# Bipartite graphs¶

This module implements bipartite graphs.

AUTHORS:

• Robert L. Miller (2008-01-20): initial version
• Ryan W. Hinton (2010-03-04): overrides for adding and deleting vertices and edges

TESTS:

sage: B = graphs.CompleteBipartiteGraph(7, 9)
True

sage: B = BipartiteGraph(graphs.CycleGraph(4))
sage: B == B.copy()
True
sage: type(B.copy())
<class 'sage.graphs.bipartite_graph.BipartiteGraph'>

class sage.graphs.bipartite_graph.BipartiteGraph(data=None, partition=None, check=True, *args, **kwds)

Bases: sage.graphs.graph.Graph

Bipartite graph.

INPUT:

• data – can be any of the following:
1. Empty or None (creates an empty graph).
2. An arbitrary graph.
4. A file in alist format.
5. From a NetworkX bipartite graph.

A reduced adjacency matrix contains only the non-redundant portion of the full adjacency matrix for the bipartite graph. Specifically, for zero matrices of the appropriate size, for the reduced adjacency matrix H, the full adjacency matrix is [[0, H'], [H, 0]].

The alist file format is described at http://www.inference.phy.cam.ac.uk/mackay/codes/alist.html

• partition – (default: None) a tuple defining vertices of the left and right partition of the graph. Partitions will be determined automatically if partition=None.
• check – (default: True) if True, an invalid input partition raises an exception. In the other case offending edges simply won’t be included.

Note

All remaining arguments are passed to the Graph constructor

EXAMPLES:

1. No inputs or None for the input creates an empty graph:

sage: B = BipartiteGraph()
sage: type(B)
<class 'sage.graphs.bipartite_graph.BipartiteGraph'>
sage: B.order()
0
sage: B == BipartiteGraph(None)
True


sage: B = BipartiteGraph(graphs.CycleGraph(4))
sage: B = BipartiteGraph(graphs.CycleGraph(5))
Traceback (most recent call last):
...
TypeError: Input graph is not bipartite!
sage: G = Graph({0:[5,6], 1:[4,5], 2:[4,6], 3:[4,5,6]})
sage: B = BipartiteGraph(G)
sage: B == G
True
sage: B.left
{0, 1, 2, 3}
sage: B.right
{4, 5, 6}
sage: B = BipartiteGraph({0:[5,6], 1:[4,5], 2:[4,6], 3:[4,5,6]})
sage: B == G
True
sage: B.left
{0, 1, 2, 3}
sage: B.right
{4, 5, 6}


You can specify a partition using partition argument. Note that if such graph is not bipartite, then Sage will raise an error. However, if one specifies check=False, the offending edges are simply deleted (along with those vertices not appearing in either list). We also lump creating one bipartite graph from another into this category:

  sage: P = graphs.PetersenGraph()
sage: partition = [range(5), range(5,10)]
sage: B = BipartiteGraph(P, partition)
Traceback (most recent call last):
...
TypeError: Input graph is not bipartite with respect to the given partition!

sage: B = BipartiteGraph(P, partition, check=False)
sage: B.left
{0, 1, 2, 3, 4}
sage: B.show()

::

sage: G = Graph({0:[5,6], 1:[4,5], 2:[4,6], 3:[4,5,6]})
sage: B = BipartiteGraph(G)
sage: B2 = BipartiteGraph(B)
sage: B == B2
True
sage: B3 = BipartiteGraph(G, [range(4), range(4,7)])
sage: B3
Bipartite graph on 7 vertices
sage: B3 == B2
True

::

sage: G = Graph({0:[], 1:[], 2:[]})
sage: part = (range(2), [2])
sage: B = BipartiteGraph(G, part)
sage: B2 = BipartiteGraph(B)
sage: B == B2
True

