For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x\neq y\in X\big\}$. 


What is an example of a connected graph $G=(V,E)$ and a subset $S\subseteq V$ such that the subgraph $$G[S]:=(S, E\cap [S]^2)$$ is connected, and there is a surjective graph homomorphism $f:G[S]\to G$, but $G[S]\not\cong G$?



Connected subgraph of infinite graph with surjective homormophism, but no graph isomorphism
Let $G$ be the graph obtained by subdividing each edge of $K\_{1,\aleph\_0}$ with one subdivision point, and let $S$ be the set obtained by removing from $V(G)$ one vertex of degree $1$.


