Bipartite graphs¶
This module implements bipartite graphs.
AUTHORS:
 Robert L. Miller (20080120): initial version
 Ryan W. Hinton (20100304): overrides for adding and deleting vertices and edges
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:Empty or
None
(creates an empty graph).An arbitrary graph.
A reduced adjacency matrix.
A reduced adjacency matrix contains only the nonredundant 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 columns correspond to vertices on the left, and the rows correspond to vertices on the right.A file in alist format.
The alist file format is described at http://www.inference.phy.cam.ac.uk/mackay/codes/alist.html
From a NetworkX bipartite graph.
partition
– (default:None
); a tuple defining vertices of the left and right partition of the graph. Partitions will be determined automatically ifpartition
isNone
.check
– boolean (default:True
); ifTrue
, 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
constructorEXAMPLES:
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
From a graph: without any more information, finds a bipartition:
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}
If a
Graph
orDiGraph
is used as data, you can specify a partition usingpartition
argument. Note that if such graph is not bipartite, then Sage will raise an error. However, if one specifiescheck=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 = [list(range(5)), list(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, [list(range(4)), list(range(4, 7))]) sage: B3 Bipartite graph on 7 vertices sage: B3 == B2 True
sage: G = Graph({0: [], 1: [], 2: []}) sage: part = (list(range(2)), [2]) sage: B = BipartiteGraph(G, part) sage: B2 = BipartiteGraph(B) sage: B == B2 True
sage: d = DiGraph(6) sage: d.add_edge(0, 1) sage: part=[[1, 2, 3], [0, 4, 5]] sage: b = BipartiteGraph(d, part) sage: b.left {1, 2, 3} sage: b.right {0, 4, 5}
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
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.is_isomorphic(H) True
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)

add_edge
(u, v=None, label=None)¶ Add an edge from \(u\) to \(v\).
INPUT:
u
– the tail of an edge.v
– (default:None
); the head of an edge. Ifv=None
, then attempt to understandu
as a edge tuple.label
– (default:None
); the label of the edge(u, v)
.
The following forms are all accepted:
G.add_edge(1, 2)
G.add_edge((1, 2))
G.add_edges([(1, 2)])
G.add_edge(1, 2, 'label')
G.add_edge((1, 2, 'label'))
G.add_edges([(1, 2, 'label')])
See
add_edge()
for more detail.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.

add_vertex
(name=None, left=False, right=False)¶ Create 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 nonnegative integer not already representing a vertex. Name must be an immutable object and cannot beNone
.left
– boolean (default:False
); ifTrue
, puts the new vertex in the left partition.right
– boolean (default:False
); ifTrue
, puts the new vertex in the right partition.
Obviously,
left
andright
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 isNone
.OUTPUT:
 If
name
isNone
, the new vertex name is returned.None
otherwise.
EXAMPLES:
sage: G = BipartiteGraph() sage: G.add_vertex(left=True) 0 sage: G.add_vertex(right=True) 1 sage: G.vertices() [0, 1] sage: G.left {0} sage: G.right {1}

add_vertices
(vertices, left=False, right=False)¶ Add vertices to the bipartite graph from an iterable container of vertices.
Vertices that already exist in the graph will not be added again.
INPUT:
vertices
– sequence of vertices to add.left
– (default:False
); eitherTrue
or sequence of same length asvertices
withTrue
/False
elements.right
– (default:False
); eitherTrue
or sequence of the same length asvertices
withTrue
/False
elements.
Only one of
left
andright
keywords should be provided. See the examples below.EXAMPLES:
sage: bg = BipartiteGraph() sage: bg.add_vertices([0, 1, 2], left=True) sage: bg.add_vertices([3, 4, 5], left=[True, False, True]) sage: bg.add_vertices([6, 7, 8], right=[True, False, True]) sage: bg.add_vertices([9, 10, 11], right=True) sage: bg.left {0, 1, 2, 3, 5, 7} sage: bg.right {4, 6, 8, 9, 10, 11}

