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"""Semiconnectedness."""
import networkx as nx
from networkx.utils import not_implemented_for, pairwise
__all__ = ["is_semiconnected"]
@not_implemented_for("undirected")
def is_semiconnected(G):
"""Returns True if the graph is semiconnected, False otherwise.
A graph is semiconnected if and only if for any pair of nodes, either one
is reachable from the other, or they are mutually reachable.
This function uses a theorem that states that a DAG is semiconnected
if for any topological sort, for node $v_n$ in that sort, there is an
edge $(v_i, v_{i+1})$. That allows us to check if a non-DAG `G` is
semiconnected by condensing the graph: i.e. constructing a new graph `H`
with nodes being the strongly connected components of `G`, and edges
(scc_1, scc_2) if there is a edge $(v_1, v_2)$ in `G` for some
$v_1 \in scc_1$ and $v_2 \in scc_2$. That results in a DAG, so we compute
the topological sort of `H` and check if for every $n$ there is an edge
$(scc_n, scc_{n+1})$.
Parameters
----------
G : NetworkX graph
A directed graph.
Returns
-------
semiconnected : bool
True if the graph is semiconnected, False otherwise.
Raises
------
NetworkXNotImplemented
If the input graph is undirected.
NetworkXPointlessConcept
If the graph is empty.
Examples
--------
>>> G = nx.path_graph(4, create_using=nx.DiGraph())
>>> print(nx.is_semiconnected(G))
True
>>> G = nx.DiGraph([(1, 2), (3, 2)])
>>> print(nx.is_semiconnected(G))
False
See Also
--------
is_strongly_connected
is_weakly_connected
is_connected
is_biconnected
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept(
"Connectivity is undefined for the null graph."
)
if not nx.is_weakly_connected(G):
return False
H = nx.condensation(G)
return all(H.has_edge(u, v) for u, v in pairwise(nx.topological_sort(H)))
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