Construction

Run SCC (Tarjan/Kosaraju). Create meta-graph vertex per SCC. For each edge (u,v) in original, add edge SCC(u) → SCC(v) if distinct.

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Applications

Reachability: u reaches v iff SCC(u) reaches SCC(v) in DAG. 2-SAT satisfiability check.

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DP on condensation

Topological order → DP. Many problems polynomial on SCC condensation even if original graph has cycles.

Complexity

O(V+E) construction. Then apply DAG algorithms.