is_biconnected

is_biconnected(G)

Return True if the graph is biconnected, False otherwise.

A graph is biconnected if, and only if, it cannot be disconnected by removing only one node (and all edges incident on that node). If removing a node increases the number of disconnected components in the graph, that node is called an articulation point, or cut vertex. A biconnected graph has no articulation points.

Parameters :

G : NetworkX Graph

An undirected graph.

Returns :

biconnected : bool

True if the graph is biconnected, False otherwise.

Raises :

NetworkXError : :

If the input graph is not undirected.

See also

biconnected_components, articulation_points, biconnected_component_edges, biconnected_component_subgraphs

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 is an articulation point if, and only if, there exists a subtree rooted at such that there is no back edge from any successor of that links to a predecessor of 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

[R165]

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

Examples

>>> G=nx.path_graph(4)
>>> print(nx.is_biconnected(G))
False
>>> G.add_edge(0,3)
>>> print(nx.is_biconnected(G))
True