If you remove any $2$ vertices from a complete graph, the chromatic number gets decreased by two. (The famous [double-critical graph conjecture](https://arxiv.org/abs/1610.00636) is about the existence of a non-complete graph such that any $2$ connected vertices can be removed such that the chromatic number of the graph decreases by $2$.)


Now there is a kind of graph where you can pick *some* $2$ points and get the chromatic number decreased. Let $C\_{2n+1}$ for $n>2$ be the "circle" graph on $2n+1$ points, and add a "top" point $t$ to it, and connect $t$ to every other point. The chromatic number of that graph is $4$. If you remove $t$ and some other point, then the chromatic number of the resulting graph is $2$. The question is if you can construct a similar example without having to resort to a "top node" $t$ as above, which is connected to everything else. More formally:


**Question.** Given a positive integer $n\in\mathbb{N}$, is there a connected graph $G = (V,E)$ with $\chi(G) \geq n$, having the following properties?


1. For every $v\in V$ there is $w\in V\setminus \{v\}$ such that $\{v,w\}\notin E$, and
2. there are $v\neq w\in V$ such that $\chi(G\setminus\{v,w\}) = \chi(G)-2$.



Decreasing the chromatic number by $2$ by removing $2$ well-chosen vertices
Let $G\_0=(V\_0,E\_0)$ be a complete graph of order $n\ge4$. Choose two distinct points $a,b\in V\_0$ and two distinct points $x,y\notin V\_0$. The graph $G=(V,E)$ with vertex set $V=V\_0\cup\{x,y\}$ and edge set $E=E\_0\cup\{\{a,x\},\{b,y\}\}$ satisfies your requirements with $\chi(G)=n$.


