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Lacon-, Shrub- and Parity-Decompositions: Characterizing Transductions of Bounded Expansion Classes

Jan Dreier

TL;DR

This work develops lacon-, shrub-, and parity-decompositions to characterize structurally bounded expansion classes via first-order transductions, enabling a dense yet tame representation of transduced sparse graphs. The authors prove equivalences between structurally bounded expansion and the existence of these decompositions with bounded generalized coloring numbers, and they show analogous characterizations for structurally bounded treewidth and treedepth. A localized Feferman–Vaught composition theorem provides a locality-based framework that underpins the decomposition approach and allows combining local FO-type information to deduce global properties. The results offer a principled route to lifting tractability from sparse graph classes to their transductions and lay groundwork for future extensions to structurally nowhere dense classes and beyond.

Abstract

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.

Lacon-, Shrub- and Parity-Decompositions: Characterizing Transductions of Bounded Expansion Classes

TL;DR

This work develops lacon-, shrub-, and parity-decompositions to characterize structurally bounded expansion classes via first-order transductions, enabling a dense yet tame representation of transduced sparse graphs. The authors prove equivalences between structurally bounded expansion and the existence of these decompositions with bounded generalized coloring numbers, and they show analogous characterizations for structurally bounded treewidth and treedepth. A localized Feferman–Vaught composition theorem provides a locality-based framework that underpins the decomposition approach and allows combining local FO-type information to deduce global properties. The results offer a principled route to lifting tractability from sparse graph classes to their transductions and lay groundwork for future extensions to structurally nowhere dense classes and beyond.

Abstract

The concept of bounded expansion provides a robust way to capture sparse graph classes with interesting algorithmic properties. Most notably, every problem definable in first-order logic can be solved in linear time on bounded expansion graph classes. First-order interpretations and transductions of sparse graph classes lead to more general, dense graph classes that seem to inherit many of the nice algorithmic properties of their sparse counterparts. In this paper, we show that one can encode graphs from a class with structurally bounded expansion via lacon-, shrub- and parity-decompositions from a class with bounded expansion. These decompositions are useful for lifting properties from sparse to structurally sparse graph classes.

Paper Structure

This paper contains 23 sections, 13 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Introducing Lacon-, Shrub- and Parity-Decompositions
  3. Generalized Coloring Numbers and Bounded Expansion
  4. Main Result
  5. Localized Feferman--Vaught Composition Theorem
  6. Techniques and Outline
  7. Preliminaries
  8. Transductions.
  9. Interpretations of Bounded Expansion Have Directed Lacon-Decompositions
  10. Constructing a Lacon-Decomposition.
  11. Correctness of Construction.
  12. Bounding the Coloring Numbers.
  13. No Apex Vertex.
  14. Directed and Undirected Lacon-Decompositions Have Same Expressive Power
  15. Construction of Lacon-Decomposition.
  16. ...and 8 more sections

Key Result

Theorem 1.6

Let $\mathcal{G}$ be a graph class. The following statements are equivalent.

Figures (3)

  • Figure 1: Hierarchy of selected properties of sparse graph classes and transductions thereof. For all of them, the first-order model-checking problem can be solved in fpt time.
  • Figure 2: Top: A graph $G$. Left: A lacon-decomposition of $G$. The hidden vertices are listed in ascending order from left to right. Middle: A shrub-decomposition of $G$ with two colors and diameter three. Vertices from $G$ are adjacent if they have distance two or distance three and the same color. Right: A parity-decomposition of $G$ with target-degree two.
  • Figure 3: Left: Part of a lacon-decomposition. Right: Corresponding part of a shrub-decomposition, including distances between round and square vertices. We assume $4{\rm col}_1(L,\pi)= 40$.

Theorems & Definitions (34)

  • Definition 1.1: Lacon-decomposition
  • Definition 1.2: Shrub-decomposition
  • Definition 1.3: Shrubdepth de2019shruboriginalshrubdepth
  • Definition 1.4: Parity-Decomposition
  • Definition 1.5: Bounded Expansion van2019uniform
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.8: q-type grohe2008logic
  • Proposition 1.9: grohe2008logic
  • ...and 24 more