allow_loops
(new, check=True)¶ Change whether loops are permitted in the (di)graph
Note
This method overwrite the
allow_loops()
method to ensure that loops are forbidden inBipartiteGraph
.INPUT:
new
– boolean
EXAMPLES:
sage: B = BipartiteGraph() sage: B.allow_loops(True) Traceback (most recent call last): ... ValueError: loops are not allowed in bipartite graphs

bipartition
()¶ Return the underlying bipartition of the bipartite graph.
EXAMPLES:
sage: B = BipartiteGraph(graphs.CycleGraph(4)) sage: B.bipartition() ({0, 2}, {1, 3})

complement
()¶ Return a complement of this graph.
EXAMPLES:
sage: B = BipartiteGraph({1: [2, 4], 3: [4, 5]}) sage: G = B.complement(); G Graph on 5 vertices sage: G.edges(labels=False) [(1, 3), (1, 5), (2, 3), (2, 4), (2, 5), (4, 5)]

delete_vertex
(vertex, in_order=False)¶ Delete vertex, removing all incident edges.
Deleting a nonexistent vertex will raise an exception.
INPUT:
vertex
– a vertex to delete.in_order
– boolean (defaultFalse
); ifTrue
, 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): ... ValueError: 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 nonexistent 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): ... ValueError: vertex (0) not in the graph

load_afile
(fname)¶ Load 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.
EXAMPLES:
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() sage: B.load_afile(file_name) 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
(value_only=False, algorithm=None, use_edge_labels=False, solver=None, verbose=0)¶ Return a maximum matching of the graph represented by the list of its edges.
Given a graph \(G\) such that each edge \(e\) has a weight \(w_e\), a maximum matching is a subset \(S\) of the edges of \(G\) of maximum weight such that no two edges of \(S\) are incident with each other.
INPUT:
value_only
– boolean (default:False
); when set toTrue
, only the cardinal (or the weight) of the matching is returnedalgorithm
– string (default:"HopcroftKarp"
ifuse_edge_labels==False
, otherwise"Edmonds"
); algorithm to use among:"HopcroftKarp"
selects the default bipartite graph algorithm as implemented in NetworkX"Eppstein"
selects Eppstein’s algorithm as implemented in NetworkX"Edmonds"
selects Edmonds’ algorithm as implemented in NetworkX"LP"
uses a Linear Program formulation of the matching problem
use_edge_labels
– boolean (default:False
) when set to
True
, computes a weighted matching where each edge is weighted by its label (if an edge has no label, \(1\) is assumed); only ifalgorithm
is"Edmonds"
,"LP"
 when set to
False
, each edge has weight \(1\)
 when set to
solver
– (default:None
) a specific Linear Program (LP) solver to be usedverbose
– integer (default:0
); sets the level of verbosity: set to 0 by default, which means quiet
EXAMPLES:
Maximum matching in a cycle graph:
sage: G = BipartiteGraph(graphs.CycleGraph(10)) sage: G.matching() [(0, 1, None), (2, 3, None), (4, 5, None), (6, 7, None), (8, 9, None)]
The size of a maximum matching in a complete bipartite graph using Eppstein:
sage: G = BipartiteGraph(graphs.CompleteBipartiteGraph(4,5)) sage: G.matching(algorithm="Eppstein", value_only=True) 4

matching_polynomial
(algorithm='Godsil', name=None)¶ Compute the matching polynomial.
The matching polynomial is defined as in [Godsil93], where \(p(G, k)\) denotes the number of \(k\)matchings (matchings with \(k\) edges) in \(G\) :
\[\mu(x)=\sum_{k \geq 0} (1)^k p(G,k) x^{n2k}\]INPUT:
algorithm
– string (default:"Godsil"
); either “Godsil” or “rook”; “rook” is usually faster for larger graphsname
– string (default:None
); name of the variable in the polynomial, set to \(x\) whenname
isNone
EXAMPLES:
sage: BipartiteGraph(graphs.CubeGraph(3)).matching_polynomial() x^8  12*x^6 + 42*x^4  44*x^2 + 9
sage: x = polygen(QQ) sage: g = BipartiteGraph(graphs.CompleteBipartiteGraph(16, 16)) sage: bool(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 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

