Contractibility under the local connectivity condition

Prove that every directed graph with terminal set T and capacities c satisfying the local connectivity condition has either a zero-capacity terminal whose removal preserves the local connectivity condition, or a pre-terminal p and adjacent terminal t such that contracting p into t and decrementing c_t by one preserves the local connectivity condition.

Background

The paper introduces the local connectivity condition as a relaxation of k-T-connectivity. Under this condition, the Győri–Lovász partition conclusion still holds, but the constructive proof inherited from the cascade framework is exponential-time, even for fixed small values of k.

A polynomial-time algorithm would follow a contraction-based approach if one could always find a capacity-preserving contraction or remove a zero-capacity terminal while maintaining local connectivity. The authors formulate this structural assertion as Conjecture A.5 and note that they do not know a counterexample. They further explain that, for fixed k, proving the conjecture would imply fixed-parameter tractability because the condition could then be checked in polynomial time.