Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph classes through the lens of logic

Michał Pilipczuk

TL;DR

This survey develops a unifying framework for structural graph theory based on First-Order transductions, recasting classic sparsity/density notions (treedepth, shrubdepth, treewidth, cliquewidth, twin-width, bounded expansion, nowhere denseness) as FO-ideals and exploring their dense/sparse dualities. It introduces monadic stability and monadic dependence as logical notions that organize graph classes via transductions, and surveys structural tools and algorithmic consequences (notably FO model-checking) across minor-free, sparse, and structurally sparse classes. The work also outlines two complementary approaches to forming FO ideals—obstruction-based and closure-based—along with the Sparsification Conjecture, which posits a deep link between monadically stable classes and nowhere dense structures. The significance lies in providing a cohesive, logic-centered perspective that connects decomposition theorems, algorithmic tractability, and model theory, guiding future research toward a comprehensive FO-centric theory of graph classes.

Abstract

Graph transformations definable in logic can be described using the notion of transductions. By understanding transductions as a basic embedding mechanism, which captures the possibility of encoding one graph in another graph by means of logical formulas, we obtain a new perspective on the landscape of graph classes and of their properties. The aim of this survey is to give a comprehensive presentation of this angle on structural graph theory. We first give a logic-focused overview of classic graph-theoretic concepts, such as treedepth, shrubdepth, treewidth, cliquewidth, twin-width, bounded expansion, and nowhere denseness. Then, we present recent developments related to notions defined purely through transductions, such as monadic stability, monadic dependence, and classes of structurally sparse graphs.

Graph classes through the lens of logic

TL;DR

This survey develops a unifying framework for structural graph theory based on First-Order transductions, recasting classic sparsity/density notions (treedepth, shrubdepth, treewidth, cliquewidth, twin-width, bounded expansion, nowhere denseness) as FO-ideals and exploring their dense/sparse dualities. It introduces monadic stability and monadic dependence as logical notions that organize graph classes via transductions, and surveys structural tools and algorithmic consequences (notably FO model-checking) across minor-free, sparse, and structurally sparse classes. The work also outlines two complementary approaches to forming FO ideals—obstruction-based and closure-based—along with the Sparsification Conjecture, which posits a deep link between monadically stable classes and nowhere dense structures. The significance lies in providing a cohesive, logic-centered perspective that connects decomposition theorems, algorithmic tractability, and model theory, guiding future research toward a comprehensive FO-centric theory of graph classes.

Abstract

Graph transformations definable in logic can be described using the notion of transductions. By understanding transductions as a basic embedding mechanism, which captures the possibility of encoding one graph in another graph by means of logical formulas, we obtain a new perspective on the landscape of graph classes and of their properties. The aim of this survey is to give a comprehensive presentation of this angle on structural graph theory. We first give a logic-focused overview of classic graph-theoretic concepts, such as treedepth, shrubdepth, treewidth, cliquewidth, twin-width, bounded expansion, and nowhere denseness. Then, we present recent developments related to notions defined purely through transductions, such as monadic stability, monadic dependence, and classes of structurally sparse graphs.
Paper Structure (30 sections, 83 theorems, 33 equations, 11 figures)

This paper contains 30 sections, 83 theorems, 33 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Overview of the content.
  3. Preliminaries
  4. Graph theory
  5. Logic on graphs
  6. Transductions
  7. Classic concepts
  8. Star-like graphs: treedepth and shrubdepth
  9. Treedepth
  10. Shrubdepth
  11. Path- and tree-like graphs: pathwidth, treewidth, and (linear) cliquewidth
  12. Treewidth and pathwidth
  13. Cliquewidth and linear cliquewidth
  14. Minor-free classes
  15. Twin-width and sparse twin-width
  16. ...and 15 more sections

Key Result

Lemma 1

Let $\varphi(x,y)\in \mathsf{FO}$ be a formula and $\psi\in \mathsf{FO}$ a sentence. Then there is a sentence $\psi[\varphi]\in \mathsf{FO}$ such that for every graph $G$,

Figures (11)

  • Figure 1: Major properties of graph classes discussed in this survey. All properties in the left-most column are weakly sparse, all properties in the remaining three columns are $\mathsf{FO}$ ideals. Arrows represent implications between the properties.
  • Figure 2: Transduction from the proof of \ref{['lem:rook-not-dependent']}. Left: a rook graph with colors $A'$, $B'$, and $F$ depicted in red, yellow, and blue, respectively. Note that every row and every column is a clique, which is depicted using gray ovals. Right: the transduced bipartite graph.
  • Figure 3: Left: A graph (in black) together with its elimination tree of depth $4$ (light blue). Right: a graph (in black) together with its tree-model of depth $3$ that uses $3$ labels. The leaves coincide with the vertices of the graph, their labels are depicted with colors (yellow, red, or blue). The internal nodes are presented as boxes containing those pairs of labels that are mapped by the corresponding function $M_t$ to $1$.
  • Figure 4: Applying a flip on a set $A$, depicted in yellow.
  • Figure 5: A half-graph of order $5$.
  • ...and 6 more figures

Theorems & Definitions (132)

  • Definition 1
  • Lemma 1: Backwards Translation Lemma
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Definition 3
  • Lemma 3
  • proof
  • Definition 4
  • ...and 122 more