Extension preservation on dense graph classes

TL;DR

This paper investigates the applicability of preservation theorems, specifically extension preservation, within dense graph classes, revealing limitations and identifying conditions under which such theorems can hold.

Contribution

It introduces the first study of preservation theorems in dense graph classes and identifies a key condition for extension preservation to be valid in this context.

Findings

Extension preservation fails on most natural dense classes.

A specific technical condition ensures extension preservation in dense classes.

Provides a dense analogue to a known sparse class result.

Abstract

Preservation theorems provide a direct correspondence between the syntactic structure of first-order sentences and the closure properties of their respective classes of models. A line of work has explored preservation theorems relativised to combinatorially tame classes of sparse structures [Atserias et al., JACM 2006; Atserias et al., SiCOMP 2008; Dawar, JCSS 2010; Dawar and Eleftheriadis, 2024]. In this article we initiate the study of preservation theorems for dense graph classes. In contrast to the sparse setting, we show that extension preservation fails on most natural dense classes of low complexity. Nonetheless, we isolate a technical condition which is sufficient for extension preservation to hold, providing a dense analogue to a result of [Atserias et al., SiCOMP 2008].