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3D-grids are not transducible from planar graphs

Jakub Gajarský, Michał Pilipczuk, Filip Pokrývka

TL;DR

It is proved that the class of 3D-grids cannot be transduced from planar graphs, and more generally from any class of graphs of bounded genus, and it is shown that edge-stable graph classes that admit slice decomposition are transducible from weakly sparse graph classes that admits slice decompositions.

Abstract

We prove that the class of 3D-grids is cannot be transduced from planar graphs, and more generally, from any class of graphs of bounded Euler genus. To prove our result, we introduce a new structural tool called slice decompositions, and show that every graph class transducible from a class of graphs of bounded Euler genus is a perturbation of a graph class that admits slice decompositions.

3D-grids are not transducible from planar graphs

TL;DR

It is proved that the class of 3D-grids cannot be transduced from planar graphs, and more generally from any class of graphs of bounded genus, and it is shown that edge-stable graph classes that admit slice decomposition are transducible from weakly sparse graph classes that admits slice decompositions.

Abstract

We prove that the class of 3D-grids is cannot be transduced from planar graphs, and more generally, from any class of graphs of bounded Euler genus. To prove our result, we introduce a new structural tool called slice decompositions, and show that every graph class transducible from a class of graphs of bounded Euler genus is a perturbation of a graph class that admits slice decompositions.
Paper Structure (14 sections, 21 theorems, 12 equations)

This paper contains 14 sections, 21 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Our results.
  3. Outline of the proof
  4. Preliminaries
  5. Logic
  6. Interpretations and transductions.
  7. Gaifman's theorem
  8. Graph theory
  9. Slice decompositions
  10. Reduction to transductions of bounded range
  11. Flip-structures
  12. Properties of flip structures with $F(G,\mathcal{P},H) = Q_N$
  13. Finishing the proof
  14. Proof Theorem \ref{['thm:main']}

Key Result

Theorem 1

The class of all cubes is not transducible from any class of graphs of bounded genus.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Gaifman's locality theorem gaifman1982local
  • Corollary 4
  • proof
  • Theorem 5: trans_bd_range, also implicit in gajarsky2022differential
  • Theorem 6: cw_sparse_tw
  • Theorem 7: cw_interp
  • Theorem 8: queue
  • Theorem 9: queue
  • ...and 25 more