If $G = (V,E)$ is a graph, then a $\omega$-*path* is an injective map $p:\omega\to V$ such that $\{p(k),p(k+1)\}\in E$ for all $k\in \omega$. In a similar fashion, we define a $\mathbb{Z}$-*path*.

Is there a graph $G$ with $V(G) = \omega$ and the following properties?

- There is no surjective $\mathbb{Z}$- or $\omega$-path on $G$;
- For every vertex $v\in V(G) = \omega$, there is a surjective $\mathbb{Z}$- or $\omega$-path on $G\setminus \{v\}$.