Let $G=(V,E)$ be a connected graph, $|V| \geq 2$.
I want to show that there are at least $2$ nodes $x,y \in V$ such that $G, G-x$ and $G-y$ have the same number of connected components.
Now, since $G$ is in itself a connected graph, it has exactly $1$ connected component. So if the statement is true, it must be that $G-x$ and $G-y$ also have only one connected component, so they are connected.
But I'm having trouble proving it, and also it seems to defy logic. If it really does turn out that in ANY connected graph there are $2$ nodes that I can remove and still get a connected graph, I would be very surprised.