  We investigate a communication setup where a source output is sent through a
free noisy channel first and an additional codeword is sent through a noiseless
but expensive channel later. With the help of the second message the decoder
should be able to decide with zero-error whether its decoding of the first
message was error-free. This scenario leads to the definition of a digraph
parameter that generalizes Witsenhausen's zero-error rate for directed graphs.
We investigate this new parameter for some specific directed graphs and explore
its relations to other digraph parameters like Sperner capacity and dichromatic
number.
  When the original problem is modified to require zero-error decoding of the
whole message then we arrive back to the Witsenhausen rate of an appropriately
defined undirected graph.
