Vertex-primitive digraphs having vertices with almost equal neighbourhoods

TL;DR

This paper investigates vertex-primitive digraphs with nearly identical neighborhoods, establishing a structural theorem and applying it to resolve a question about synchronising groups.

Contribution

It provides a new structural characterization of certain vertex-primitive digraphs and applies this to a problem in group theory related to synchronization.

Findings

Structural theorem for vertex-primitive digraphs with similar neighborhoods

Resolution of a question on synchronising groups by Araújo and Cameron

Insights into the structure of digraphs with almost equal neighborhoods

Abstract

We consider vertex-primitive digraphs having two vertices with almost equal neighbourhoods (that is, the set of vertices that are neighbours of one but not the other is small). We prove a structural result about such digraphs and then apply it to answer a question of Ara\'ujo and Cameron about synchronising groups.