# Fast dense graphs¶

For an overview of graph data structures in sage, see overview.

## Usage Introduction¶

sage: from sage.graphs.base.dense_graph import DenseGraph


Dense graphs are initialized as follows:

sage: D = DenseGraph(nverts=10, extra_vertices=10)


This example initializes a dense graph with room for twenty vertices, the first ten of which are in the graph. In general, the first nverts are “active.” For example, see that 9 is already in the graph:

sage: D.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
9
sage: D.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]


But 10 is not, until we add it:

sage: D.add_vertex(10)
10
sage: D.verts()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]


You can begin working right away as follows:

sage: D.add_arc(0, 1)
sage: D.has_arc(7, 3)
False
sage: D.has_arc(0, 1)
True
sage: D.in_neighbors(1)

sage: D.out_neighbors(1)
[0, 2]
sage: D.del_all_arcs(0, 1)
sage: D.has_arc(0, 1)
False
sage: D.has_arc(1, 2)
True
sage: D.del_vertex(7)
sage: D.has_arc(7, 3)
False


Dense graphs do not support multiple or labeled edges.

sage: T = DenseGraph(nverts=3, extra_vertices=2)
sage: T.has_arc(0, 1)
True

sage: for _ in range(10): D.add_arc(5, 4)
sage: D.has_arc(5,4 )
True


Dense graphs are by their nature directed. As of this writing, you need to do operations in pairs to treat the undirected case (or use a backend or a Sage graph):

sage: T.has_arc(1, 0)
False


The curious developer is encouraged to check out the unsafe functions, which do not check input but which run in pure C.

## Underlying Data Structure¶

The class DenseGraph contains the following variables which are inherited from CGraph (for explanation, refer to the documentation there):

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


It also contains the following variables:

cdef int num_longs
cdef unsigned long *edges


The array edges is a series of bits which are turned on or off, and due to this, dense graphs only support graphs without edge labels and with no multiple edges. num_longs stores the length of the edges array. Recall that this length reflects the number of available vertices, not the number of “actual” vertices. For more details about this, refer to the documentation for CGraph.

class sage.graphs.base.dense_graph.DenseGraph

Compiled dense graphs.

sage: from sage.graphs.base.dense_graph import DenseGraph


Dense graphs are initialized as follows:

sage: D = DenseGraph(nverts=10, extra_vertices=10)


INPUT:

• nverts – non-negative integer; the number of vertices
• extra_vertices – non-negative integer (default: 10); how many extra vertices to allocate
• verts – list (default: None); optional list of vertices to add
• arcs – list (default: None); optional list of arcs to add

The first nverts are created as vertices of the graph, and the next extra_vertices can be freely added without reallocation. See top level documentation for more details. The input verts and arcs are mainly for use in pickling.

add_arc(u, v)

Add arc (u, v) to the graph.

INPUT:

• u, v – non-negative integers, must be in self

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
Traceback (most recent call last):
...
LookupError: vertex (7) is not a vertex of the graph
sage: G.has_arc(1, 0)
False
sage: G.has_arc(0, 1)
True

complement()

Replace the graph with its complement

Note

Assumes that the graph has no loop.

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.has_arc(0, 1)
True
sage: G.complement()
sage: G.has_arc(0, 1)
False

del_all_arcs(u, v)

Delete the arc from u to v.

INPUT:

• u, v – integers

Note

The naming of this function is for consistency with SparseGraph. Of course, there can be at most one arc for a DenseGraph.

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.has_arc(0, 1)
True
sage: G.del_all_arcs(0, 1)
sage: G.has_arc(0, 1)
False

has_arc(u, v)

Check whether arc (u, v) is in the graph.

INPUT:

• u, v – integers

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.has_arc(1, 0)
False
sage: G.has_arc(0, 1)
True

in_neighbors(v)

Return the list of in-neighbors of u.

INPUT:

• v – integer

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.in_neighbors(1)
[0, 3]
sage: G.in_neighbors(3)


out_neighbors(u)

Return the list of out-neighbors of u.

INPUT:

• u – integer

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: G = DenseGraph(5)
sage: G.out_neighbors(0)

sage: G.out_neighbors(1)
[2, 3]

realloc(total_verts)

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

INPUT:

• total – integer; the total size to make the array

Returns -1 and fails if reallocation would destroy any active vertices.

EXAMPLES:

sage: from sage.graphs.base.dense_graph import DenseGraph
sage: D = DenseGraph(nverts=4, extra_vertices=4)
sage: D.current_allocation()
8
6
sage: D.current_allocation()
8
10
sage: D.current_allocation()
16
Traceback (most recent call last):
...
RuntimeError: requested vertex is past twice the allocated range: use realloc
sage: D.realloc(50)
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

class sage.graphs.base.dense_graph.DenseGraphBackend

Backend for Sage graphs using DenseGraphs.

sage: from sage.graphs.base.dense_graph import DenseGraphBackend


This class is only intended for use by the Sage Graph and DiGraph class. If you are interested in using a DenseGraph, you probably want to do something like the following example, which creates a Sage Graph instance which wraps a DenseGraph object:

sage: G = Graph(30, implementation="c_graph", sparse=False)
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)]


Note that Sage graphs using the backend are more flexible than DenseGraphs themselves. This is because DenseGraphs (by design) do not deal with Python objects:

sage: G.add_vertex((0, 1, 2))
sage: sorted(list(G),
....:        key=lambda x: (isinstance(x, tuple), x))
[0,
...
29,
(0, 1, 2)]
sage: from sage.graphs.base.dense_graph import DenseGraph
sage: DG = DenseGraph(30)
sage: DG.add_vertex((0, 1, 2))
Traceback (most recent call last):
...
TypeError: an integer is required

add_edge(u, v, l, directed)

Add edge (u, v) to self.

