Let G be a connected graph with at least two vertices. Prove that G has a vertex v such that if v is removed from G (along with all edges incident with it), the resulting graph is also connected.
Hint: Consider a spanning tree and one of its leaves.
So I know that (using the hint), if the graph is a tree, one of such vertices that has this feature would obviously be a leaf vertex. The problem is, this problem doesn't specifically state to prove this for Trees, but for graphs in general. Any help here as to where to start would be appreciated.