Condensation graph

Contract each strongly connected component of a directed graph to one vertex, retaining edges between different components. A directed cycle between contracted vertices would make those components mutually reachable, contradicting their distinctness; thus the condensation graph is a directed acyclic graph. Parallel edges can be retained by algorithms that count them consistently.

Assign each vertex of a finite directed graph a label in a total order. To find the least label reachable from each vertex, first compress its strongly connected components. Give each component its own least label, then process the condensation graph in reverse topological order, replacing each label by the minimum of itself and its successors' labels. Every reachable component is either the current one or reached through a first outgoing edge; induction over this order proves the recurrence. The graph-processing work is $O(|V|+|E|)$ with constant-cost label comparisons. Store a representative vertex as well when a witness is required.

A positive-integer labeling with $a⇝b\Rightarrow \ell (a)\le \ell (b)$ and $a/⇝b\Rightarrow \ell (a)>\ell (b)$ exists exactly when the condensation graph is totally ordered by reachability. Mutual paths force equal labels, and equal labels force mutual paths. Two incomparable components would require strict inequalities in both directions. Conversely number the components in topological order; total comparability forces every earlier component to reach every later one, proving both conditions. Compute the labeling in linear time. Reachability here includes length-zero paths.