This question was posted a few days ago on the Mathematics StackExchange, but so far it has not been answered. 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? Thx.