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Extension preservation on dense graph classes

Ioannis Eleftheriadis

TL;DR

This work initiates preservation-theoretic study for dense graph classes by (i) showing extension preservation fails on natural dense families with low complexity, notably graphs of cliquewidth at least $4$, and (ii) proving a positive dense analogue using strongly flip-flat classes and flip-sums. The key method blends Gaifman locality with a new flip-based wideness notion, enabling extension preservation to hold over hereditary strongly flip-flat classes closed under flip-sums over bottleneck partitions. The results also connect to transductions of uniformly almost-wide classes, providing a broad positive landscape, while clarifying the limits imposed by interwoven linear orders that drive the negative examples. Overall, the paper extends the sparsity-driven preservation program into the dense regime, outlining concrete structural conditions and offering directions for extending to further dense-tame graph classes and resolving open questions about cliquewidth thresholds and twin-width.

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].

Extension preservation on dense graph classes

TL;DR

This work initiates preservation-theoretic study for dense graph classes by (i) showing extension preservation fails on natural dense families with low complexity, notably graphs of cliquewidth at least , and (ii) proving a positive dense analogue using strongly flip-flat classes and flip-sums. The key method blends Gaifman locality with a new flip-based wideness notion, enabling extension preservation to hold over hereditary strongly flip-flat classes closed under flip-sums over bottleneck partitions. The results also connect to transductions of uniformly almost-wide classes, providing a broad positive landscape, while clarifying the limits imposed by interwoven linear orders that drive the negative examples. Overall, the paper extends the sparsity-driven preservation program into the dense regime, outlining concrete structural conditions and offering directions for extending to further dense-tame graph classes and resolving open questions about cliquewidth thresholds and twin-width.

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].
Paper Structure (5 sections, 13 theorems, 44 equations, 3 figures)

This paper contains 5 sections, 13 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Failure of preservation on graphs of cliquewidth 4
  4. Extension preservation on strongly flip-flat classes
  5. 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 (3)

  • Figure 1: The gadget induced by $I(\bar{v},\bar{u},a,b)$.
  • Figure 2: The graph $\mathcal{H}_7$.
  • Figure 3: The graph $H\star_{(F,P)}H$, where $H$ is the half-graph of order $4$, $P$ is the partition into red and blue vertices and $F=\{(1,2),(2,1)\}$.

Theorems & Definitions (31)

  • Theorem 2.1: Gaifman Locality
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 21 more