Graph Isomorphism is GI-complete, being complete for its own degree, and so are Digraph Isomorphism and DAG Isomorphism: the three problems reduce to one another by first-order reductions, and GI is also the degree of Digraph Isomorphism. A directed graph becomes a simple graph by subdividing every arc three times and attaching a pendant that carries its direction; a directed graph becomes an acyclic one by subdividing every arc twice, the direction surviving as the difference of the two levels. In the other direction simplicity and the validity of the acyclicity witnesses are first-order, so a reduction only tests them. These are problems complete for a class defined by no logic, and conjectured to be neither in PTIME nor NP-complete.