1. From a reduced adjacency matrix:

sage: M = Matrix([(1,1,1,0,0,0,0), (1,0,0,1,1,0,0),
...               (0,1,0,1,0,1,0), (1,1,0,1,0,0,1)])
sage: M
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[0 1 0 1 0 1 0]
[1 1 0 1 0 0 1]
sage: H = BipartiteGraph(M); H
Bipartite graph on 11 vertices
sage: H.edges()
[(0, 7, None),
(0, 8, None),
(0, 10, None),
(1, 7, None),
(1, 9, None),
(1, 10, None),
(2, 7, None),
(3, 8, None),
(3, 9, None),
(3, 10, None),
(4, 8, None),
(5, 9, None),
(6, 10, None)]

sage: M = Matrix([(1, 1, 2, 0, 0), (0, 2, 1, 1, 1), (0, 1, 2, 1, 1)])
sage: B = BipartiteGraph(M, multiedges=True, sparse=True)
sage: B.edges()
[(0, 5, None),
(1, 5, None),
(1, 6, None),
(1, 6, None),
(1, 7, None),
(2, 5, None),
(2, 5, None),
(2, 6, None),
(2, 7, None),
(2, 7, None),
(3, 6, None),
(3, 7, None),
(4, 6, None),
(4, 7, None)]

sage: F.<a> = GF(4)
sage: MS = MatrixSpace(F, 2, 3)
sage: M = MS.matrix([[0, 1, a+1], [a, 1, 1]])
sage: B = BipartiteGraph(M, weighted=True, sparse=True)
sage: B.edges()
[(0, 4, a), (1, 3, 1), (1, 4, 1), (2, 3, a + 1), (2, 4, 1)]
sage: B.weighted()
True

1. From an alist file:

sage: file_name = os.path.join(SAGE_TMP, 'deleteme.alist.txt')
sage: fi = open(file_name, 'w')
sage: fi.write("7 4 \n 3 4 \n 3 3 1 3 1 1 1 \n 3 3 3 4 \n\
1 2 4 \n 1 3 4 \n 1 0 0 \n 2 3 4 \n\
2 0 0 \n 3 0 0 \n 4 0 0 \n\
1 2 3 0 \n 1 4 5 0 \n 2 4 6 0 \n 1 2 4 7 \n")
sage: fi.close();
sage: B = BipartiteGraph(file_name)
sage: B == H
True

2. From a NetworkX bipartite graph:

sage: import networkx
sage: G = graphs.OctahedralGraph()
sage: N = networkx.make_clique_bipartite(G.networkx_graph())
sage: B = BipartiteGraph(N)


TESTS:

Make sure we can create a BipartiteGraph with keywords but no positional arguments (trac #10958).

sage: B = BipartiteGraph(multiedges=True)
sage: B.allows_multiple_edges()
True


Ensure that we can construct a BipartiteGraph with isolated vertices via the reduced adjacency matrix (trac #10356):

sage: a=BipartiteGraph(matrix(2,2,[1,0,1,0]))
sage: a
Bipartite graph on 4 vertices
sage: a.vertices()
[0, 1, 2, 3]
sage: g = BipartiteGraph(matrix(4,4,[1]*4+[0]*12))
sage: g.vertices()
[0, 1, 2, 3, 4, 5, 6, 7]
sage: sorted(g.left.union(g.right))
[0, 1, 2, 3, 4, 5, 6, 7]


Adds an edge from u and v.

INPUT:

• u – the tail of an edge.
• v – (default: None) the head of an edge. If v=None, then attempt to understand u as a edge tuple.
• label – (default: None) the label of the edge (u, v).

The following forms are all accepted:

This method simply checks that the edge endpoints are in different partitions. If a new vertex is to be created, it will be added to the proper partition. If both vertices are created, the first one will be added to the left partition, the second to the right partition.

TEST:

sage: bg = BipartiteGraph()
Traceback (most recent call last):
...
RuntimeError: Edge vertices must lie in different partitions.
[1, 3]
sage: bg.add_edge(5,6); 5 in bg.left; 6 in bg.right
True
True


Creates an isolated vertex. If the vertex already exists, then nothing is done.

