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Existential Positive Transductions of Sparse Graphs

Nikolas Mählmann, Sebastian Siebertz

TL;DR

The paper advances the sparsification program for graph classes by introducing existential positive transductions (EP-transductions) and a new combinatorial operation, subflips, to capture the semi-ladder-free fragment of monadic dependence. It shows that co-matching-free, monadically stable classes admit EP-sparsifications from nowhere dense classes, with the dense preimages realized as subgraphs, and provides strong structural tools (subflip-flatness, subflipper-rank) to characterize and compute these sparsifications. The results recover and extend known cases (shrub-depth, bounded width notions, and merge-width) and deliver a canonical sparsification into subgraphs with bounded tree-depth or related parameters, enabling algorithmic applications such as kernelization for domination and independence problems. Additionally, the work reveals a MSO-to-FO collapse for existentially positive fragments on relational structures, highlighting a unifying simplification in expressive power. Overall, the paper strengthens the link between logical transductions, sparsity, and algorithmic tractability in both sparse and certain dense graph classes.

Abstract

Monadic stability generalizes many tameness notions from structural graph theory such as planarity, bounded degree, bounded tree-width, and nowhere density. The sparsification conjecture predicts that the (possibly dense) monadically stable graph classes are exactly those that can be logically encoded by first-order (FO) transductions in the (always sparse) nowhere dense classes. So far this conjecture has been verified for several special cases, such as for classes of bounded shrub-depth, and for the monadically stable fragments of bounded (linear) clique-width, twin-width, and merge-width. In this work we propose the existential positive sparsification conjecture, predicting that the more restricted co-matching-free, monadically stable classes are exactly those that can be transduced from nowhere dense classes using only existential positive FO formulas. While the general conjecture remains open, we verify its truth for all known special cases of the original conjecture. Even stronger, we find the sparse preimages as subgraphs of the dense input graphs. As a key ingredient, we introduce a new combinatorial operation, called subflip, that arises as the natural co-matching-free analog of the flip operation, which is a central tool in the characterization of monadic stability. Using subflips, we characterize the co-matching-free fragment of monadic stability by appropriate strengthenings of the known flip-flatness and flipper game characterizations for monadic stability. In an attempt to generalize our results to the more expressive MSO logic, we discover (rediscover?) that on relational structures (existential) positive MSO has the same expressive power as (existential) positive FO.

Existential Positive Transductions of Sparse Graphs

TL;DR

The paper advances the sparsification program for graph classes by introducing existential positive transductions (EP-transductions) and a new combinatorial operation, subflips, to capture the semi-ladder-free fragment of monadic dependence. It shows that co-matching-free, monadically stable classes admit EP-sparsifications from nowhere dense classes, with the dense preimages realized as subgraphs, and provides strong structural tools (subflip-flatness, subflipper-rank) to characterize and compute these sparsifications. The results recover and extend known cases (shrub-depth, bounded width notions, and merge-width) and deliver a canonical sparsification into subgraphs with bounded tree-depth or related parameters, enabling algorithmic applications such as kernelization for domination and independence problems. Additionally, the work reveals a MSO-to-FO collapse for existentially positive fragments on relational structures, highlighting a unifying simplification in expressive power. Overall, the paper strengthens the link between logical transductions, sparsity, and algorithmic tractability in both sparse and certain dense graph classes.

Abstract

Monadic stability generalizes many tameness notions from structural graph theory such as planarity, bounded degree, bounded tree-width, and nowhere density. The sparsification conjecture predicts that the (possibly dense) monadically stable graph classes are exactly those that can be logically encoded by first-order (FO) transductions in the (always sparse) nowhere dense classes. So far this conjecture has been verified for several special cases, such as for classes of bounded shrub-depth, and for the monadically stable fragments of bounded (linear) clique-width, twin-width, and merge-width. In this work we propose the existential positive sparsification conjecture, predicting that the more restricted co-matching-free, monadically stable classes are exactly those that can be transduced from nowhere dense classes using only existential positive FO formulas. While the general conjecture remains open, we verify its truth for all known special cases of the original conjecture. Even stronger, we find the sparse preimages as subgraphs of the dense input graphs. As a key ingredient, we introduce a new combinatorial operation, called subflip, that arises as the natural co-matching-free analog of the flip operation, which is a central tool in the characterization of monadic stability. Using subflips, we characterize the co-matching-free fragment of monadic stability by appropriate strengthenings of the known flip-flatness and flipper game characterizations for monadic stability. In an attempt to generalize our results to the more expressive MSO logic, we discover (rediscover?) that on relational structures (existential) positive MSO has the same expressive power as (existential) positive FO.
Paper Structure (27 sections, 66 theorems, 71 equations, 10 figures)

This paper contains 27 sections, 66 theorems, 71 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On self-loops
  4. Self-loops in general transductions
  5. Self-loops in positive transductions
  6. Subflips
  7. Flips and partitions
  8. Subflips
  9. Symmetry
  10. Transitivity
  11. Hereditariness
  12. Approximating flips through subflips
  13. Subflip-flatness
  14. Subflipper-rank
  15. Definition and basic properties
  16. ...and 12 more sections

Key Result

Theorem 1.3

conjecture:stable-refined is true for the class property bounded $\mathcal{P}$ for all In particular, for every half-graph-free class $\mathscr{C}$, all of the following hold.

Figures (10)

  • Figure 1: Example of a transduction. On the very left: the graph $G$. On the very right: a graph $H \in \mathsf{T}_\varphi(G)$ for the formula $\varphi(x,y) = (\textnormal{dist}(x,y) = 3) \vee (\textnormal{Red}(x) \wedge \textnormal{Red}(y))$.
  • Figure 2: A hierarchy of properties of graph classes. An arrow $P_1 \rightarrow P_2$ between two properties means that every graph class that has property $P_1$ also has property $P_2$.
  • Figure 3: A hierarchy of properties of graph classes. An arrow $P_1 \rightarrow P_2$ between two properties means that every graph class that has property $P_1$ also has property $P_2$. The highlighted semi-ladder-free column represents our results on the existential positive sparsification conjecture.
  • Figure 4: Two graphs that are $3$-flips of each other. This is witnessed by the partition $\{ P_1,P_2,P_3 \}$ where the pairs $(P_1,P_2)$, $(P_2,P_3)$, and $(P_3,P_3)$ were flipped.
  • Figure 5: A star and the graph obtained by applying the transduction $\mathsf{T}_\varphi$ to it.
  • ...and 5 more figures

Theorems & Definitions (134)

  • Definition 1.0
  • Conjecture 1.1: see POM21
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5: $\circlearrowleft$
  • Conjecture 1.6: $\circlearrowleft$
  • Theorem 1.7
  • Corollary 1.8
  • proof
  • ...and 124 more