INPUT:

• u,v – the vertices of the edge
• l – the edge label (ignored)
• directed – if False, also add (v, u)

Note

The input l is for consistency with other backends.

EXAMPLES:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_edge(0, 1, None, False)
sage: list(D.iterator_edges(range(9), True))
[(0, 1, None)]

add_edges(edges, directed)

Add edges from a list.

INPUT:

• edges – an iterable of edges to be added; each edge can either be
of the form (u, v) or (u, v, l)
• directed – if False, adds (v, u) as well as (u, v)

EXAMPLES:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_edges([(0, 1), (2, 3), (4, 5), (5, 6)], False)
sage: list(D.iterator_edges(range(9), True))
[(0, 1, None),
(2, 3, None),
(4, 5, None),
(5, 6, None)]

del_edge(u, v, l, directed)

Delete edge (u, v).

INPUT:

• u,v – the vertices of the edge
• l – the edge label (ignored)
• directed – if False, also delete (v, u, l)

Note

The input l is for consistency with other backends.

EXAMPLES:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_edges([(0, 1), (2, 3), (4, 5), (5, 6)], False)
sage: list(D.iterator_edges(range(9), True))
[(0, 1, None),
(2, 3, None),
(4, 5, None),
(5, 6, None)]
sage: D.del_edge(0, 1, None, True)
sage: list(D.iterator_out_edges(range(9), True))
[(1, 0, None),
(2, 3, None),
(3, 2, None),
(4, 5, None),
(5, 4, None),
(5, 6, None),
(6, 5, None)]

get_edge_label(u, v)

Return the edge label for (u, v).

Always None, since dense graphs do not support edge labels.

INPUT:

• u,v – the vertices of the edge

EXAMPLES:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_edges([(0, 1), (2, 3, 7), (4, 5), (5, 6)], False)
sage: list(D.iterator_edges(range(9), True))
[(0, 1, None),
(2, 3, None),
(4, 5, None),
(5, 6, None)]
sage: D.del_edge(0, 1, None, True)
sage: list(D.iterator_out_edges(range(9), True))
[(1, 0, None),
(2, 3, None),
(3, 2, None),
(4, 5, None),
(5, 4, None),
(5, 6, None),
(6, 5, None)]
sage: D.get_edge_label(2, 3)
sage: D.get_edge_label(2, 4)
Traceback (most recent call last):
...
LookupError: (2, 4) is not an edge of the graph

has_edge(u, v, l)

Check whether this graph has edge (u, v).

Note

The input l is for consistency with other backends.

INPUT:

• u,v – the vertices of the edge
• l – the edge label (ignored)

EXAMPLES:

sage: D = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: D.add_edges([(0, 1), (2, 3), (4, 5), (5, 6)], False)
sage: D.has_edge(0, 1, None)
True

iterator_edges(vertices, labels)

Return an iterator over the edges incident to a sequence of vertices.

Edges are assumed to be undirected.

INPUT:

• vertices – a list of vertex labels
• labels – boolean; whether to return edge labels as well

EXAMPLES:

sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: G.add_edge(1, 2, None, False)
sage: list(G.iterator_edges(range(9), False))
[(1, 2)]
sage: list(G.iterator_edges(range(9), True))
[(1, 2, None)]

iterator_in_edges(vertices, labels)

Return an iterator over the incoming edges incident to a sequence of vertices.

INPUT:

• vertices – a list of vertex labels
• labels – boolean; whether to return labels as well

EXAMPLES:

sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: G.add_edge(1, 2, None, True)
sage: list(G.iterator_in_edges(, False))
[]
sage: list(G.iterator_in_edges(, False))
[(1, 2)]
sage: list(G.iterator_in_edges(, True))
[(1, 2, None)]

iterator_out_edges(vertices, labels)

Return an iterator over the outbound edges incident to a sequence of vertices.

INPUT:

• vertices – a list of vertex labels
• labels – boolean; whether to return labels as well

EXAMPLES:

sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: G.add_edge(1, 2, None, True)
sage: list(G.iterator_out_edges(, False))
[]
sage: list(G.iterator_out_edges(, False))
[(1, 2)]
sage: list(G.iterator_out_edges(, True))
[(1, 2, None)]

multiple_edges(new)

Get/set whether or not self allows multiple edges.

INPUT:

• new – boolean (to set) or None (to get)

EXAMPLES:

sage: import sage.graphs.base.dense_graph
sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: G.multiple_edges(True)
Traceback (most recent call last):
...
NotImplementedError: dense graphs do not support multiple edges
sage: G.multiple_edges(None)
False

set_edge_label(u, v, l, directed)

Label the edge (u, v) by l.

INPUT:

• u,v – the vertices of the edge
• l – the edge label
• directed – if False, also set (v, u) with label l

EXAMPLES:

sage: import sage.graphs.base.dense_graph
sage: G = sage.graphs.base.dense_graph.DenseGraphBackend(9)
sage: G.set_edge_label(1, 2, 'a', True)
Traceback (most recent call last):
...
NotImplementedError: dense graphs do not support edge labels