On the Automorphism Group of the Substructure Ordering of Finite Directed Graphs

TL;DR

This paper explores the automorphism group of the substructure ordering of finite directed graphs, providing computational evidence supporting a specific conjecture about its structure.

Contribution

It confirms the conjectured structure of the automorphism group on initial levels using computer calculations, advancing understanding of the automorphism group of directed graph substructure posets.

Findings

Automorphism group behaves as conjectured on initial levels

Computer analysis of 3160 directed graphs supports the conjecture

Provides partial confirmation of the automorphism group's structure

Abstract

We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group $(\mathbb{Z}_2^4 \times S_4)\rtimes_{\alpha} \mathbb{Z}_2$. Though unable to prove it, we solidify this conjecture by showing that the automorphism group behaves as expected by the conjecture on the first few levels of the poset in question. With the use of computer calculation we analyze the first four levels holding 3160 directed graphs.