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
Enjeck M. Cleopatra (2022): fixes incorrect partite sets and adds graph creation from graph6 string
- class sage.graphs.bipartite_graph.BipartiteGraph(data=None, partition=None, check=True, hash_labels=None, *args, **kwds)#
Bases:
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 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 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
A
graph6
string (see documentation ofgraph6_string()
).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.loops
– ignored; bipartite graphs cannot have loopsmultiedges
– boolean (default:None
); whether to allow multiple edgesweighted
– boolean (default:None
); whether graph thinks of itself as weighted or not. Seeself.weighted()
hash_labels
– boolean (default:None
); whether to include edge labels during hashing. This parameter defaults toTrue
if the graph is weighted. This parameter is ignored if the graph is mutable. Beware that trying to hash unhashable labels will raise an error.
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): ... ValueError: 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() # needs sage.plot
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: # needs sage.modules 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(sort=True) [(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)]) # needs sage.modules sage: B = BipartiteGraph(M, multiedges=True, sparse=True) # needs sage.modules sage: B.edges(sort=True) # needs sage.modules [(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: # needs sage.modules sage.rings.finite_rings 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(sort=True) [(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: import tempfile sage: with tempfile.NamedTemporaryFile(mode="w+t") as f: ....: _ = f.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") ....: f.flush() ....: B = BipartiteGraph(f.name) sage: B.is_isomorphic(H) # needs sage.modules True
From a
graph6
string:sage: B = BipartiteGraph('Bo') sage: B Bipartite graph on 3 vertices sage: B.left {0} sage: B.right {1, 2}
sage: B = BipartiteGraph('F?^T_\n', format='graph6') sage: B.vertices(sort=True) [0, 1, 2, 3, 4, 5, 6] sage: B.edges(sort=True) [(0, 5, None), (0, 6, None), (1, 4, None), (1, 5, None), (2, 4, None), (2, 6, None), (3, 4, None), (3, 5, None), (3, 6, None)] sage: B.left {0, 1, 2, 3} sage: B.right {4, 5, 6}
- ::
sage: B = BipartiteGraph(‘Bo’, partition=[[0], [1, 2]]) sage: B.left {0} sage: B.right {1, 2}
sage: B = BipartiteGraph('F?^T_\n', partition=[[0, 1, 2], [3, 4, 5, 6]]) Traceback (most recent call last): ... TypeError: input graph is not bipartite with respect to the given partition sage: B = BipartiteGraph('F?^T_\n', partition=[[0, 1, 2], [3, 4, 5, 6]], check=False) sage: B.left {0, 1, 2} sage: B.show() # needs sage.plot
From a NetworkX bipartite graph:
sage: # needs networkx 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. If both vertices are in the same partition but different connected components, one of the components will be “flipped”, i.e. each vertex will be put into whichever partition it’s not currently in. This will allow for the graph to remain bipartite, without changing the edges or vertices.
- add_edges(edges, loops=True)#
Add edges from an iterable container.
INPUT:
edges
– an iterable of edges, given either as(u, v)
or(u, v, label)
.loops
– ignored
See
add_edges()
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. If both vertices are in the same partition but different connected components, one of the components will be “flipped”, i.e. each vertex will be put into whichever partition it’s not currently in. This will allow for the graph to remain bipartite, without changing the edges or vertices.
EXAMPLES:
sage: bg = BipartiteGraph() sage: bg.add_vertices([0, 1, 2], left=[True, False, True]) sage: bg.add_edges([(0, 1), (2, 1)]) sage: bg.add_edges([[0, 2]]) Traceback (most recent call last): ... ValueError: the specified set of edges cannot be added while still preserving the bipartition property sage: G = BipartiteGraph() sage: G.add_edges([(0, 1), (3, 2), (1, 2)]) sage: G.bipartition() ({0, 2}, {1, 3})
Loops will raise an error:
sage: bg.add_edges([[0, 3], [3, 3]]) Traceback (most recent call last): ... ValueError: the specified set of edges cannot be added while still preserving the bipartition property
Adding edges is fine as long as there exists a valid bipartition. Otherwise an error is raised without modifyiong the graph:
sage: G = BipartiteGraph() sage: G.add_edges([(0, 1), (2, 3)]) sage: G.bipartition() ({0, 2}, {1, 3}) sage: G.add_edges([(0,2), (0,3)]) Traceback (most recent call last): ... ValueError: the specified set of edges cannot be added while still preserving the bipartition property sage: G.bipartition() ({0, 2}, {1, 3}) sage: G.edges(labels=False, sort=True) [(0, 1), (2, 3)]
- 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 non-negative 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(sort=True) [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})
- canonical_label(partition=None, certificate=False, edge_labels=False, algorithm=None, return_graph=True)#
Return the canonical graph.
