  An identifying code of a (di)graph $G$ is a dominating subset $C$ of the
vertices of $G$ such that all distinct vertices of $G$ have distinct
(in)neighbourhoods within $C$. In this paper, we classify all finite digraphs
which only admit their whole vertex set in any identifying code. We also
classify all such infinite oriented graphs. Furthermore, by relating this
concept to a well known theorem of A. Bondy on set systems we classify the
extremal cases for this theorem.
