Two characterisations of accessible quasi-transitive graphs

We prove two characterisations of accessibility of locally finite quasi-transitive connected graphs. First, we prove that any such graph $G$ is accessible if and only if its set of separations of finite order is an ${\rm Aut}(G)$-finitely generated semiring. The second characterisation says that $G$ is accessible if and only if every process of splittings in terms of tree amalgamations stops after finitely many steps.