An infinite highly arc-transitive digraph
European Journal of Combinatorics
A shorter model theory
Descendants in highly arc transitive digraphs
Discrete Mathematics
Highly arc transitive digraphs: reachability, topological groups
European Journal of Combinatorics
Reachability relations in digraphs
European Journal of Combinatorics
Some constructions of highly arc-transitive digraphs
Combinatorica
Hi-index | 0.00 |
The descendant setdesc(@a) of a vertex @a in a digraph D is the set of vertices which can be reached by a directed path from @a. 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 (Evans (1997) [6]) 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 descendant-homogeneous 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.