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Merge-width and First-Order Model Checking

Jan Dreier, Szymon Toruńczyk

TL;DR

This work introduces merge-width, a unifying spectrum of graph-structural parameters defined via construction sequences, capturing treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. The central technical advance is a locality theorem enabling first-order model checking to be fixed-parameter tractable on graph classes with bounded radius-$r$ merge-width, provided a witnessing construction sequence is given. The authors show that bounded merge-width classes subsume bounded expansion and bounded twin-width (and are closed under first-order interpretations/transductions), while also connecting to flip-width and monadic dependence through almost-bounded variants. The paper combines a Gaifman-type locality framework for distance-augmented logics with a construction-sequence algorithm, yielding a unifying algorithmic meta-theorem with broad implications for structural and algorithmic graph theory. The work also lays groundwork for future research on approximating merge-width, the potential equivalence with flip-width, and the role of merge-width in monadically dependent graph classes.

Abstract

We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositions called construction sequences. These are sequences of ever coarser partitions of the vertex set, where each pair of parts has a specified default connection, and all vertex pairs of the graph that differ from the default are marked as resolved. The radius-$r$ merge-width is the maximum number of parts reached from a vertex by following a path of at most $r$ resolved edges. Graph classes of bounded merge-width -- for which the radius-$r$ merge-width parameter can be bounded by a constant, for each fixed $r=1,2,3,\ldots$ -- include all classes of bounded expansion or of bounded twin-width, thus unifying two central notions from the Sparsity and Twin-width frameworks. Furthermore, they are preserved under first-order transductions, which attests to their robustness. We conjecture that classes of bounded merge-width are equivalent to the previously introduced classes of bounded flip-width. As our main result, we show that the model checking problem for first-order logic is fixed-parameter tractable on graph classes of bounded merge-width, assuming the input includes a witnessing construction sequence. This unites and extends two previous model checking results: the result of Dvořák, Král, and Thomas for classes of bounded expansion, and the result of Bonnet, Kim, Thomassé, and Watrigant for classes of bounded twin-width. Finally, we suggest future research directions that could impact the study of structural and algorithmic graph theory, in particular of monadically dependent graph classes, which we conjecture to coincide with classes of almost bounded merge-width.

Merge-width and First-Order Model Checking

TL;DR

This work introduces merge-width, a unifying spectrum of graph-structural parameters defined via construction sequences, capturing treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. The central technical advance is a locality theorem enabling first-order model checking to be fixed-parameter tractable on graph classes with bounded radius- merge-width, provided a witnessing construction sequence is given. The authors show that bounded merge-width classes subsume bounded expansion and bounded twin-width (and are closed under first-order interpretations/transductions), while also connecting to flip-width and monadic dependence through almost-bounded variants. The paper combines a Gaifman-type locality framework for distance-augmented logics with a construction-sequence algorithm, yielding a unifying algorithmic meta-theorem with broad implications for structural and algorithmic graph theory. The work also lays groundwork for future research on approximating merge-width, the potential equivalence with flip-width, and the role of merge-width in monadically dependent graph classes.

Abstract

We introduce merge-width, a family of graph parameters that unifies several structural graph measures, including treewidth, degeneracy, twin-width, clique-width, and generalized coloring numbers. Our parameters are based on new decompositions called construction sequences. These are sequences of ever coarser partitions of the vertex set, where each pair of parts has a specified default connection, and all vertex pairs of the graph that differ from the default are marked as resolved. The radius- merge-width is the maximum number of parts reached from a vertex by following a path of at most resolved edges. Graph classes of bounded merge-width -- for which the radius- merge-width parameter can be bounded by a constant, for each fixed -- include all classes of bounded expansion or of bounded twin-width, thus unifying two central notions from the Sparsity and Twin-width frameworks. Furthermore, they are preserved under first-order transductions, which attests to their robustness. We conjecture that classes of bounded merge-width are equivalent to the previously introduced classes of bounded flip-width. As our main result, we show that the model checking problem for first-order logic is fixed-parameter tractable on graph classes of bounded merge-width, assuming the input includes a witnessing construction sequence. This unites and extends two previous model checking results: the result of Dvořák, Král, and Thomas for classes of bounded expansion, and the result of Bonnet, Kim, Thomassé, and Watrigant for classes of bounded twin-width. Finally, we suggest future research directions that could impact the study of structural and algorithmic graph theory, in particular of monadically dependent graph classes, which we conjecture to coincide with classes of almost bounded merge-width.

