Infinite primitive directed graphs

A group G of permutations of a set Ω is primitive if it acts transitively on Ω, and the only G-invariant equivalence relations on Ω are the trivial and universal relations. A digraph Γ is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of Γ, such that the induced digraph Γ∖{α} is not connected. If Γ has connectivity one, a lobe of Γ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Primitive graphs (and thus digraphs) with connectivity one are necessarily infinite. The primitive graphs with connectivity one have been fully classified by Jung and Watkins: the lobes of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a primitive digraph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these digraphs, and obtain a complete characterisation