Sometimes it's still unknown for how to determine homeomorphism or isomorphism of graph in infinite case. I have two problem that I don't know how to ask.
1) Consider two graphs $G$ and $H$. The first one is $V(G) = \{0,1,2,3\dots\}$ and $E(G)=\{ (i,i+1)\}_{i \in \mathbb{Z}_{\ge 0}}$. The second one is $V(H) = V(G)$ and $E(H) = \{(0,i)\}_{i \in \mathbb{N}}$
2) Consider the same graph $G$ and $M$, such as $V(G)=V(M)$ and $E(M) = E(G)/(1,2) \cup (0,2)$
Now we want to know if graphs are homeomorphic or not ?
I guess in first case the question is no. Their plane picture looks like line and infinite star. So we can't make even a isomorphism between them.
For a second part I have a doubt. They have so much in common, but this erasing of edge make me believe that make some sense. However there are subgraphs that $M' \cong G$ and $G' \cong M$.
Any help will be appreciated.
In the second case, the graphs are isomorphic. The isomorphism exchanges $0$ and $1$ and leaves all other points fixed.