Characterization of directed graphs attaining the extremal orientation bound

Determine for which directed graphs H the equality D(n,H)=2^{ex(n,H)} holds.

Background

The paper studies D(G,H), the number of orientations of a graph G that avoid a directed graph H, and D(n,H), the maximum of this quantity over all n-vertex graphs. The universal lower bound D(n,H)≥2^{ex(n,H)} follows by taking an H-free underlying graph and orienting all its edges arbitrarily. The problem asks for a complete characterization of when this elementary lower bound is sharp. The paper establishes equality for several classes, including tournaments, certain orientations of edge-critical graphs, and selected fans, but does not resolve the characterization in general.