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?



Connected graphs on $\omega$ that "doesn't change" whenever 2 points are collapsed
There are many such graphs:


* The [(countable) **random graph**](https://en.wikipedia.org/wiki/Rado_graph).
* The "infinite depth-$1$ tree" given by $E=\{\{0,i\}: i>0\}$.
* The "majestic sunflower" consisting of infinitely many $n$-cycles, for each $n$, joined at a common point.


