  We consider the following problem for oriented graphs and digraphs: Given an
oriented graph (digraph) $G$, does it contain an induced subdivision of a
prescribed digraph $D$? The complexity of this problem depends on $D$ and on
whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We
give a number of examples of polynomial instances as well as several
NP-completeness proofs.
