Additive‑congestion constraints in single‑source unsplittable flow enforce a stable‑set structure. This note isolates and generalises that mechanism. We introduce a path‑closed notion of realising an independence system by the zero‑cost choices of primary terminals in a directed acyclic flow instance. The definition quantifies over every directed source‑terminal path, so it remains valid under prefix borrowing, suffix splicing, and hybrid routes.
Our main result extends the triangle mechanism: every finite loopless independence system admits a polynomial‑size realisation, measured by the incidence size of its minimal forbidden sets. Consequently, every finite simple graph, and more generally every hypergraph independence system without singleton forbidden hyperedges, can be represented by an acyclic single‑source gadget. We then specialise the construction to odd cycles. For $C_{2k+1}$, a uniform rational family yields a fractional cheap‑selection vector that violates the odd‑cycle inequality. A potential shift converts a signed connector separator into non‑negative arc costs and gives the exact cost‑preserving additive‑congestion threshold $\tau = 1 - bq$. Within the symmetric family, the supremum threshold is $(k+2)/(2(k+1))$, which tends to $1/2$. For $C_5$, an exact certificate independently derives all source‑terminal paths and enumerates all $3^{10}=59049$ unsplittable routings using rational arithmetic.
Review