  Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced
the Anchored k-Core problem, where the task is for a given graph G and integers
b, k, and p to find an induced subgraph H with at least p vertices (the core)
such that all but at most b vertices (called anchors) of H are of degree at
least k. In this paper, we extend the notion of k-core to directed graphs and
provide a number of new algorithmic and complexity results for the directed
version of the problem. We show that
  - The decision version of the problem is NP-complete for every k>=1 even if
the input graph is restricted to be a planar directed acyclic graph of maximum
degree at most k+2.
  - The problem is fixed parameter tractable (FPT) parameterized by the size of
the core p for k=1, and W[1]-hard for k>=2.
  - When the maximum degree of the graph is at most \Delta, the problem is FPT
parameterized by p+\Delta if k>= \Delta/2.
