Complete classification of infinite directed length sets

Complete the complexity classification of the directed Hamiltonian cycle and Hamiltonian path problems for all fixed infinite sets of permitted walk lengths.

Background

The paper studies Hamiltonian cycles and paths in directed graphs subject to the requirement that consecutive vertices be connected by directed walks whose lengths belong to a fixed set S. It completely classifies finite nonempty sets, cofinite sets, and several particular infinite sets.

For infinite directed sets, the paper establishes distinct behaviours: cofinite sets yield linear-time algorithms, the fixed set {2} ∪ O∞ yields NP-completeness, and a fixed recursive infinite set yields undecidability. These results do not determine the complexity for every fixed infinite set, leaving the general classification unresolved.