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Separability Properties of Monadically Dependent Graph Classes

Édouard Bonnet, Samuel Braunfeld, Ioannis Eleftheriadis, Colin Geniet, Nikolas Mählmann, Michał Pilipczuk, Wojciech Przybyszewski, Szymon Toruńczyk

TL;DR

The paper characterizes monadically dependent graph classes as exactly the flip-separable ones, introducing a dense-graph analogue of nowhere-dense separability through a bounded set of vertex flips. It develops a robust toolbox of flip metrics, partition flips, and definable flips, and proves a metric-conversion result showing partition-based locality can be captured by definable flips, enabling local sparsification. The main technical advance is a locality-driven sparsification argument: monadic dependence implies the existence of small sparsifying families that bound radius-$r$ balls by a factor $\varepsilon$, accomplished via flip-breakability, Gaifman locality, and the sunflower lemma. Conversely, flip-separability implies monadic dependence, establishing a tight equivalence. The result provides a foundational link between logical definability and local sparsification in dense graph classes, with potential implications for FO-limits and modelling limits in dense settings.

Abstract

A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every $r\in \mathbb{N}$, $\varepsilon>0$, and every graph $G\in \mathcal{C}$ equipped with a weight function on vertices, one can apply a bounded (in terms of $\mathcal{C},r,\varepsilon$) number of flips (complementations of the adjacency relation on a subset of vertices) to $G$ so that in the resulting graph, every radius-$r$ ball contains at most an $\varepsilon$-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.

Separability Properties of Monadically Dependent Graph Classes

TL;DR

The paper characterizes monadically dependent graph classes as exactly the flip-separable ones, introducing a dense-graph analogue of nowhere-dense separability through a bounded set of vertex flips. It develops a robust toolbox of flip metrics, partition flips, and definable flips, and proves a metric-conversion result showing partition-based locality can be captured by definable flips, enabling local sparsification. The main technical advance is a locality-driven sparsification argument: monadic dependence implies the existence of small sparsifying families that bound radius- balls by a factor , accomplished via flip-breakability, Gaifman locality, and the sunflower lemma. Conversely, flip-separability implies monadic dependence, establishing a tight equivalence. The result provides a foundational link between logical definability and local sparsification in dense graph classes, with potential implications for FO-limits and modelling limits in dense settings.

Abstract

A graph class is monadically dependent if one cannot interpret all graphs in colored graphs from using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every , , and every graph equipped with a weight function on vertices, one can apply a bounded (in terms of ) number of flips (complementations of the adjacency relation on a subset of vertices) to so that in the resulting graph, every radius- ball contains at most an -fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.
Paper Structure (11 sections, 19 theorems, 19 equations, 2 figures)

This paper contains 11 sections, 19 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard graph-theoretic notation
  4. Flips
  5. VC-dimension, transductions, and monadic dependence
  6. Metric conversion
  7. Flip-separability
  8. Locality for various metrics
  9. Flip-breakability
  10. Monadic dependence implies flip-separability
  11. Flip-separability implies monadic dependence

Key Result

Theorem 1

For every nowhere dense graph class $\mathcal{C}$, $r\in\mathrm{\mathbb{N}}$, and $\varepsilon>0$ there is some $k\in\mathrm{\mathbb{N}}$ with the following property. For every $n$-vertex graph $G\in \mathcal{C}$ there is a set $S$ consisting of at most $k$ vertices of $G$ such that every ball of ra

Figures (2)

  • Figure 2: The two types of bipartite graphs $B$ with both $B$ and $\overline{B}$ disconnected (\ref{['lem:bipartite-disco-classification']}).
  • Figure 3: Impossible situations in the proof of \ref{['lem:bipartite-disco-classification']}, in which enough non-edges (drawn with dashes) are found for $\overline{B}$ to be connected.

Theorems & Definitions (21)

  • Theorem 1: structural-sparsity
  • Definition 2
  • Theorem 3
  • Lemma 3: boundedLocalCliquewidthflip-width
  • Lemma 3
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 11 more