1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
|
# -*- coding: utf-8 -*-
# bridges.py - bridge-finding algorithms
#
# Copyright 2004-2017 NetworkX developers.
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Bridge-finding algorithms."""
from itertools import chain
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ['bridges', 'has_bridges', 'local_bridges']
@not_implemented_for('multigraph')
@not_implemented_for('directed')
def bridges(G, root=None):
"""Generate all bridges in a graph.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase. Equivalently, a bridge is an
edge that does not belong to any cycle.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be returned.
Yields
------
e : edge
An edge in the graph whose removal disconnects the graph (or
causes the number of connected components to increase).
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
The barbell graph with parameter zero has a single bridge:
>>> G = nx.barbell_graph(10, 0)
>>> list(nx.bridges(G))
[(9, 10)]
Notes
-----
This is an implementation of the algorithm described in _[1]. An edge is a
bridge iff it is not contained in any chain. Chains are found using the
:func:`networkx.chain_decomposition` function.
Ignoring polylogarithmic factors, the worst-case time complexity is the
same as the :func:`networkx.chain_decomposition` function,
$O(m + n)$, where $n$ is the number of nodes in the graph and $m$ is
the number of edges.
References
----------
.. [1] https://en.wikipedia.org/wiki/Bridge_(graph_theory)#Bridge-Finding_with_Chain_Decompositions
"""
chains = nx.chain_decomposition(G, root=root)
chain_edges = set(chain.from_iterable(chains))
for u, v in G.edges():
if (u, v) not in chain_edges and (v, u) not in chain_edges:
yield u, v
@not_implemented_for('multigraph')
@not_implemented_for('directed')
def has_bridges(G, root=None):
"""Decide whether a graph has any bridges.
A *bridge* in a graph is an edge whose removal causes the number of
connected components of the graph to increase.
Parameters
----------
G : undirected graph
root : node (optional)
A node in the graph `G`. If specified, only the bridges in the
connected component containing this node will be considered.
Returns
-------
bool
Whether the graph (or the connected component containing `root`)
has any bridges.
Raises
------
NodeNotFound
If `root` is not in the graph `G`.
Examples
--------
The barbell graph with parameter zero has a single bridge::
>>> G = nx.barbell_graph(10, 0)
>>> nx.has_bridges(G)
True
On the other hand, the cycle graph has no bridges::
>>> G = nx.cycle_graph(5)
>>> nx.has_bridges(G)
False
Notes
-----
This implementation uses the :func:`networkx.bridges` function, so
it shares its worst-case time complexity, $O(m + n)$, ignoring
polylogarithmic factors, where $n$ is the number of nodes in the
graph and $m$ is the number of edges.
"""
try:
next(bridges(G))
except StopIteration:
return False
else:
return True
@not_implemented_for('multigraph')
@not_implemented_for('directed')
def local_bridges(G, with_span=True, weight=None):
"""Iterate over local bridges of `G` optionally computing the span
A *local bridge* is an edge whose endpoints have no common neighbors.
That is, the edge is not part of a triangle in the graph.
The *span* of a *local bridge* is the shortest path length between
the endpoints if the local bridge is removed.
Parameters
----------
G : undirected graph
with_span : bool
If True, yield a 3-tuple `(u, v, span)`
weight : function, string or None (default: None)
If function, used to compute edge weights for the span.
If string, the edge data attribute used in calculating span.
If None, all edges have weight 1.
Yields
------
e : edge
The local bridges as an edge 2-tuple of nodes `(u, v)` or
as a 3-tuple `(u, v, span)` when `with_span is True`.
Examples
--------
A cycle graph has every edge a local bridge with span N-1.
>>> G = nx.cycle_graph(9)
>>> (0, 8, 8) in set(nx.local_bridges(G))
True
"""
if with_span is not True:
for u, v in G.edges:
if not (set(G[u]) & set(G[v])):
yield u, v
else:
wt = nx.weighted._weight_function(G, weight)
for u, v in G.edges:
if not (set(G[u]) & set(G[v])):
enodes = {u, v}
def hide_edge(n, nbr, d):
if n not in enodes or nbr not in enodes:
return wt(n, nbr, d)
return None
try:
span = nx.shortest_path_length(G, u, v, weight=hide_edge)
yield u, v, span
except nx.NetworkXNoPath:
yield u, v, float('inf')
|
