Descendant-homogeneous digraphs

The descendant set desc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant-homogeneous digraphs with finite out-valency, specially those which are also highly arc-transitive. We show that these digraphs must be imprimitive. In particular, we study those which can be mapped homomorphically onto Z and show that their descendant sets have only one end. There are examples of descendant-homogeneous digraphs whose descendant sets are rooted trees. We show that these are highly arc-transitive and do not admit a homomorphism onto Z. The first example [5] known to the authors of a descendant-homogeneous digraph (which led us to formulate the definition) is of this type. We construct infinitely many other descendanthomogeneous digraphs, and also uncountably many digraphs whose descendant sets are rooted trees but which are descendant-homogeneous only in a weaker sense, and give a number of other examples.