Classification of underlying graphs admitting a sharp orientation

Determine for which graphs H there exists an orientation H such that D(n,H)=2^{ex(n,H)}.

Background

The preceding results show that some underlying graphs admit orientations for which the extremal orientation count equals the ordinary Turán lower bound, whereas the wheel example in Proposition \ref{wheel_prop} shows that certain graphs admit no such orientation. This problem asks for a classification at the level of undirected underlying graphs, rather than a classification of individual directed graphs.

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}