# Source code for networkx.algorithms.bridges

```
"""Bridge-finding algorithms."""
from itertools import chain
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["bridges", "has_bridges", "local_bridges"]
[docs]@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. Bridges are also known as cut-edges,
isthmuses, or cut arcs.
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`.
NetworkXNotImplemented
If `G` is a directed graph.
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 if and only if it is not contained in any chain. Chains are found
using the :func:`networkx.chain_decomposition` function.
The algorithm described in [1]_ requires a simple graph. If the provided
graph is a multigraph, we convert it to a simple graph and verify that any
bridges discovered by the chain decomposition algorithm are not multi-edges.
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_%28graph_theory%29#Bridge-Finding_with_Chain_Decompositions
"""
multigraph = G.is_multigraph()
H = nx.Graph(G) if multigraph else G
chains = nx.chain_decomposition(H, root=root)
chain_edges = set(chain.from_iterable(chains))
H_copy = H.copy()
if root is not None:
H = H.subgraph(nx.node_connected_component(H, root)).copy()
for u, v in H.edges():
if (u, v) not in chain_edges and (v, u) not in chain_edges:
if multigraph and len(G[u][v]) > 1:
continue
yield u, v
[docs]@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`.
NetworkXNotImplemented
If `G` is a directed graph.
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, root=root))
except StopIteration:
return False
else:
return True
[docs]@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`.
Raises
------
NetworkXNotImplemented
If `G` is a directed graph or multigraph.
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")
```