Bipartite¶

This module provides functions and operations for bipartite graphs. Bipartite graphs have two node sets and edges in that only connect nodes from opposite sets. It is common in the literature to use an spatial analogy referring to the two node sets as top and bottom nodes.

The bipartite algorithms are not imported into the networkx namespace at the top level so the easiest way to use them is with:

>>> import networkx as nx
>>> from networkx.algorithms import bipartite

NetworkX does not have a custom bipartite graph class but the Graph() or DiGraph() classes can be used to represent bipartite graphs. However, you have to keep track of which set each node belongs to, and make sure that there is no edge between nodes of the same set. The convention used in NetworkX is to use a node attribute named “bipartite” with values 0 or 1 to identify the sets each node belongs to.

For example:

>>> B = nx.Graph()
>>> B.add_nodes_from([1,2,3,4], bipartite=0) # Add the node attribute "bipartite"
>>> B.add_nodes_from(['a','b','c'], bipartite=1)
>>> B.add_edges_from([(1,'a'), (1,'b'), (2,'b'), (2,'c'), (3,'c'), (4,'a')])

Many algorithms of the bipartite module of NetworkX require, as an argument, a container with all the nodes that belong to one set, in addition to the bipartite graph . If is connected, you can find the node sets using a two-coloring algorithm:

>>> nx.is_connected(B)
True
>>> bottom_nodes, top_nodes = bipartite.sets(B)

list(top_nodes) [1, 2, 3, 4] list(bottom_nodes) [‘a’, ‘c’, ‘b’]

However, if the input graph is not connected, there are more than one possible colorations. Thus, the following result is correct:

>>> B.remove_edge(2,'c')
>>> nx.is_connected(B)
False
>>> bottom_nodes, top_nodes = bipartite.sets(B)

list(top_nodes) [1, 2, 4, ‘c’] list(bottom_nodes) [‘a’, 3, ‘b’]

Using the “bipartite” node attribute, you can easily get the two node sets:

>>> top_nodes = set(n for n,d in B.nodes(data=True) if d['bipartite']==0)
>>> bottom_nodes = set(B) - top_nodes

list(top_nodes) [1, 2, 3, 4] list(bottom_nodes) [‘a’, ‘c’, ‘b’]

So you can easily use the bipartite algorithms that require, as an argument, a container with all nodes that belong to one node set:

>>> print(round(bipartite.density(B, bottom_nodes),2))
0.42
>>> G = bipartite.projected_graph(B, top_nodes)
>>> G.edges()
[(1, 2), (1, 4)]

All bipartite graph generators in NetworkX build bipartite graphs with the “bipartite” node attribute. Thus, you can use the same approach:

>>> RB = bipartite.random_graph(5, 7, 0.2)
>>> RB_top = set(n for n,d in RB.nodes(data=True) if d['bipartite']==0)
>>> RB_bottom = set(RB) - RB_top
>>> list(RB_top)
[0, 1, 2, 3, 4]
>>> list(RB_bottom)
[5, 6, 7, 8, 9, 10, 11]

For other bipartite graph generators see the bipartite section of Graph generators.

Matching¶

Provides functions for computing a maximum cardinality matching in a bipartite graph.

If you don’t care about the particular implementation of the maximum matching algorithm, simply use the maximum_matching(). If you do care, you can import one of the named maximum matching algorithms directly.

For example, to find a maximum matching in the complete bipartite graph with two vertices on the left and three vertices on the right:

>>> import networkx as nx
>>> G = nx.complete_bipartite_graph(2, 3)
>>> left, right = nx.bipartite.sets(G)
>>> list(left)
[0, 1]
>>> list(right)
[2, 3, 4]
>>> nx.bipartite.maximum_matching(G)
{0: 2, 1: 3, 2: 0, 3: 1}

The dictionary returned by maximum_matching() includes a mapping for vertices in both the left and right vertex sets.

eppstein_matching(G)	Returns the maximum cardinality matching of the bipartite graph .
hopcroft_karp_matching(G)	Returns the maximum cardinality matching of the bipartite graph .
to_vertex_cover(G, matching)	Returns the minimum vertex cover corresponding to the given maximum matching of the bipartite graph .

Generators¶

Generators and functions for bipartite graphs.

complete_bipartite_graph(n1, n2[, create_using])	Return the complete bipartite graph .
configuration_model(aseq, bseq[, ...])	Return a random bipartite graph from two given degree sequences.
havel_hakimi_graph(aseq, bseq[, create_using])	Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction.
reverse_havel_hakimi_graph(aseq, bseq[, ...])	Return a bipartite graph from two given degree sequences using a Havel-Hakimi style construction.
alternating_havel_hakimi_graph(aseq, bseq[, ...])	Return a bipartite graph from two given degree sequences using an alternating Havel-Hakimi style construction.
preferential_attachment_graph(aseq, p[, ...])	Create a bipartite graph with a preferential attachment model from a given single degree sequence.
random_graph(n, m, p[, seed, directed])	Return a bipartite random graph.
gnmk_random_graph(n, m, k[, seed, directed])	Return a random bipartite graph G_{n,m,k}.