Finding an induced subdivision of a digraph

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.

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