# Term for the contraction of all the edges of a (connected) induced subgraph?

Let G be a graph that contains at least one connected subset of vertices.

Let S be a connected subset of vertices of G.

Let G[S] denote the induced subgraph of S in G.

Let's consider an operation that transforms G to another graph G' by contracting all the edges of G[S].

Informally speaking, it consists of "condensing" or "contracting" G[S] to one vertex (always possible as S is connected), while conserving all the edges that connect a vertex in S to a vertex outside of S.

Is there a particular name used to call such a type of operation? May one say this is the contraction of S in G, or is there any other more appropriate term, if any?

Supplementary question: Is there a particular name used to call the relationship between G and G'?

• Welcome to MSE! Let me know if there's an itch my answer doesn't scratch. Commented Apr 7, 2021 at 11:25

This is called a quotient map in other contexts, and I would write it as $$G/S$$ to indicate that I've used an equivalence relation to identify all vertices in $$S$$, or $$G/{\sim}$$ if instead I specify an equivalence relation denoted $${\sim}$$.
As for what to call $$G/S$$? In topology, you might see it called the quotient space; in graph theory, it seems it is called the quotient graph.