articulation_points¶

articulation_points(G)[source]¶

Return a generator of articulation points, or cut vertices, of a graph.

An articulation point or cut vertex is any node whose removal (along with all its incident edges) increases the number of connected components of a graph. An undirected connected graph without articulation points is biconnected. Articulation points belong to more than one biconnected component of a graph.

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

Parameters:	G : NetworkX Graph An undirected graph.
Returns:	articulation points : generator generator of nodes
Raises:	NetworkXNotImplemented : If the input graph is not undirected.

See also

is_biconnected, biconnected_components, 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 \(n\) 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

[R206]

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.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> list(nx.articulation_points(G))
[6, 5, 4, 3]
>>> G.add_edge(2,8)
>>> print(nx.is_biconnected(G))
True
>>> list(nx.articulation_points(G))
[]