plot
(*args, **kwds)¶ Override Graph’s plot function, to illustrate the bipartite nature.
EXAMPLES:
sage: B = BipartiteGraph(graphs.CycleGraph(20)) sage: B.plot() Graphics object consisting of 41 graphics primitives

project_left
()¶ Project
self
onto left vertices. Edges are 2paths in the original.EXAMPLES:
sage: B = BipartiteGraph(graphs.CycleGraph(20)) sage: G = B.project_left() sage: G.order(), G.size() (10, 10)

project_right
()¶ Project
self
onto right vertices. Edges are 2paths in the original.EXAMPLES:
sage: B = BipartiteGraph(graphs.CycleGraph(20)) sage: G = B.project_right() sage: G.order(), G.size() (10, 10)

reduced_adjacency_matrix
(sparse=True)¶ Return the reduced adjacency matrix for the given graph.
A reduced adjacency matrix contains only the nonredundant 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]]
.INPUT:
sparse
– boolean (default:True
); whether to return a sparse matrix
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 = B.reduced_adjacency_matrix() 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
Multiedge 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 = B.reduced_adjacency_matrix() 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 = B.reduced_adjacency_matrix(sparse=False) sage: N == M True sage: N[0,0].parent() Finite Field in a of size 2^2

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.
EXAMPLES:
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.is_isomorphic(b2) True

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

vertex_cover
(algorithm='Konig', value_only=False, reduction_rules=True, solver=None, verbosity=0)¶ Return a minimum vertex cover of self represented by a set of vertices.
A minimum vertex cover of a graph is a set \(S\) of vertices such that each edge is incident to at least one element of \(S\), and such that \(S\) is of minimum cardinality. For more information, see Wikipedia article Vertex_cover.
Equivalently, a vertex cover is defined as the complement of an independent set.
As an optimization problem, it can be expressed as follows:
\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall (u,v) \in G.edges(), b_u+b_v\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]INPUT:
algorithm
– string (default:"Konig"
); algorithm to use among:"Konig"
will compute a minimum vertex cover using Konig’s algorithm (Wikipedia article Kőnig’s_theorem_(graph_theory))"Cliquer"
will compute a minimum vertex cover using the Cliquer package"MILP"
will compute a minimum vertex cover through a mixed integer linear program"mcqd"
will use the MCQD solver (http://www.sicmm.org/~konc/maxclique/), and the MCQD package must be installed
value_only
– boolean (default:False
); if set toTrue
, only the size of a minimum vertex cover is returned. Otherwise, a minimum vertex cover is returned as a list of vertices.reduction_rules
– (default:True
); specify if the reductions rules from kernelization must be applied as preprocessing or not. See [ACFLSS04] for more details. Note that depending on the instance, it might be faster to disable reduction rules. This parameter is currently ignored whenalgorithm == "Konig"
.solver
– (default:None
); specify a Linear Program (LP) solver to be used. If set toNone
, the default one is used. For more information on LP solvers and which default solver is used, see the methodsage.numerical.mip.MixedIntegerLinearProgram.solve()
of the classsage.numerical.mip.MixedIntegerLinearProgram
.verbosity
– nonnegative integer (default:0
); set the level of verbosity you want from the linear program solver. Since the problem of computing a vertex cover is \(NP\)complete, its solving may take some time depending on the graph. A value of 0 means that there will be no message printed by the solver. This option is only useful ifalgorithm="MILP"
.
EXAMPLES:
On the Cycle Graph:
sage: B = BipartiteGraph(graphs.CycleGraph(6)) sage: len(B.vertex_cover()) 3 sage: B.vertex_cover(value_only=True) 3
The two algorithms should return the same result:
sage: g = BipartiteGraph(graphs.RandomBipartite(10, 10, .5)) sage: vc1 = g.vertex_cover(algorithm="Konig") sage: vc2 = g.vertex_cover(algorithm="Cliquer") sage: len(vc1) == len(vc2) True