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- """Semiconnectedness."""
- import networkx as nx
- from networkx.utils import not_implemented_for, pairwise
- __all__ = ["is_semiconnected"]
- @not_implemented_for("undirected")
- @nx._dispatchable
- def is_semiconnected(G):
- r"""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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