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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.
G : NetworkX Graph
An undirected graph.
biconnected : bool
True if the graph is biconnected, False otherwise.
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.
[R210] Hopcroft, J.; Tarjan, R. (1973). “Efficient algorithms for graph manipulation”. Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
>>> G=nx.path_graph(4) >>> print(nx.is_biconnected(G)) False >>> G.add_edge(0,3) >>> print(nx.is_biconnected(G)) True