Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Preservation theorems on sparse classes revisited

Anuj Dawar, Ioannis Eleftheriadis

TL;DR

This work revisits homomorphism preservation for sparse classes and shows that the classical addability requirement is not sufficient for the previously used proof strategies. It introduces a stronger closure, amalgamation over bottlenecks, and proves a corrected theorem: homomorphism preservation holds for hereditary, quasi-wide classes that are closed under bottleneck amalgamation. The authors construct a counterexample with bounded treewidth to demonstrate the necessity of the stronger condition and also prove non-preservation for planar (and more generally $K_5$-minor-free) graphs, clarifying the limits of the approach. The results refine our understanding of preservation phenomena in sparse classes and connect to extension preservation, offering guidance for identifying natural classes where these properties hold.

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].

Preservation theorems on sparse classes revisited

TL;DR

This work revisits homomorphism preservation for sparse classes and shows that the classical addability requirement is not sufficient for the previously used proof strategies. It introduces a stronger closure, amalgamation over bottlenecks, and proves a corrected theorem: homomorphism preservation holds for hereditary, quasi-wide classes that are closed under bottleneck amalgamation. The authors construct a counterexample with bounded treewidth to demonstrate the necessity of the stronger condition and also prove non-preservation for planar (and more generally -minor-free) graphs, clarifying the limits of the approach. The results refine our understanding of preservation phenomena in sparse classes and connect to extension preservation, offering guidance for identifying natural classes where these properties hold.

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].
Paper Structure (6 sections, 17 theorems, 25 equations, 4 figures)

This paper contains 6 sections, 17 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preservation can fail on classes of small treewidth
  4. Preservation under bottleneck amalgamation
  5. Homomorphism preservation fails on planar graphs
  6. Conclusion

Key Result

Theorem 2.1

Every first-order sentence of quantifier rank $q$ is equivalent to a Boolean combination of basic local sentences of locality radius $7^q$.

Figures (4)

  • Figure 1: The bouquet of cycles of type $(6,9,10)$ and the wheel of order $9$ respectively.
  • Figure 2: Planar embeddings of $G_9$ and $D_9$ respectively.
  • Figure 3: Planar embeddings of $A_6, B_6,$ and $C_6$ respectively.
  • Figure :

Theorems & Definitions (34)

  • Theorem 2.1: Gaifman Locality
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 24 more