INPUT:

• name – (default: None) name of the new vertex. If no name is specified, then the vertex will be represented by the least non-negative integer not already representing a vertex. Name must be an immutable object and cannot be None.
• left – (default: False) if True, puts the new vertex in the left partition.
• right – (default: False) if True, puts the new vertex in the right partition.

Obviously, left and right are mutually exclusive.

As it is implemented now, if a graph $$G$$ has a large number of vertices with numeric labels, then G.add_vertex() could potentially be slow, if name is None.

OUTPUT:

• If name=None, the new vertex name is returned. None otherwise.

EXAMPLES:

sage: G = BipartiteGraph()
0
1
sage: G.vertices()
[0, 1]
sage: G.left
{0}
sage: G.right
{1}


TESTS:

Exactly one of left and right must be true:

sage: G = BipartiteGraph()
Traceback (most recent call last):
...
RuntimeError: Partition must be specified (e.g. left=True).
Traceback (most recent call last):
...
RuntimeError: Only one partition may be specified.


Adding the same vertex must specify the same partition:

sage: bg = BipartiteGraph()
sage: bg.vertices()
[0]
Traceback (most recent call last):
...
RuntimeError: Cannot add duplicate vertex to other partition.


Add vertices to the bipartite graph from an iterable container of vertices. Vertices that already exist in the graph will not be added again.

INPUTS:

• vertices – sequence of vertices to add.
• left – (default: False) either True or sequence of same length as vertices with True/False elements.
• right – (default: False) either True or sequence of the same length as vertices with True/False elements.

Only one of left and right keywords should be provided. See the examples below.

EXAMPLES:

sage: bg = BipartiteGraph()
sage: bg.left
{0, 1, 2, 3, 5, 7}
sage: bg.right
{4, 6, 8, 9, 10, 11}


TEST:

sage: bg = BipartiteGraph()
Traceback (most recent call last):
...
RuntimeError: Partition must be specified (e.g. left=True).
Traceback (most recent call last):
...
RuntimeError: Only one partition may be specified.
Traceback (most recent call last):
...
RuntimeError: Cannot add duplicate vertex to other partition.
sage: (bg.left, bg.right)
({0, 1, 2}, set())

bipartition()

Returns the underlying bipartition of the bipartite graph.

EXAMPLE:

sage: B = BipartiteGraph(graphs.CycleGraph(4))
sage: B.bipartition()
({0, 2}, {1, 3})

delete_vertex(vertex, in_order=False)

Deletes vertex, removing all incident edges. Deleting a non-existent vertex will raise an exception.

INPUT:

• vertex – a vertex to delete.
• in_order – (default False) if True, this deletes the $$i$$-th vertex in the sorted list of vertices, i.e. G.vertices()[i].

EXAMPLES:

sage: B = BipartiteGraph(graphs.CycleGraph(4))
sage: B
Bipartite cycle graph: graph on 4 vertices
sage: B.delete_vertex(0)
sage: B
Bipartite cycle graph: graph on 3 vertices
sage: B.left
{2}
sage: B.edges()
[(1, 2, None), (2, 3, None)]
sage: B.delete_vertex(3)
sage: B.right
{1}
sage: B.edges()
[(1, 2, None)]
sage: B.delete_vertex(0)
Traceback (most recent call last):
...
RuntimeError: Vertex (0) not in the graph.

sage: g = Graph({'a':['b'], 'c':['b']})
sage: bg = BipartiteGraph(g)  # finds bipartition
sage: bg.vertices()
['a', 'b', 'c']
sage: bg.delete_vertex('a')
sage: bg.edges()
[('b', 'c', None)]
sage: bg.vertices()
['b', 'c']
sage: bg2 = BipartiteGraph(g)
sage: bg2.delete_vertex(0, in_order=True)
sage: bg2 == bg
True

delete_vertices(vertices)

Remove vertices from the bipartite graph taken from an iterable sequence of vertices. Deleting a non-existent vertex will raise an exception.

INPUT:

• vertices – a sequence of vertices to remove.

