380 lines
12 KiB
Python
380 lines
12 KiB
Python
"""Biconnected components and articulation points."""
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from itertools import chain
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from networkx.utils.decorators import not_implemented_for
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__all__ = [
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"biconnected_components",
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"biconnected_component_edges",
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"is_biconnected",
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"articulation_points",
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]
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@not_implemented_for("directed")
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def is_biconnected(G):
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"""Returns True if the graph is biconnected, False otherwise.
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A graph is biconnected if, and only if, it cannot be disconnected by
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removing only one node (and all edges incident on that node). If
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removing a node increases the number of disconnected components
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in the graph, that node is called an articulation point, or cut
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vertex. A biconnected graph has no articulation points.
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Parameters
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----------
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G : NetworkX Graph
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An undirected graph.
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Returns
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-------
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biconnected : bool
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True if the graph is biconnected, False otherwise.
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Raises
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------
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NetworkXNotImplemented
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If the input graph is not undirected.
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Examples
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--------
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>>> G = nx.path_graph(4)
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>>> print(nx.is_biconnected(G))
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False
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>>> G.add_edge(0, 3)
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>>> print(nx.is_biconnected(G))
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True
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See Also
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--------
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biconnected_components
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articulation_points
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biconnected_component_edges
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is_strongly_connected
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is_weakly_connected
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is_connected
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is_semiconnected
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Notes
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-----
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The algorithm to find articulation points and biconnected
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components is implemented using a non-recursive depth-first-search
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(DFS) that keeps track of the highest level that back edges reach
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in the DFS tree. A node `n` is an articulation point if, and only
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if, there exists a subtree rooted at `n` such that there is no
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back edge from any successor of `n` that links to a predecessor of
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`n` in the DFS tree. By keeping track of all the edges traversed
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by the DFS we can obtain the biconnected components because all
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edges of a bicomponent will be traversed consecutively between
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articulation points.
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References
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----------
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.. [1] Hopcroft, J.; Tarjan, R. (1973).
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"Efficient algorithms for graph manipulation".
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Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
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"""
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bcc = list(biconnected_components(G))
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if len(bcc) == 1:
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return len(bcc[0]) == len(G)
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return False # Multiple bicomponents or No bicomponents (empty graph?)
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# if len(bcc) == 0: # No bicomponents (it could be an empty graph)
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# return False
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# return len(bcc[0]) == len(G)
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@not_implemented_for("directed")
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def biconnected_component_edges(G):
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"""Returns a generator of lists of edges, one list for each biconnected
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component of the input graph.
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Biconnected components are maximal subgraphs such that the removal of a
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node (and all edges incident on that node) will not disconnect the
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subgraph. Note that nodes may be part of more than one biconnected
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component. Those nodes are articulation points, or cut vertices.
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However, each edge belongs to one, and only one, biconnected component.
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Notice that by convention a dyad is considered a biconnected component.
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Parameters
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----------
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G : NetworkX Graph
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An undirected graph.
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Returns
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-------
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edges : generator of lists
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Generator of lists of edges, one list for each bicomponent.
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Raises
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------
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NetworkXNotImplemented
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If the input graph is not undirected.
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Examples
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--------
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>>> G = nx.barbell_graph(4, 2)
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>>> print(nx.is_biconnected(G))
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False
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>>> bicomponents_edges = list(nx.biconnected_component_edges(G))
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>>> len(bicomponents_edges)
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5
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>>> G.add_edge(2, 8)
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>>> print(nx.is_biconnected(G))
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True
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>>> bicomponents_edges = list(nx.biconnected_component_edges(G))
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>>> len(bicomponents_edges)
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1
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See Also
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--------
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is_biconnected,
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biconnected_components,
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articulation_points,
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Notes
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-----
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The algorithm to find articulation points and biconnected
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components is implemented using a non-recursive depth-first-search
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(DFS) that keeps track of the highest level that back edges reach
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in the DFS tree. A node `n` is an articulation point if, and only
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if, there exists a subtree rooted at `n` such that there is no
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back edge from any successor of `n` that links to a predecessor of
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`n` in the DFS tree. By keeping track of all the edges traversed
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by the DFS we can obtain the biconnected components because all
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edges of a bicomponent will be traversed consecutively between
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articulation points.
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References
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----------
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.. [1] Hopcroft, J.; Tarjan, R. (1973).
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"Efficient algorithms for graph manipulation".
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Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
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"""
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yield from _biconnected_dfs(G, components=True)
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@not_implemented_for("directed")
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def biconnected_components(G):
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"""Returns a generator of sets of nodes, one set for each biconnected
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component of the graph
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Biconnected components are maximal subgraphs such that the removal of a
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node (and all edges incident on that node) will not disconnect the
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subgraph. Note that nodes may be part of more than one biconnected
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component. Those nodes are articulation points, or cut vertices. The
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removal of articulation points will increase the number of connected
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components of the graph.
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Notice that by convention a dyad is considered a biconnected component.
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Parameters
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----------
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G : NetworkX Graph
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An undirected graph.
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Returns
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-------
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nodes : generator
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Generator of sets of nodes, one set for each biconnected component.
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Raises
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------
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NetworkXNotImplemented
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If the input graph is not undirected.
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See Also
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--------
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k_components : this function is a special case where k=2
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bridge_components : similar to this function, but is defined using
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2-edge-connectivity instead of 2-node-connectivity.
