Locally-finite connected-homogeneous digraphs

TL;DR

This paper classifies locally-finite connected-homogeneous digraphs with multiple ends, revealing their structure varies based on the presence of triangles and the number of ends, including tree-like and highly arc-transitive examples.

Contribution

It provides a complete classification of such digraphs, introducing new constructions and characterizing their reachability graphs across different cases.

Findings

Classified digraphs with triangles as tree-like structures.

Proved highly arc-transitivity in triangle-free cases.

Identified reachability graphs for infinite-end digraphs.

Abstract

A digraph is connected-homogeneous if any isomorphism between finite connected induced subdigraphs extends to an automorphism of the digraph. We consider locally-finite connected-homogeneous digraphs with more than one end. In the case that the digraph embeds a triangle we give a complete classification, obtaining a family of tree-like graphs constructed by gluing together directed triangles. In the triangle-free case we show that these digraphs are highly arc-transitive. We give a classification in the two-ended case, showing that all examples arise from a simple construction given by gluing along a directed line copies of some fixed finite directed complete bipartite graph. When the digraph has infinitely many ends we show that the descendants of a vertex form a tree, and the reachability graph (which is one of the basic building blocks of the digraph) is one of: an even cycle, a complete bipartite graph, the complement of a perfect matching, or an infinite semiregular tree. We give examples showing that each of these possibilities is realised as the reachability graph of some connected-homogeneous digraph, and in the process we obtain a new family of highly arc-transitive digraphs without property Z.