EXAMPLES:

sage: B = BipartiteGraph(graphs.CycleGraph(4))
sage: B
Bipartite cycle graph: graph on 4 vertices
sage: B.delete_vertices([0,3])
sage: B
Bipartite cycle graph: graph on 2 vertices
sage: B.left
{2}
sage: B.right
{1}
sage: B.edges()
[(1, 2, None)]
sage: B.delete_vertices([0])
Traceback (most recent call last):
...
RuntimeError: Vertex (0) not in the graph.


Loads into the current object the bipartite graph specified in the given file name. This file should follow David MacKay’s alist format, see http://www.inference.phy.cam.ac.uk/mackay/codes/data.html for examples and definition of the format.

EXAMPLE:

sage: file_name = os.path.join(SAGE_TMP, 'deleteme.alist.txt')
sage: fi = open(file_name, 'w')
sage: fi.write("7 4 \n 3 4 \n 3 3 1 3 1 1 1 \n 3 3 3 4 \n\
1 2 4 \n 1 3 4 \n 1 0 0 \n 2 3 4 \n\
2 0 0 \n 3 0 0 \n 4 0 0 \n\
1 2 3 0 \n 1 4 5 0 \n 2 4 6 0 \n 1 2 4 7 \n")
sage: fi.close();
sage: B = BipartiteGraph()
Bipartite graph on 11 vertices
sage: B.edges()
[(0, 7, None),
(0, 8, None),
(0, 10, None),
(1, 7, None),
(1, 9, None),
(1, 10, None),
(2, 7, None),
(3, 8, None),
(3, 9, None),
(3, 10, None),
(4, 8, None),
(5, 9, None),
(6, 10, None)]
sage: B2 = BipartiteGraph(file_name)
sage: B2 == B
True

matching_polynomial(algorithm='Godsil', name=None)

Computes the matching polynomial.

If $$p(G, k)$$ denotes the number of $$k$$-matchings (matchings with $$k$$ edges) in $$G$$, then the matching polynomial is defined as [Godsil93]:

$\mu(x)=\sum_{k \geq 0} (-1)^k p(G,k) x^{n-2k}$

INPUT:

• algorithm - a string which must be either “Godsil” (default) or “rook”; “rook” is usually faster for larger graphs.
• name - optional string for the variable name in the polynomial.

EXAMPLE:

sage: BipartiteGraph(graphs.CubeGraph(3)).matching_polynomial()
x^8 - 12*x^6 + 42*x^4 - 44*x^2 + 9

sage: x = polygen(ZZ)
sage: g = BipartiteGraph(graphs.CompleteBipartiteGraph(16, 16))
sage: factorial(16)*laguerre(16,x^2) == g.matching_polynomial(algorithm='rook')
True


Compute the matching polynomial of a line with $$60$$ vertices:

sage: from sage.functions.orthogonal_polys import chebyshev_U
sage: g = next(graphs.trees(60))
sage: chebyshev_U(60, x/2) == BipartiteGraph(g).matching_polynomial(algorithm='rook')
True


The matching polynomial of a tree graphs is equal to its characteristic polynomial:

sage: g = graphs.RandomTree(20)
sage: p = g.characteristic_polynomial()
sage: p == BipartiteGraph(g).matching_polynomial(algorithm='rook')
True


TESTS:

sage: g = BipartiteGraph(matrix.ones(4,3))
sage: g.matching_polynomial()
x^7 - 12*x^5 + 36*x^3 - 24*x
sage: g.matching_polynomial(algorithm="rook")
x^7 - 12*x^5 + 36*x^3 - 24*x

plot(*args, **kwds)

Overrides Graph’s plot function, to illustrate the bipartite nature.

EXAMPLE:

sage: B = BipartiteGraph(graphs.CycleGraph(20))
sage: B.plot()
Graphics object consisting of 41 graphics primitives

project_left()

Projects self onto left vertices. Edges are 2-paths in the original.

EXAMPLE:

sage: B = BipartiteGraph(graphs.CycleGraph(20))
sage: G = B.project_left()
sage: G.order(), G.size()
(10, 10)

project_right()

Projects self onto right vertices. Edges are 2-paths in the original.

