Descendant-homogeneous digraphs (Q618299)

The paper begins by giving the basic definitions and some background on the reachability relation and property \(Z\) and recalls material on descendant sets in infinite highly arc-transitive digraphs with finite out-valency which are vertex-primitive. It then collects together those parts of the general theory that has been developed, beginning by remarking that any infinite descendant-homogeneous digraph in which the descendant sets are rooted trees of finite valency does not have property \(Z\), has the universal reachability relation, and is imprimitive. It shows how to construct an infinite family of descendant-homogeneous digraphs with descendant sets rooted \(q\)-valent trees omitting a 4-crown and generalizations of this. It gives examples of descendant-homogeneous digraphs, in particular describing a new family of highly arc-transitive digraphs whose descendant sets need not be trees. Some examples of descendant-homogeneous digraphs having directed cycles, based on locally finite distance transitive digraphs, are also given.