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Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

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networkx.algorithms.components.biconnected_components¶

biconnected_components(G)[source]¶

Returns a generator of sets of nodes, one set for each biconnected component of the graph

Biconnected components are maximal subgraphs such that the removal of a node (and all edges incident on that node) will not disconnect the subgraph. Note that nodes may be part of more than one biconnected component. Those nodes are articulation points, or cut vertices. The removal of articulation points will increase the number of connected components of the graph.

Notice that by convention a dyad is considered a biconnected component.

Parameters: G (NetworkX Graph) – An undirected graph.
Returns: nodes – Generator of sets of nodes, one set for each biconnected component.
Return type: generator
Raises: NetworkXNotImplemented – If the input graph is not undirected.

See also

k_components(): this function is a special case where k=2
bridge_components(): similar to this function, but is defined using 2-edge-connectivity instead of 2-node-connectivity.

Examples

>>> G = nx.lollipop_graph(5, 1)
>>> print(nx.is_biconnected(G))
False
>>> bicomponents = list(nx.biconnected_components(G))
>>> len(bicomponents)
2
>>> G.add_edge(0, 5)
>>> print(nx.is_biconnected(G))
True
>>> bicomponents = list(nx.biconnected_components(G))
>>> len(bicomponents)
1

You can generate a sorted list of biconnected components, largest first, using sort.

>>> G.remove_edge(0, 5)
>>> [len(c) for c in sorted(nx.biconnected_components(G), key=len, reverse=True)]
[5, 2]

If you only want the largest connected component, it’s more efficient to use max instead of sort.

>>> Gc = max(nx.biconnected_components(G), key=len)

To create the components as subgraphs use: (G.subgraph(c).copy() for c in biconnected_components(G))

Notes

The algorithm to find articulation points and biconnected components is implemented using a non-recursive depth-first-search (DFS) that keeps track of the highest level that back edges reach in the DFS tree. A node n is an articulation point if, and only if, there exists a subtree rooted at n such that there is no back edge from any successor of n that links to a predecessor of n in the DFS tree. By keeping track of all the edges traversed by the DFS we can obtain the biconnected components because all edges of a bicomponent will be traversed consecutively between articulation points.

References

1: Hopcroft, J.; Tarjan, R. (1973). “Efficient algorithms for graph manipulation”. Communications of the ACM 16: 372–378. doi:10.1145/362248.362272