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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.
G : NetworkX Graph
An undirected graph.
articulation points : generator
generator of nodes
If the input graph is not undirected.
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.
[R203] Hopcroft, J.; Tarjan, R. (1973). “Efficient algorithms for graph manipulation”. Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
>>> 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))