A canonical graph is the representative graph of an isomorphism class by some canonization function \(c\). If \(G\) and \(H\) are graphs, then \(G \cong c(G)\), and \(c(G) == c(H)\) if and only if \(G \cong H\).
See the Wikipedia article Graph_canonization for more information.
INPUT:
partition
– if given, the canonical label with respect to this set partition will be computed. The default is the unit set partition.certificate
– boolean (default:False
). When set toTrue
, a dictionary mapping from the vertices of the (di)graph to its canonical label will also be returned.edge_labels
– boolean (default:False
). When set toTrue
, allows only permutations respecting edge labels.algorithm
– a string (default:None
). The algorithm to use; currently available:'bliss'
: use the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html);'sage'
: always use Sage’s implementation.None
(default): use bliss when available and possibleNote
Make sure you always compare canonical forms obtained by the same algorithm.
return_graph
– boolean (default:True
). When set toFalse
, returns the list of edges of the canonical graph instead of the canonical graph; only available when'bliss'
is explicitly set as algorithm.
EXAMPLES:
sage: B = BipartiteGraph( [(0, 4), (0, 5), (0, 6), (0, 8), (1, 5), ....: (1, 7), (1, 8), (2, 6), (2, 7), (2, 8), ....: (3, 4), (3, 7), (3, 8), (4, 9), (5, 9), ....: (6, 9), (7, 9)] ) sage: C = B.canonical_label(partition=(B.left,B.right), algorithm='sage') sage: C Bipartite graph on 10 vertices sage: C.left {0, 1, 2, 3, 4} sage: C.right {5, 6, 7, 8, 9}
sage: B = BipartiteGraph( [(0, 4), (0, 5), (0, 6), (0, 8), (1, 5), ....: (1, 7), (1, 8), (2, 6), (2, 7), (2, 8), ....: (3, 4), (3, 7), (3, 8), (4, 9), (5, 9), ....: (6, 9), (7, 9)] ) sage: C, cert = B.canonical_label(partition=(B.left,B.right), certificate=True, algorithm='sage') sage: C Bipartite graph on 10 vertices sage: C.left {0, 1, 2, 3, 4} sage: C.right {5, 6, 7, 8, 9} sage: cert == {0: 3, 1: 0, 2: 1, 3: 2, 4: 5, 5: 7, 6: 6, 7: 8, 8: 9, 9: 4} True
sage: G = Graph({0: [5, 6], 1: [4, 5], 2: [4, 6], 3: [4, 5, 6]}) sage: B = BipartiteGraph(G) sage: C = B.canonical_label(partition=(B.left,B.right), edge_labels=True, algorithm='sage') sage: C.left {0, 1, 2, 3} sage: C.right {4, 5, 6}
See also
- complement()#
Return a complement of this graph.
Given a simple
BipartiteGraph
\(G = (L, R, E)\) with vertex set \(L\cup R\) and edge set \(E\), this method returns aGraph
\(H = (V, F)\), where \(V = L\cup R\) and \(F\) is the set of edges of a complete graph of order \(|V|\) minus the edges in \(E\).Warning
This method returns the complement of a bipartite graph \(G = (V = L \cup R, E)\) with respect the complete graph of order \(|V|\). If looking for the complement with respect the complete bipartite graph \(K = (L, R, L\times R)\), use method
complement_bipartite()
.See also
EXAMPLES:
sage: B = BipartiteGraph({1: [2, 4], 3: [4, 5]}) sage: G = B.complement(); G Graph on 5 vertices sage: G.edges(sort=True, labels=False) [(1, 3), (1, 5), (2, 3), (2, 4), (2, 5), (4, 5)] sage: B.size() + G.size() == graphs.CompleteGraph(B.order()).size() True
- complement_bipartite()#
Return the bipartite complement of this bipartite graph.
