Dichotomy for CSPs under arbitrary digraph-free restrictions

Establish whether, for every finite digraph H and finite digraph F, the problem CSP(H) restricted to F-free digraphs admits a P-versus-NP-complete dichotomy, and whether the analogous dichotomy holds when the input is restricted to F-subgraph-free digraphs.

Background

The paper studies complexity dichotomies for digraph homomorphism problems under restrictions forbidding a fixed digraph either as an induced subgraph or as a subgraph. The authors settle this question for directed paths, digraphs on three vertices, and a family of smooth tournaments, but leave the general case unresolved.

A negative answer would, under P ≠ NP, produce finite digraphs whose restricted CSPs are NP-intermediate, making the question relevant both to restricted homomorphism algorithmics and to the structure of intermediate complexity classes.