# Connected subgraph of infinite graph with surjective homormophism, but no graph isomorphism

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 $$G[S]\not\cong G$$?

• Let $S$ be the infinite tree where each node has $3$ descendants; let $G$ be the tree where the first node has $2$ descendants, each isomorphic to $S$. – user44191 Feb 25 at 22:41

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 $$1$$.