3D-grids are not transducible from planar graphs

TL;DR

This paper proves that 3D-grid graphs cannot be obtained from planar graphs or graphs of bounded Euler genus through transduction, introducing slice decompositions as a key structural tool.

Contribution

The paper establishes a fundamental limitation on transducing 3D-grid graphs from planar graphs and introduces slice decompositions for structural analysis.

Findings

3D-grid graphs are not transducible from planar graphs

Slice decompositions are introduced as a new structural tool

Graph classes transducible from bounded Euler genus are perturbations of classes with 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.