Let $G$ be a connected graph s.t. no two nodes of degree 1 share a neighbour.
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In this graph there are two adjacent edges, s.t. you can remove them and your resulting graph still has only one non-trivial component.
We even have proved a bunch of interesting properties for graphs. (Say correlations of the numbers of edges and vertices, also equivalent statements to being a tree..) It seems to me that this should be a relatively easy consequence of them. But I have no good idea. Maybe one can reformulate the problem so that its a little clearer...
Any ideas/hints are appreciated! :)