Is complexity of $GI_{di}$ same as $GI_{un}$?
The Wikipedia page mentions that digraph isomorphism is GI-complete, and gives a reference. For the proof, something along the following lines should work. Replace each $u\to v$ edges with a path $u\to x\to y\to v$, and attack a clique of size $A$ to $x$, and a clique of size $B \neq A$ to $y$. Choose $A,B &gt; n$ to ensure to these are the only cliques of that size. Instead of a clique you can have a long path or a large star.