Classification of graphs admitting a bound-attaining orientation

Characterize 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) concerns orientations avoiding a directed graph, whereas ex(n,H) is the ordinary extremal number of the underlying graph. The paper gives examples of orientations for which equality with 2^{ex(n,H)} holds and also constructs a family of graphs, namely sufficiently large even-parameter odd wheels, for which no orientation has this property. The stated problem asks for a complete classification between these two behaviors.