I am stuck on a problem on connectivity (problem 3.10 in Diestel). Given a three connected graph $G$ not $K⁴$ and any edge $e$ of $G$ I want to show that either $G∸e$ or $G/e$ is again 3-connected.
Following the hint from Diestel I would try to show it by contradiction. So assuming both $G∸e$ and $G/e$ are not 3-connected we get that there are vertices $x,y\in V(G∸e)$ and $u,w\in V(G/e)$ such that $G∸e-\{x,y\}$ and $G/e-\{u,w\}$ are disconnected. Then I think we can say that either $u$ or $w$ must be the contracted vertex $v_e$ because else $\{u,w\}$ would be a separating set in $G$. Also, neither $x$ nor $y$ can be an end point of $e$ because then $\{x,y\}$ would be a separating set in $G$. But here I'm stuck.
If anyone knows how to continue from here or has another approach please share, this question has been bugging me for a while now.