Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.components.biconnected

# -*- coding: utf-8 -*-
"""
Biconnected components and articulation points.
"""
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
from itertools import chain
import networkx as nx
from networkx.utils.decorators import not_implemented_for
__author__ = '\n'.join(['Jordi Torrents <jtorrents@milnou.net>',
'Dan Schult <dschult@colgate.edu>',
'Aric Hagberg <aric.hagberg@gmail.com>'])
__all__ = ['biconnected_components',
'biconnected_component_edges',
'biconnected_component_subgraphs',
'is_biconnected',
'articulation_points',
]

@not_implemented_for('directed')
[docs]def 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
------
NetworkXNotImplemented :
If the input graph is not undirected.

Examples
--------
>>> G=nx.path_graph(4)
>>> print(nx.is_biconnected(G))
False
>>> print(nx.is_biconnected(G))
True

--------
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 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
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
"""
bcc = list(biconnected_components(G))
if not bcc: # No bicomponents (it could be an empty graph)
return False
return len(bcc[0]) == len(G)

@not_implemented_for('directed')
[docs]def biconnected_component_edges(G):
"""Return a generator of lists of edges, one list for each biconnected
component of the input graph.

Biconnected components are maximal subgraphs such that the removal of a
node (and all edges incident on that node) will not disconnect the
subgraph. Note that nodes may be part of more than one biconnected
component. Those nodes are articulation points, or cut vertices. However,
each edge belongs to one, and only one, biconnected component.

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

Parameters
----------
G : NetworkX Graph
An undirected graph.

Returns
-------
edges : generator of lists
Generator of lists of edges, one list for each bicomponent.

Raises
------
NetworkXNotImplemented :
If the input graph is not undirected.

Examples
--------
>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> components = nx.biconnected_component_edges(G)
>>> print(nx.is_biconnected(G))
True
>>> components = nx.biconnected_component_edges(G)

--------
is_biconnected,
biconnected_components,
articulation_points,
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
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
"""
for comp in _biconnected_dfs(G,components=True):
yield comp

@not_implemented_for('directed')
[docs]def biconnected_components(G):
"""Return a generator of sets of nodes, one set for each biconnected
component of the graph

Biconnected components are maximal subgraphs such that the removal of a
node (and all edges incident on that node) will not disconnect the
subgraph. Note that nodes may be part of more than one biconnected
component. Those nodes are articulation points, or cut vertices. The
removal of articulation points will increase the number of connected
components of the graph.

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

Parameters
----------
G : NetworkX Graph
An undirected graph.

Returns
-------
nodes : generator
Generator of sets of nodes, one set for each biconnected component.

Raises
------
NetworkXNotImplemented :
If the input graph is not undirected.

Examples
--------
>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> components = nx.biconnected_components(G)
>>> print(nx.is_biconnected(G))
True
>>> components = nx.biconnected_components(G)

--------
is_biconnected,
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 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
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
"""
for comp in _biconnected_dfs(G,components=True):
yield set(chain.from_iterable(comp))

@not_implemented_for('directed')
[docs]def biconnected_component_subgraphs(G, copy=True):
"""Return a generator of graphs, one graph for each biconnected component
of the input graph.

Biconnected components are maximal subgraphs such that the removal of a
node (and all edges incident on that node) will not disconnect the
subgraph. Note that nodes may be part of more than one biconnected
component. Those nodes are articulation points, or cut vertices. The
removal of articulation points will increase the number of connected
components of the graph.

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

Parameters
----------
G : NetworkX Graph
An undirected graph.

Returns
-------
graphs : generator
Generator of graphs, one graph for each biconnected component.

Raises
------
NetworkXNotImplemented :
If the input graph is not undirected.

Examples
--------
>>> G = nx.barbell_graph(4,2)
>>> print(nx.is_biconnected(G))
False
>>> subgraphs = list(nx.biconnected_component_subgraphs(G))

--------
is_biconnected,
articulation_points,
biconnected_component_edges,
biconnected_components

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.

Graph, node, and edge attributes are copied to the subgraphs.

References
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
"""
for comp_nodes in biconnected_components(G):
if copy:
yield G.subgraph(comp_nodes).copy()
else:
yield G.subgraph(comp_nodes)

@not_implemented_for('directed')
[docs]def articulation_points(G):
"""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.

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

--------
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
----------
.. [1] Hopcroft, J.; Tarjan, R. (1973).
"Efficient algorithms for graph manipulation".
Communications of the ACM 16: 372–378. doi:10.1145/362248.362272
"""
return _biconnected_dfs(G,components=False)

@not_implemented_for('directed')
def _biconnected_dfs(G, components=True):
# depth-first search algorithm to generate articulation points
# and biconnected components
visited = set()
for start in G:
if start in visited:
continue
discovery = {start:0} # "time" of first discovery of node during search
low = {start:0}
root_children = 0
edge_stack = []
stack = [(start, start, iter(G[start]))]
while stack:
grandparent, parent, children = stack[-1]
try:
child = next(children)
if grandparent == child:
continue
if child in visited:
if discovery[child] <= discovery[parent]: # back edge
low[parent] = min(low[parent],discovery[child])
if components:
edge_stack.append((parent,child))
else:
low[child] = discovery[child] = len(discovery)
stack.append((parent, child, iter(G[child])))
if components:
edge_stack.append((parent,child))
except StopIteration:
stack.pop()
if len(stack) > 1:
if low[parent] >= discovery[grandparent]:
if components:
ind = edge_stack.index((grandparent,parent))
yield edge_stack[ind:]
edge_stack=edge_stack[:ind]
else:
yield grandparent
low[grandparent] = min(low[parent], low[grandparent])
elif stack: # length 1 so grandparent is root
root_children += 1
if components:
ind = edge_stack.index((grandparent,parent))
yield edge_stack[ind:]
if not components:
# root node is articulation point if it has more than 1 child
if root_children > 1:
yield start