I am looking for standard terminology for the following operation on a directed graph $G=(V,E)$.
Intuitively, the operation removes a node $v$ while preserving paths via $v$, by adding edges that skip $v$.
Formally, the operation takes $G=(V,E)$ to $G'=(V',E')$ with $V'=V-\{v\}$ and $E'=\{(v_1,v_2) \in E \mid \{v_1,v_2\} \subseteq V'\} \cup \{ (v_1,v_2) \in (V')^2 \mid (v_1,v) \in E \land (v,v_2) \in E \}$.
What is the name of this operation? It sounds a lot like edge contraction, but it is not actually the same.