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Examples
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--------
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>>> G = nx.lollipop_graph(5, 1)
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>>> print(nx.is_biconnected(G))
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False
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>>> bicomponents = list(nx.biconnected_components(G))
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>>> len(bicomponents)
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2
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>>> G.add_edge(0, 5)
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>>> print(nx.is_biconnected(G))
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True
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>>> bicomponents = list(nx.biconnected_components(G))
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>>> len(bicomponents)
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1
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You can generate a sorted list of biconnected components, largest
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first, using sort.
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>>> G.remove_edge(0, 5)
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>>> [len(c) for c in sorted(nx.biconnected_components(G), key=len, reverse=True)]
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[5, 2]
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If you only want the largest connected component, it's more
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efficient to use max instead of sort.
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>>> Gc = max(nx.biconnected_components(G), key=len)
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To create the components as subgraphs use:
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``(G.subgraph(c).copy() for c in biconnected_components(G))``
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See Also
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--------
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is_biconnected
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articulation_points
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biconnected_component_edges
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Notes
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-----
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The algorithm to find articulation points and biconnected
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components is implemented using a non-recursive depth-first-search
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(DFS) that keeps track of the highest level that back edges reach
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in the DFS tree. A node `n` is an articulation point if, and only
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if, there exists a subtree rooted at `n` such that there is no
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back edge from any successor of `n` that links to a predecessor of
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`n` in the DFS tree. By keeping track of all the edges traversed
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by the DFS we can obtain the biconnected components because all
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edges of a bicomponent will be traversed consecutively between
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articulation points.
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References
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----------
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.. [1] Hopcroft, J.; Tarjan, R. (1973).
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"Efficient algorithms for graph manipulation".
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Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
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"""
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for comp in _biconnected_dfs(G, components=True):
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yield set(chain.from_iterable(comp))
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@not_implemented_for("directed")
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def articulation_points(G):
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"""Yield the articulation points, or cut vertices, of a graph.
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An articulation point or cut vertex is any node whose removal (along with
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all its incident edges) increases the number of connected components of
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a graph. An undirected connected graph without articulation points is
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biconnected. Articulation points belong to more than one biconnected
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component of a graph.
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Notice that by convention a dyad is considered a biconnected component.
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Parameters
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----------
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G : NetworkX Graph
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An undirected graph.
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Yields
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------
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node
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An articulation point in the graph.
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Raises
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------
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NetworkXNotImplemented
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If the input graph is not undirected.
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Examples
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--------
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>>> G = nx.barbell_graph(4, 2)
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>>> print(nx.is_biconnected(G))
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False
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>>> len(list(nx.articulation_points(G)))
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4
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>>> G.add_edge(2, 8)
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>>> print(nx.is_biconnected(G))
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True
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>>> len(list(nx.articulation_points(G)))
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0
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See Also
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--------
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is_biconnected
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biconnected_components
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biconnected_component_edges
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Notes
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-----
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The algorithm to find articulation points and biconnected
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components is implemented using a non-recursive depth-first-search
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(DFS) that keeps track of the highest level that back edges reach
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in the DFS tree. A node `n` is an articulation point if, and only
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if, there exists a subtree rooted at `n` such that there is no
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back edge from any successor of `n` that links to a predecessor of
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`n` in the DFS tree. By keeping track of all the edges traversed
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by the DFS we can obtain the biconnected components because all
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edges of a bicomponent will be traversed consecutively between
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articulation points.
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References
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----------
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.. [1] Hopcroft, J.; Tarjan, R. (1973).
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"Efficient algorithms for graph manipulation".
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Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
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"""
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seen = set()
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for articulation in _biconnected_dfs(G, components=False):
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if articulation not in seen:
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seen.add(articulation)
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yield articulation
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@not_implemented_for("directed")
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def _biconnected_dfs(G, components=True):
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# depth-first search algorithm to generate articulation points
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# and biconnected components
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visited = set()
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for start in G:
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if start in visited:
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continue
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discovery = {start: 0} # time of first discovery of node during search
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low = {start: 0}
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root_children = 0
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visited.add(start)
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edge_stack = []
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stack = [(start, start, iter(G[start]))]
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while stack:
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grandparent, parent, children = stack[-1]
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try:
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child = next(children)
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if grandparent == child:
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continue
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if child in visited:
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if discovery[child] <= discovery[parent]: # back edge
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low[parent] = min(low[parent], discovery[child])
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if components:
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edge_stack.append((parent, child))
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else:
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low[child] = discovery[child] = len(discovery)
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visited.add(child)
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stack.append((parent, child, iter(G[child])))
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if components:
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edge_stack.append((parent, child))
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except StopIteration:
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stack.pop()
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if len(stack) > 1:
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if low[parent] >= discovery[grandparent]:
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if components:
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ind = edge_stack.index((grandparent, parent))
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yield edge_stack[ind:]
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edge_stack = edge_stack[:ind]
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else:
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yield grandparent
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low[grandparent] = min(low[parent], low[grandparent])
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elif stack: # length 1 so grandparent is root
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root_children += 1
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if components:
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ind = edge_stack.index((grandparent, parent))
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yield edge_stack[ind:]
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if not components:
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# root node is articulation point if it has more than 1 child
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if root_children > 1:
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yield start
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