Definability in the substructure ordering of finite directed graphs

TL;DR

This paper explores the first-order definability of the substructure ordering of finite directed graphs, revealing complex automorphism groups and demonstrating how adding constants enhances expressive power.

Contribution

It shows that with finitely many constants, the first-order language of substructure ordering can express the embeddability ordering, and it investigates the automorphism group structure.

Findings

First-order language with constants can express the embeddability ordering.

Automorphism group of the substructure ordering is finite and more complex than previously known.

Conjecture on the automorphism group's isomorphism to a specific semidirect product.

Abstract

We deal with first-order definability in the substructure ordering $(\mathcal{D}; \sqsubseteq)$ of finite directed graphs. In two papers, the author has already investigated the first-order language of the embeddability ordering $( \mathcal{D}; \leq)$. The latter has turned out to be quite strong, e.g., it has been shown that, modulo edge-reversing (on the whole graphs), it can express the full second-order language of directed graphs. Now we show that, with finitely many directed graphs added as constants, the first order language of $( \mathcal{D}; \sqsubseteq)$ can express that of $( \mathcal{D}; \leq)$. The limits of the expressive power of such languages are intimately related to the automorphism groups of the orderings. Previously, analogue investigations have found the concerning automorphism groups to be quite trivial, e.g., the automorphism group of $( \mathcal{D}; \leq)$ is isomorphic to $\mathbb{Z}_2$. Here, unprecedentedly, this is not the case. Even though we conjecture that the automorphism group is isomorphic to $(\mathbb{Z}_2^4 \times S_4)\rtimes_{\alpha} \mathbb{Z}_2$, with a particular $\alpha$ in the semidirect product, we only prove it is finite.