EXAMPLE:

sage: B = BipartiteGraph(graphs.CycleGraph(20))
sage: G = B.project_right()
sage: G.order(), G.size()
(10, 10)


Return the reduced adjacency matrix for the given graph.

A reduced adjacency matrix contains only the non-redundant portion of the full adjacency matrix for the bipartite graph. Specifically, for zero matrices of the appropriate size, for the reduced adjacency matrix H, the full adjacency matrix is [[0, H'], [H, 0]].

This method supports the named argument ‘sparse’ which defaults to True. When enabled, the returned matrix will be sparse.

EXAMPLES:

Bipartite graphs that are not weighted will return a matrix over ZZ:

sage: M = Matrix([(1,1,1,0,0,0,0), (1,0,0,1,1,0,0),
...               (0,1,0,1,0,1,0), (1,1,0,1,0,0,1)])
sage: B = BipartiteGraph(M)
sage: N
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[0 1 0 1 0 1 0]
[1 1 0 1 0 0 1]
sage: N == M
True
sage: N[0,0].parent()
Integer Ring


Multi-edge graphs also return a matrix over ZZ:

sage: M = Matrix([(1,1,2,0,0), (0,2,1,1,1), (0,1,2,1,1)])
sage: B = BipartiteGraph(M, multiedges=True, sparse=True)
sage: N == M
True
sage: N[0,0].parent()
Integer Ring


Weighted graphs will return a matrix over the ring given by their (first) weights:

sage: F.<a> = GF(4)
sage: MS = MatrixSpace(F, 2, 3)
sage: M = MS.matrix([[0, 1, a+1], [a, 1, 1]])
sage: B = BipartiteGraph(M, weighted=True, sparse=True)
sage: N == M
True
sage: N[0,0].parent()
Finite Field in a of size 2^2


TESTS:

sage: B = BipartiteGraph()
[]
sage: M = Matrix([[0,0], [0,0]])
True
sage: M = Matrix([[10,2/3], [0,0]])
sage: B = BipartiteGraph(M, weighted=True, sparse=True)
True

save_afile(fname)

Save the graph to file in alist format.

Saves this graph to file in David MacKay’s alist format, see http://www.inference.phy.cam.ac.uk/mackay/codes/data.html for examples and definition of the format.

EXAMPLE:

sage: M = Matrix([(1,1,1,0,0,0,0), (1,0,0,1,1,0,0),
...               (0,1,0,1,0,1,0), (1,1,0,1,0,0,1)])
sage: M
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[0 1 0 1 0 1 0]
[1 1 0 1 0 0 1]
sage: b = BipartiteGraph(M)
sage: file_name = os.path.join(SAGE_TMP, 'deleteme.alist.txt')
sage: b.save_afile(file_name)
sage: b2 = BipartiteGraph(file_name)
sage: b == b2
True


TESTS:

sage: file_name = os.path.join(SAGE_TMP, 'deleteme.alist.txt')
sage: for order in range(3, 13, 3):
....:     num_chks = int(order / 3)
....:     num_vars = order - num_chks
....:     partition = (range(num_vars), range(num_vars, num_vars+num_chks))
....:     for idx in range(100):
....:         g = graphs.RandomGNP(order, 0.5)
....:         try:
....:             b = BipartiteGraph(g, partition, check=False)
....:             b.save_afile(file_name)
....:             b2 = BipartiteGraph(file_name)
....:             if b != b2:
....:                 print "Load/save failed for code with edges:"
....:                 print b.edges()
....:                 break
....:         except Exception:
....:             print "Exception encountered for graph of order "+ str(order)
....:             print "with edges: "
....:             g.edges()
....:             raise

to_undirected()

Return an undirected Graph (without bipartite constraint) of the given object.

EXAMPLES:

sage: BipartiteGraph(graphs.CycleGraph(6)).to_undirected()
Cycle graph: Graph on 6 vertices


Directed graphs

Common Graphs