Show by counterexample that filtering based on Theorem 3.41 is incomplete, even when all separators are used.
Theorem 3.41
If S is a separator of directed graph G, then G contains a hamiltonian cycle only if GS contains a permissible hamiltonian cycle.
Furthermore, an edge of G connecting vertices in S is hamiltonian only if it is part of a permissible hamiltonian cycle of GS.