Is there a term for this graph construction? I replace several vertices by a single one. I want to study a graph $\mathcal G=(\mathcal V,\mathcal E)$, which emerges from a simple graph $G=(V,E)$ by considering certain vertex sets in $G$ as single vertices. In detail: I choose a partition $V_1\,\dot\cup\,\cdots\,\dot\cup\, V_n=V$ and define
$$\mathcal V=\{V_1,...,V_n\},\qquad \mathcal E=\{\{V_1,V_2\}\mid \text{$E$ contains an edge between $V_1$ and $V_2$}\}.$$
To me, $\mathcal G$ looks like a graph minor of $G$, but 1) we can contract also edges which are not there, 2) I am not allowed to delete edges. Is there a term for such a process or such a graph $\mathcal G$ (cluster graph, block graph, ...)?

The term you are looking for is Quotient Graph.
