Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected component.
So, obviously, the Cayley graph for the group $\mathbb Z \times \mathbb Z$ w.r.t. the standard generating set is an example. Obviously, the Cayley graph for a free group is not an example.
I have a question: what is the name of such a property? Has it been studied?
And the next question: which are the Cayley graphs with this property?