-loop permission of this graph according to the boolean value of new.

EXAMPLE:

sage: G = Graph(implementation='c_graph')
sage: G._backend.loops()
False
sage: G._backend.loops(True)
sage: G._backend.loops()
True

name(new=None)¶

Returns the name of this graph.

INPUT:

new – (default: None); boolean (to set) or None (to get).

OUTPUT:

If new=None, return the name of this graph. Otherwise, set the name of this graph to the value of new.

EXAMPLE:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: G._backend.name()
'Petersen graph'
sage: G._backend.name("Peter Pan's graph")
sage: G._backend.name()
"Peter Pan's graph"

num_edges(directed)¶

Returns the number of edges in self.

INPUT:

directed – boolean; whether to count (u,v) and (v,u) as one or two edges.

OUTPUT:

If directed=True, counts the number of directed edges in this graph. Otherwise, return the size of this graph.

See also

num_verts() – return the order of this graph.

EXAMPLE:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: G._backend.num_edges(False)
15

num_verts()¶

Returns the number of vertices in self.

INPUT:

None.

OUTPUT:

The order of this graph.

See also

num_edges() – return the number of (directed) edges in this graph.

EXAMPLE:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: G._backend.num_verts()
10

relabel(perm, directed)¶

Relabels the graph according to perm.

INPUT:

perm – anything which represents a permutation as v --> perm[v], for example a dict or a list.
directed – ignored (this is here for compatibility with other backends).

EXAMPLES:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: G._backend.relabel(range(9,-1,-1), False)
sage: G.edges()
[(0, 2, None),
 (0, 3, None),
 (0, 5, None),
 (1, 3, None),
 (1, 4, None),
 (1, 6, None),
 (2, 4, None),
 (2, 7, None),
 (3, 8, None),
 (4, 9, None),
 (5, 6, None),
 (5, 9, None),
 (6, 7, None),
 (7, 8, None),
 (8, 9, None)]

shortest_path(x, y)¶

Returns the shortest path between x and y.

INPUT:

x – the starting vertex in the shortest path from x to y.
y – the end vertex in the shortest path from x to y.

OUTPUT:

A list of vertices in the shortest path from x to y.

EXAMPLE:

sage: G = Graph(graphs.PetersenGraph(), implementation="c_graph")
sage: G.shortest_path(0, 1)
[0, 1]

shortest_path_all_vertices(v, cutoff=None)¶

Returns for each vertex u a shortest v-u path.

INPUT:

v – a starting vertex in the shortest path.
cutoff – maximal distance. Longer paths will not be returned.

OUTPUT:

A list which associates to each vertex u the shortest path between u and v if there is one.

Note

The weight of edges is not taken into account.

ALGORITHM:

This is just a breadth-first search.

EXAMPLES:

On the Petersen Graph:

sage: g = graphs.PetersenGraph()
sage: paths = g._backend.shortest_path_all_vertices(0)
sage: all([ len(paths[v]) == 0 or len(paths[v])-1 == g.distance(0,v) for v in g])
True

On a disconnected graph

sage: g = 2*graphs.RandomGNP(20,.3)
sage: paths = g._backend.shortest_path_all_vertices(0)
sage: all([ (not paths.has_key(v) and g.distance(0,v) == +Infinity) or len(paths[v])-1 == g.distance(0,v) for v in g])
True

strongly_connected_component_containing_vertex(v)¶

Returns the strongly connected component containing the given vertex.

INPUT:

v – a vertex

EXAMPLES:

The digraph obtained from the PetersenGraph has an unique strongly connected component:

sage: g = DiGraph(graphs.PetersenGraph())
sage: g.strongly_connected_component_containing_vertex(0)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

In the Butterfly DiGraph, each vertex is a strongly connected component:

sage: g = digraphs.ButterflyGraph(3)
sage: all([[v] == g.strongly_connected_component_containing_vertex(v) for v in g])
True

class sage.graphs.base.c_graph.Search_iterator¶

Bases: object

An iterator for traversing a (di)graph.

This class is commonly used to perform a depth-first or breadth-first search. The class does not build all at once in memory the whole list of visited vertices. The class maintains the following variables:

graph – a graph whose vertices are to be iterated over.
direction – integer; this determines the position at which vertices to be visited are removed from the list stack. For breadth-first search (BFS), element removal occurs at the start of the list, as signified by the value direction=0. This is because in implementations of BFS, the list of vertices to visit are usually maintained by a queue, so element insertion and removal follow a first-in first-out (FIFO) protocol. For depth-first search (DFS), element removal occurs at the end of the list, as signified by the value direction=-1. The reason is that DFS is usually implemented using a stack to maintain the list of vertices to visit. Hence, element insertion and removal follow a last-in first-out (LIFO) protocol.
stack – a list of vertices to visit.
seen – a list of vertices that are already visited.
test_out – boolean; whether we want to consider the out-neighbors of the graph to be traversed. For undirected graphs, we consider both the in- and out-neighbors. However, for digraphs we only traverse along out-neighbors.
test_in – boolean; whether we want to consider the in-neighbors of the graph to be traversed. For undirected graphs, we consider both the in- and out-neighbors.

next()¶: x.next() -> the next value, or raise StopIteration

Fast compiled graphs¶

Data structure¶

Classes and methods¶

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