Transductions of Graph Classes Admitting Product Structure

TL;DR

This paper investigates the properties of graph classes that are first-order transductions of classes with product structures, providing structural characterizations and demonstrating limitations for certain classes like 3D grids.

Contribution

It introduces a new structural description for graph classes that are first-order transductions of classes with product structures, extending understanding of the transduction hierarchy.

Findings

3D grids are not first-order transducible from classes with product structure.

Certain modifications of 2D grids cannot be obtained via first-order transductions from classes with product structure.

Classes with product structure include classes transducible from planar graphs.

Abstract

In a quest to thoroughly understand the first-order transduction hierarchy of hereditary graph classes, some questions in particular stand out; such as, what properties hold for graph classes that are first-order transductions of planar graphs (and of similar classes)? When addressing this (so-far wide open) question, we turn to the concept of a product structure - being a subgraph of the strong product of a path and a graph of bounded tree-width, introduced by Dujmovic et al. [JACM 2020]. Namely, we prove that any graph class which is a first-order transduction of a class admitting such product structure, up to perturbations also meets a structural description generalizing the concept of a product structure in a dense hereditary way - the latter concept being introduced just recently by Hlineny and Jedelsky under the name of H-clique-width [MFCS 2024]. Using this characterization, we show that the class of the 3D grids, as well as a class of certain modifications of 2D grids, are not first-order transducible from classes admitting a product structure, and in particular not from the class of planar graphs.