Given a simple
BipartiteGraph
\(G = (L, R, E)\) with vertex set \(L\cup R\) and edge set \(E\), this method returns aBipartiteGraph
\(H = (L\cup R, F)\), where \(F\) is the set of edges of a complete bipartite graph between vertex sets \(L\) and \(R\) minus the edges in \(E\).See also
EXAMPLES:
sage: B = BipartiteGraph({0: [1, 2, 3]}) sage: C = B.complement_bipartite() sage: C Bipartite graph on 4 vertices sage: C.is_bipartite() True sage: B.left == C.left and B.right == C.right True sage: C.size() == len(B.left)*len(B.right) - B.size() True sage: G = B.complement() sage: G.is_bipartite() False
- delete_vertex(vertex, in_order=False)#
Delete vertex, removing all incident edges.
Deleting a non-existent 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(sort=True)[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(sort=True) [(1, 2, None), (2, 3, None)] sage: B.delete_vertex(3) sage: B.right {1} sage: B.edges(sort=True) [(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(sort=True) ['a', 'b', 'c'] sage: bg.delete_vertex('a') sage: bg.edges(sort=True) [('b', 'c', None)] sage: bg.vertices(sort=True) ['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(sort=True) [(1, 2, None)] sage: B.delete_vertices([0]) Traceback (most recent call last): ... ValueError: vertex (0) not in the graph
- is_bipartite(certificate=False)#
Check whether the graph is bipartite.
This method always returns
True
as first value, plus a certificate whencertificate == True
.INPUT:
certificate
– boolean (default:False
); whether to return a certificate. If set toTrue
, the certificate returned is a proper 2-coloring of the vertices.
See also
EXAMPLES:
sage: g = BipartiteGraph(graphs.RandomBipartite(3, 3, .5)) # needs numpy sage: g.is_bipartite() # needs numpy True sage: g.is_bipartite(certificate=True) # random # needs numpy (True, {(0, 0): 0, (0, 1): 0, (0, 2): 0, (1, 0): 1, (1, 1): 1, (1, 2): 1})
- 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: import tempfile sage: with tempfile.NamedTemporaryFile(mode="w+t") as f: ....: _ = f.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") ....: f.flush() ....: B = BipartiteGraph() ....: B2 = BipartiteGraph(f.name) ....: B.load_afile(f.name) Bipartite graph on 11 vertices sage: B.edges(sort=True) [(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 == B True
- matching(value_only, algorithm=False, use_edge_labels=None, solver=False, verbose=None, integrality_tolerance=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:"Hopcroft-Karp"
ifuse_edge_labels==False
, otherwise"Edmonds"
); algorithm to use among:"Hopcroft-Karp"
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 \(1\)
solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
Maximum matching in a cycle graph:
sage: G = BipartiteGraph(graphs.CycleGraph(10)) sage: G.matching() # needs networkx [(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) # needs networkx 4
- matching_polynomial(algorithm='Godsil', name=None)#
Compute the matching polynomial.
The matching polynomial is defined as in [God1993], where \(p(G, k)\) denotes the number of \(k\)-matchings (matchings with \(k\) edges) in \(G\) :
\[\mu(x)=\sum_{k \geq 0} (-1)^k p(G,k) x^{n-2k}\]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() # needs sage.libs.flint 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) # needs sage.symbolic ....: == 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 # needs sage.symbolic sage: g = next(graphs.trees(60)) sage: (chebyshev_U(60, x/2) # needs sage.symbolic ....: == 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() # needs sage.modules sage: p == BipartiteGraph(g).matching_polynomial(algorithm='rook') # needs sage.modules True
- perfect_matchings(labels=False)#
Return an iterator over all perfect matchings of the bipartite graph.
ALGORITHM:
Choose a vertex \(v\) in the right set of vertices, then recurse through all edges incident to \(v\), removing one edge at a time whenever an edge is added to a matching.
INPUT:
labels
– boolean (default:False
); whenTrue
, the edges in each perfect matching are triples (containing the label as the third element), otherwise the edges are pairs.
