A Stallings type theorem for quasi-transitive graphs

We consider locally finite, connected, quasi-transitive graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. Our result also leads to a characterisation of accessible graphs. We obtain applications of our results for planar graphs (answering a variant of a question by Mohar in the affirmative) and graphs without thick ends.