Persistent construction of 3-colourability

Characterize the finite digraphs H that rpp-construct the 3-colouring template for every possibly infinite digraph R that does not admit a homomorphism to H.

Background

A finite structure H is defined to persistently construct 3-colourability when every restriction R failing to map to H yields an RCSP template (H,R) that rpp-constructs 3-colourability, and hence is NP-hard under the paper’s reductions.

The paper notes that every hereditarily hard digraph has this property, but does not characterize all finite digraphs with it.

References

Characterize the class of finite digraphs (structures) that persistently construct $_3$.

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