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

Let $G=(\omega,E)$ be a connected simple, undirected graph, where $E\subseteq [\omega]^2$.. For $m<n\in \omega$ we let $G/_{\{m,n\}}$ be the graph that we get from removing $n$, and $m$ gets all the neighbours from $n$. Formally, let $$G/_{\{m,n\}} = \big(\omega\setminus \{n\},\; (E\cap [\omega\setminus\{n\}]^2)\cup\{\{m, x\}: \{n, x\}\in E\} \land x\neq m\big).$$

We say that $G=(\omega, V)$ has the *collapse-property* if $G\cong G/_{\{m,n\}}$ for all $m<n\in\omega$.

What is an example of a non-complete, connected graph $G=(\omega,E)$ having the collapse-property?