Finite obstruction sets for oriented graphs

Determine whether every oriented graph H has finitely many minimal H-homomorphism obstructions among digraphs that are directed-3-cycle-free.

Background

For the smooth tournament family considered in the paper, the authors construct finite obstruction sets for CSP(H) restricted to directed-3-cycle-free digraphs. They ask whether this phenomenon extends to all oriented graphs.

A positive result would generalize the finite-obstruction theorem beyond tournaments and clarify when restricted homomorphism classes admit finite forbidden descriptions.

References

Is is true that for every oriented graph $$ there a finitely many $\vec{}_3$-free minimal obstructions to $\CSP()$? (Compare to Theorem~\ref{thm:TCn-min-obs}).

— Restricted CSPs and F-free Digraph Algorithmics  (2502.17596 - Guzmán-Pro et al., 24 Feb 2025) in Conclusion and outlook, Section 8