Characterization of universally NP-hard restricted CSP templates

Characterize the finite digraphs H such that RCSP(H,R) is NP-hard for every finite digraph R that does not admit a homomorphism to H, assuming P ≠ NP.

Background

Theorem 3 of Brewster and Graves establishes this universal hardness property for hereditarily hard digraphs. The authors ask for the converse classification or, more generally, a complete characterization of all finite digraphs with the property.

This formulation is the complexity-theoretic counterpart of persistent rpp-construction and is explicitly conditional on P ≠ NP.

References

Characterize the class of finite digraphs (structures) $$ such that $\RCSP(,')$ is $\NP$-hard whenever $'\not\to $ (assuming $\cP\neq \NP$).

— Restricted CSPs and F-free Digraph Algorithmics  (2502.17596 - Guzmán-Pro et al., 24 Feb 2025) in Problem, Section 8, subsection “Persistent structures”