Two-ended quasi-transitive graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation $A{\ast }_{C}B$ or an HNN-extension ${\ast }_{\varphi }C$, where $C$ is a finite group and $[A:C]=[B:C]=2$ and $\phi\in Aut©$. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.