Set-homogeneous directed graphs

TL;DR

This paper classifies finite set-homogeneous directed graphs, extending previous work on homogeneous digraphs, and provides initial results on infinite cases under certain restrictions.

Contribution

It offers a complete classification of finite set-homogeneous digraphs and initial insights into infinite cases with specific constraints.

Findings

Finite set-homogeneous digraphs classified

Extension of Lachlan's work on homogeneous digraphs

Initial classification results for infinite set-homogeneous digraphs

Abstract

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption that such pairs of vertices are not allowed, we obtain initial results on countably infinite set-homogeneous digraphs, classifying those which are not 2-homogeneous.