We characterize valid inequalities for bilevel programs in which the leader makes binary decisions, and the follower solves a shortest-path problem. We exploit a dualize-and-combine reformulation and show that even with ideal formulations for the leader and the follower primal solution sets, fractional solutions may still appear in the continuous relaxation. From that reformulation, we identify that the interaction between the leader variables and the follower dual variables, which we capture through a leader-potential set, is the key structure to be convexified to guarantee that a continuous relaxation recovers the optimal value of the original problem, closing the objective gap. Furthermore, we characterize the valid inequalities for the leader-potential set through network flows, and identify a family of inequalities we name root cuts that suffice to recover the optimal value. Complementary to the structural results, we prove that the cut generation problem is NP-hard even with partially fixed coefficients and a network with two nodes and two parallel arcs. Motivated by the complexity result, we propose a constructive method to fix coefficients in the valid inequalities while preserving separation guarantees, by exploiting the network structure.