# Source code for networkx.algorithms.components.strongly_connected

```
"""Strongly connected components."""
import networkx as nx
from networkx.utils.decorators import not_implemented_for
__all__ = [
"number_strongly_connected_components",
"strongly_connected_components",
"is_strongly_connected",
"strongly_connected_components_recursive",
"kosaraju_strongly_connected_components",
"condensation",
]
[docs]@nx._dispatch
@not_implemented_for("undirected")
def strongly_connected_components(G):
"""Generate nodes in strongly connected components of graph.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
comp : generator of sets
A generator of sets of nodes, one for each strongly connected
component of G.
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
Generate a sorted list of strongly connected components, largest first.
>>> G = nx.cycle_graph(4, create_using=nx.DiGraph())
>>> nx.add_cycle(G, [10, 11, 12])
>>> [
... len(c)
... for c in sorted(nx.strongly_connected_components(G), key=len, reverse=True)
... ]
[4, 3]
If you only want the largest component, it's more efficient to
use max instead of sort.
>>> largest = max(nx.strongly_connected_components(G), key=len)
See Also
--------
connected_components
weakly_connected_components
kosaraju_strongly_connected_components
Notes
-----
Uses Tarjan's algorithm[1]_ with Nuutila's modifications[2]_.
Nonrecursive version of algorithm.
References
----------
.. [1] Depth-first search and linear graph algorithms, R. Tarjan
SIAM Journal of Computing 1(2):146-160, (1972).
.. [2] On finding the strongly connected components in a directed graph.
E. Nuutila and E. Soisalon-Soinen
Information Processing Letters 49(1): 9-14, (1994)..
"""
preorder = {}
lowlink = {}
scc_found = set()
scc_queue = []
i = 0 # Preorder counter
neighbors = {v: iter(G[v]) for v in G}
for source in G:
if source not in scc_found:
queue = [source]
while queue:
v = queue[-1]
if v not in preorder:
i = i + 1
preorder[v] = i
done = True
for w in neighbors[v]:
if w not in preorder:
queue.append(w)
done = False
break
if done:
lowlink[v] = preorder[v]
for w in G[v]:
if w not in scc_found:
if preorder[w] > preorder[v]:
lowlink[v] = min([lowlink[v], lowlink[w]])
else:
lowlink[v] = min([lowlink[v], preorder[w]])
queue.pop()
if lowlink[v] == preorder[v]:
scc = {v}
while scc_queue and preorder[scc_queue[-1]] > preorder[v]:
k = scc_queue.pop()
scc.add(k)
scc_found.update(scc)
yield scc
else:
scc_queue.append(v)
[docs]@not_implemented_for("undirected")
def kosaraju_strongly_connected_components(G, source=None):
"""Generate nodes in strongly connected components of graph.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
comp : generator of sets
A generator of sets of nodes, one for each strongly connected
component of G.
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
Generate a sorted list of strongly connected components, largest first.
>>> G = nx.cycle_graph(4, create_using=nx.DiGraph())
>>> nx.add_cycle(G, [10, 11, 12])
>>> [
... len(c)
... for c in sorted(
... nx.kosaraju_strongly_connected_components(G), key=len, reverse=True
... )
... ]
[4, 3]
If you only want the largest component, it's more efficient to
use max instead of sort.
>>> largest = max(nx.kosaraju_strongly_connected_components(G), key=len)
See Also
--------
strongly_connected_components
Notes
-----
Uses Kosaraju's algorithm.
"""
post = list(nx.dfs_postorder_nodes(G.reverse(copy=False), source=source))
seen = set()
while post:
r = post.pop()
if r in seen:
continue
c = nx.dfs_preorder_nodes(G, r)
new = {v for v in c if v not in seen}
seen.update(new)
yield new
[docs]@not_implemented_for("undirected")
def strongly_connected_components_recursive(G):
"""Generate nodes in strongly connected components of graph.
Recursive version of algorithm.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
comp : generator of sets
A generator of sets of nodes, one for each strongly connected
component of G.
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
Generate a sorted list of strongly connected components, largest first.
>>> G = nx.cycle_graph(4, create_using=nx.DiGraph())
>>> nx.add_cycle(G, [10, 11, 12])
>>> [
... len(c)
... for c in sorted(
... nx.strongly_connected_components_recursive(G), key=len, reverse=True
... )
... ]
[4, 3]
If you only want the largest component, it's more efficient to
use max instead of sort.
>>> largest = max(nx.strongly_connected_components_recursive(G), key=len)
To create the induced subgraph of the components use:
>>> S = [G.subgraph(c).copy() for c in nx.weakly_connected_components(G)]
See Also
--------
connected_components
Notes
-----
Uses Tarjan's algorithm[1]_ with Nuutila's modifications[2]_.
References
----------
.. [1] Depth-first search and linear graph algorithms, R. Tarjan
SIAM Journal of Computing 1(2):146-160, (1972).
.. [2] On finding the strongly connected components in a directed graph.
