# 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, ...)?

• If each subgraph $V_j$ is connected, this is the same thing as contracting all edges in $V_j$ and removing loops and multiedges. – N. S. Mar 28 '18 at 16:34
• Replacing any one set of vertices with a single vertex (and collapsing multiple edges into a single edge) is call vertex identification (or vertex contraction) – Joffan Apr 2 '18 at 21:29

In topology, if you have a topological space, you can create a new space by considering some points in the space to be the same. (For example, if you have $[0,1]$ with standard subspace topology, then you form a new space, say by considering $0$ and $1$ to be the same. This will be homeomorphic to a circle $S^1$)