Diameter of undirected graph

Question

Let $G$ be a strongly connected directed graph of diameter $D$, and suppose that we remove the orientation of the arcs, thus getting an undirected graph $G'$ with diameter $D'$. Obviously, $D' \leq D$. What else can be said about $D$ and $D'$?. In particular, what can be said about $D$ and $D'$ if we know that $G$ is regular, vertex-transitive, or a Cayley graph?

Answer 1

Some answers are in Babai's paper https://people.cs.uchicago.edu/~laci/papers/eulerian-soda06.pdf. In particular, you have a very good bound if $G$ is Cayley, a quite good bound if $G$ is vertex-transitive, and negative results if $G$ is only regular.