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

Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets