Special form of 3 vertex-connectedness for Graphs with every edge contained in a perfect matching

Question

I am currently struggling with the following problem:

Given a simple, connected Graph $G = (V,E)$ such that every edge is contained in a perfect matching of $G$. Show that for each edge $e \in E$ (of the form $e = \{ v,w\}$), the graph $G[V \setminus \{v,w\}]$ is connected.

The above is quite similar to this however in the linked question we only show 2 connectedness which seems way easier. I have tried using the same technique of using the parity in the components (of the supposed not-connected Graph $G [V \setminus \{v,w\}]$), however I am struggling with the fact, that $v$ (and $w$ with the same reasoning) could be connected with the component $w$ "belongs" to.

Answer 1

You're struggling because the claim is false. The $2\times 3$ grid graph

*----*----*
|    |    |
|    |    |
*----*----*

has the property you want: for every edge, there is some perfect matching that contains it. However, if you delete the endpoints of the middle edge, you're left with $2$ components.