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 typ...

• Type: Journal
• Year: 2011
• Where: JCT
• Authors: Daniela Amato, John K. Truss
• Keywords: Descendant Sets, Descendant-homogeneous Digraph, Descendanthomogeneous Digraphs, JCT 2011