Classification of graphs admitting an optimal orientation bound

Classify all undirected graphs H for which there exists an orientation \vec{H} satisfying D(n,\vec{H})=2^{ex(n,H}) for sufficiently large n.

Background

The quantity D(n,H) measures the largest number of H-free orientations over all n-vertex host graphs, while ex(n,H) is the ordinary extremal number of the underlying undirected graph. The baseline construction always gives D(n,H) at least 2{ex(n,H)}, but the paper shows that equality can fail for some directed graphs.

The stated problem asks for a classification at the level of underlying graphs: determine precisely which graphs possess at least one orientation for which the baseline extremal bound is attained. The paper does not resolve this general classification.

References

For which graphs $H$ is there an orientation $H$ such that $$D(n, H) = 2{ex(n, H)}?$$

— Orientations of graphs omitting non-edge-critical directed graphs  (2502.21287 - Sheats, 28 Feb 2025) in Section 5, subsection “Questions and open problems,” Problem \ref{classification_prob}