See also
EXAMPLES:
sage: B = BipartiteGraph({0: [5, 7], 1: [4, 6, 7], 2: [4, 5, 8], 3: [4, 5, 6], 6: [9], 8: [9]}) sage: len(list(B.perfect_matchings())) 6 sage: G = Graph(B.edges(sort=False)) sage: len(list(G.perfect_matchings())) 6
The algorithm ensures that for any edge of a perfect matching, the first vertex is on the left set of vertices and the second vertex in the right set:
sage: B = BipartiteGraph({0: [5, 7], 1: [4, 6, 7], 2: [4, 5, 8], 3: [4, 5, 6], 6: [9], 8: [9]}) sage: m = next(B.perfect_matchings(labels=False)) sage: B.left {0, 1, 2, 3, 9} sage: B.right {4, 5, 6, 7, 8} sage: sorted(m) [(0, 7), (1, 4), (2, 5), (3, 6), (9, 8)] sage: all((u in B.left and v in B.right) for u, v in m) True
Multiple edges are taken into account:
sage: B = BipartiteGraph({0: [5, 7], 1: [4, 6, 7], 2: [4, 5, 8], 3: [4, 5, 6], 6: [9], 8: [9]}) sage: B.allow_multiple_edges(True) sage: B.add_edge(0, 7) sage: len(list(B.perfect_matchings())) 10
Empty graph:
sage: list(BipartiteGraph().perfect_matchings()) [[]]
Bipartite graph without perfect matching:
sage: B = BipartiteGraph(graphs.CompleteBipartiteGraph(3, 4)) sage: list(B.perfect_matchings()) []
Check that the number of perfect matchings of a complete bipartite graph is consistent with the matching polynomial:
sage: B = BipartiteGraph(graphs.CompleteBipartiteGraph(4, 4)) sage: len(list(B.perfect_matchings())) 24 sage: B.matching_polynomial(algorithm='rook')(0) # needs sage.modules 24
- plot(*args, **kwds)#
Override Graph’s plot function, to illustrate the bipartite nature.
EXAMPLES:
sage: B = BipartiteGraph(graphs.CycleGraph(20)) sage: B.plot() # needs sage.plot Graphics object consisting of 41 graphics primitives
- project_left()#
Project
self
onto left vertices. Edges are 2-paths 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 2-paths 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, base_ring=True, **kwds)#
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]]
.By default, the matrix returned is over the integers.
INPUT:
sparse
– boolean (default:True
); whether to return a sparse matrixbase_ring
– a ring (default:None
); the base ring of the matrix space to use. By default, the base ring isZZ
if the graph is not weighted and otherwise the same ring as the (first) weights.**kwds
– other keywords to pass tomatrix()
EXAMPLES:
Bipartite graphs that are not weighted will return a matrix over ZZ, unless a base ring is specified:
sage: # needs sage.modules 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(); 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 sage: N2 = B.reduced_adjacency_matrix(base_ring=RDF); N2 [1.0 1.0 1.0 0.0 0.0 0.0 0.0] [1.0 0.0 0.0 1.0 1.0 0.0 0.0] [0.0 1.0 0.0 1.0 0.0 1.0 0.0] [1.0 1.0 0.0 1.0 0.0 0.0 1.0] sage: N2[0, 0].parent() Real Double Field
Multi-edge graphs also return a matrix over ZZ, unless a base ring is specified:
sage: # needs sage.modules 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 sage: N2 = B.reduced_adjacency_matrix(base_ring=RDF) sage: N2[0, 0].parent() Real Double Field
Weighted graphs will return a matrix over the ring given by their (first) weights, unless a base ring is specified:
sage: # needs sage.modules sage.rings.finite_rings 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 sage: N2 = B.reduced_adjacency_matrix(base_ring=F) sage: N2[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: # needs sage.modules 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: import tempfile sage: with tempfile.NamedTemporaryFile() as f: ....: b.save_afile(f.name) ....: b2 = BipartiteGraph(f.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, value_only='Konig', reduction_rules=False, solver=True, verbose=None, integrality_tolerance=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:
\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall (u,v) \in G.edges(sort=True), b_u+b_v\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]INPUT:
algorithm
– string (default:"Konig"
); algorithm to use among:"Konig"
will compute a minimum vertex cover using Konig’s algorithm (Wikipedia article Kőnig%27s_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 pre-processing 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
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
On the Cycle Graph:
sage: B = BipartiteGraph(graphs.CycleGraph(6)) sage: len(B.vertex_cover()) # needs networkx 3 sage: B.vertex_cover(value_only=True) # needs networkx 3
The two algorithms should return the same result:
sage: # needs networkx numpy 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