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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)))