Preservation theorems on sparse classes revisited

TL;DR

This paper revisits homomorphism preservation in sparse classes, showing that previous assumptions are insufficient and proposing a stronger condition for the property to hold.

Contribution

It demonstrates that addability alone does not guarantee homomorphism preservation and introduces a stronger amalgamation condition necessary for the property.

Findings

Homomorphism preservation fails in certain classes of bounded treewidth graphs.

Homomorphism preservation fails in the class of planar graphs.

A stronger amalgamation condition restores homomorphism preservation.

Abstract

We revisit the work studying homomorphism preservation for first-order logic in sparse classes of structures initiated in [Atserias et al., JACM 2006] and [Dawar, JCSS 2010]. These established that first-order logic has the homomorphism preservation property in any sparse class that is monotone and addable. It turns out that the assumption of addability is not strong enough for the proofs given. We demonstrate this by constructing classes of graphs of bounded treewidth which are monotone and addable but fail to have homomorphism preservation. We also show that homomorphism preservation fails on the class of planar graphs. On the other hand, the proofs of homomorphism preservation can be recovered by replacing addability by a stronger condition of amalgamation over bottlenecks. This is analogous to a similar condition formulated for extension preservation in [Atserias et al., SiCOMP 2008].