Paper Structure

This paper contains 48 sections, 65 theorems, 74 equations, 6 figures.

Table of Contents

  1. Acknowledgements.
  2. Introduction
  3. First-order model checking.
  4. Flip-width.
  5. Merge-width.
  6. Main result.
  7. Closure under interpretations and transductions.
  8. Almost bounded merge-width.
  9. Combinatorial properties.
  10. Algorithmic simplicity.
  11. Approximating merge-width.
  12. Merge-width and flip-width.
  13. Model checking and monadic dependence.
  14. Proof overview
  15. Outline.
  16. ...and 33 more sections

Key Result

Theorem 1.4

Graph classes of bounded twin-width have bounded merge-width.

Figures (6)

  • Figure 1: Properties of graph classes, and implications ($\mathrel{}$) among them. Each property in the upper row implies the graph class is weakly-sparse -- excludes some biclique $K_{t,t}$ as a subgraph. Each property in the lower row, restricted to weakly sparse graph classes, yields the property in the upper row directly above it. Also, each property in the lower row, apart from almost bounded merge-width/flip-width, is known to be preserved under first-order transductions. Within each of the two larger red boxes, we conjecture that also the converse implications hold, and thus that the three notions are equivalent.
  • Figure 2: Excerpt of a construction sequence of a graph, with a partition $\mathcal{P}=\{A,B,C,D,E,F\}$. The resolved pairs are indicated as (edges) and (non-edges). Recall that unresolved pairs between any two parts are either all adjacent or all non-adjacent (\ref{['remark:maindef']}, item 2). Thus, resolved pairs are indicated in an aggregated way as (edges) and (non-edges) connecting parts, or as fill colors within each part. In this and other pictures, each arrow labeled "resolve" represents a sequence of either positive or negative resolve operations, while each arrow labeled "merge" usually represents a single merge operation. For visual clarity, this figure only shows default connections within or involving parts $C$ and $D$. As discussed in item 3 of \ref{['remark:maindef']}, before we can merge the two parts $C$ and $D$, every part that has a different default connection to $C$ than to $D$ must be fully resolved with either $C$ or $D$. As $E$ has the same default connection () to both $C$ and $D$, no vertex pairs incident to $E$ need to be resolved before the merge. The same holds for $F$. However, both $A$ and $B$ have different default connections to $C$ and to $D$. Thus, to enable the merge, each of the parts $A$ and $B$ need to commit to a default connection to the new part $C \cup D$. The part $A$ commits to and thus has to resolve all missing edges to $C$. Similarly, $B$ commits to and resolves all missing non-edges to $D$. As $D$ has a default self-connection before the merge, but $C \cup D$ commits to default self-connection , the part $D$ also resolves all missing edges within itself. Afterwards, $C$ and $D$ can be merged.
  • Figure 3: A full construction sequence of a graph $G$ witnessing $\mathop{\mathrm{mw}}\nolimits_1(G)\leqslant 3$, using notation from Fig. \ref{['fig:a']}.
  • Figure 4: Left: A graph $G$ consisting of a matching and a complement of a matching. Right: An excerpt from a construction sequence of $G$. First the symmetric difference to the empty default connection (for the matching) and the full default connection (for the complement) is resolved, in a sequence of steps. Each vertex reaches three parts by a path of arbitrary length in the resolved graph. Then the middle column is merged, in a sequence of steps, into a single part. For clarity, default connections are only drawn between neighboring columns. Note that in the context of twin-width (see \ref{['ex:tww']}), such a contraction sequence would result in many parts that are inhomogeneous towards the middle part.
  • Figure 5: A step in a construction sequence of a graph with degeneracy one.
  • ...and 1 more figures

Theorems & Definitions (137)

  • Remark 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.8
  • ...and 127 more