Suppose $G$ is a finite simple graph. Let $h(G)$ denote the *Hadwiger number* of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$. 


What is an example of a graph $G\_0=(V\_0, E\_0)$ containing two non-adjacent vertices $v, w\in V\_0$ such that when $v$ and $w$ are identified, the Hadwiger number of the resulting graph is smaller than $h(G\_0)$?



Identifying two non-adjacent vertices and the effect on the Hadwiger number
Identify two opposite vertices of the cycle graph $C\_4$. This reduces the Hadwiger number from 3 to 2.