E. Nuutila and E. Soisalon-Soinen
Information Processing Letters 49(1): 9-14, (1994)..
"""
def visit(v, cnt):
root[v] = cnt
visited[v] = cnt
cnt += 1
stack.append(v)
for w in G[v]:
if w not in visited:
yield from visit(w, cnt)
if w not in component:
root[v] = min(root[v], root[w])
if root[v] == visited[v]:
component[v] = root[v]
tmpc = {v} # hold nodes in this component
while stack[-1] != v:
w = stack.pop()
component[w] = root[v]
tmpc.add(w)
stack.remove(v)
yield tmpc
visited = {}
component = {}
root = {}
cnt = 0
stack = []
for source in G:
if source not in visited:
yield from visit(source, cnt)
[docs]@not_implemented_for("undirected")
def number_strongly_connected_components(G):
"""Returns number of strongly connected components in graph.
Parameters
----------
G : NetworkX graph
A directed graph.
Returns
-------
n : integer
Number of strongly connected components
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
>>> G = nx.DiGraph([(0, 1), (1, 2), (2, 0), (2, 3), (4, 5), (3, 4), (5, 6), (6, 3), (6, 7)])
>>> nx.number_strongly_connected_components(G)
3
See Also
--------
strongly_connected_components
number_connected_components
number_weakly_connected_components
Notes
-----
For directed graphs only.
"""
return sum(1 for scc in strongly_connected_components(G))
[docs]@not_implemented_for("undirected")
def is_strongly_connected(G):
"""Test directed graph for strong connectivity.
A directed graph is strongly connected if and only if every vertex in
the graph is reachable from every other vertex.
Parameters
----------
G : NetworkX Graph
A directed graph.
Returns
-------
connected : bool
True if the graph is strongly connected, False otherwise.
Examples
--------
>>> G = nx.DiGraph([(0, 1), (1, 2), (2, 3), (3, 0), (2, 4), (4, 2)])
>>> nx.is_strongly_connected(G)
True
>>> G.remove_edge(2, 3)
>>> nx.is_strongly_connected(G)
False
Raises
------
NetworkXNotImplemented
If G is undirected.
See Also
--------
is_weakly_connected
is_semiconnected
is_connected
is_biconnected
strongly_connected_components
Notes
-----
For directed graphs only.
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept(
"""Connectivity is undefined for the null graph."""
)
return len(next(strongly_connected_components(G))) == len(G)
[docs]@not_implemented_for("undirected")
def condensation(G, scc=None):
"""Returns the condensation of G.
The condensation of G is the graph with each of the strongly connected
components contracted into a single node.
Parameters
----------
G : NetworkX DiGraph
A directed graph.
scc: list or generator (optional, default=None)
Strongly connected components. If provided, the elements in
`scc` must partition the nodes in `G`. If not provided, it will be
calculated as scc=nx.strongly_connected_components(G).
Returns
-------
C : NetworkX DiGraph
The condensation graph C of G. The node labels are integers
corresponding to the index of the component in the list of
strongly connected components of G. C has a graph attribute named
'mapping' with a dictionary mapping the original nodes to the
nodes in C to which they belong. Each node in C also has a node
attribute 'members' with the set of original nodes in G that
form the SCC that the node in C represents.
Raises
------
NetworkXNotImplemented
If G is undirected.
Examples
--------
Contracting two sets of strongly connected nodes into two distinct SCC
using the barbell graph.
>>> G = nx.barbell_graph(4, 0)
>>> G.remove_edge(3, 4)
>>> G = nx.DiGraph(G)
>>> H = nx.condensation(G)
>>> H.nodes.data()
NodeDataView({0: {'members': {0, 1, 2, 3}}, 1: {'members': {4, 5, 6, 7}}})
>>> H.graph['mapping']
{0: 0, 1: 0, 2: 0, 3: 0, 4: 1, 5: 1, 6: 1, 7: 1}
Contracting a complete graph into one single SCC.
>>> G = nx.complete_graph(7, create_using=nx.DiGraph)
>>> H = nx.condensation(G)
>>> H.nodes
NodeView((0,))
>>> H.nodes.data()
NodeDataView({0: {'members': {0, 1, 2, 3, 4, 5, 6}}})
Notes
-----
After contracting all strongly connected components to a single node,
the resulting graph is a directed acyclic graph.
"""
if scc is None:
scc = nx.strongly_connected_components(G)
mapping = {}
members = {}
C = nx.DiGraph()
# Add mapping dict as graph attribute
C.graph["mapping"] = mapping
if len(G) == 0:
return C
for i, component in enumerate(scc):
members[i] = component
mapping.update((n, i) for n in component)
number_of_components = i + 1
C.add_nodes_from(range(number_of_components))
C.add_edges_from(
(mapping[u], mapping[v]) for u, v in G.edges() if mapping[u] != mapping[v]
)
# Add a list of members (ie original nodes) to each node (ie scc) in C.
nx.set_node_attributes(C, members, "members")
return C
```