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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?

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    $\begingroup$ For a graph with finite valency, it is called "one-ended graph" and it's been extensively studied (in the case of Cayley graphs, see for instance en.wikipedia.org/wiki/Stallings_theorem_about_ends_of_groups). If one allows infinite valency vertices, it is sometimes called "one-ended graph" (but there's then an inequivalent alternative definition of one-ended, replacing "finite" with "bounded). $\endgroup$ – YCor Jul 27 '18 at 20:49
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    $\begingroup$ Also, if a group has a finite generating set, and if the Cayley graph for some (equivalently, any) finite generating set is one-ended, then the group is called a one-ended group. $\endgroup$ – Lee Mosher Jul 28 '18 at 0:46
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    $\begingroup$ @YCor You should make this into an answer. The question is legitimate and well-posed, and you give an essentially complete answer, there is no reason for this to remain as a comment. $\endgroup$ – Gro-Tsen Aug 2 '18 at 11:39
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Quoting the answer given by YCor in the comments.

For a graph with finite valency, it is called "one-ended graph" and it's been extensively studied (in the case of Cayley graphs, see for instance Stallings theorem). If one allows infinite valency vertices, it is sometimes called "one-ended graph" (but there's then an inequivalent alternative definition of one-ended, replacing "finite" with "bounded").

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