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.

翻译：在深入理解遗传图类的一阶转导层次结构的探索中，某些问题尤为突出；例如，作为平面图（及类似图类）的一阶转导的图类具有哪些性质？在探讨这个（迄今为止仍非常开放的）问题时，我们转向乘积结构的概念——即由 Dujmovic 等人 [JACM 2020] 引入的，作为一条路径与一个有界树宽图的强乘积的子图。具体而言，我们证明：任何图类，如果它是某个承认此类乘积结构的图类的一阶转导，那么在扰动意义下，该图类也满足一个结构描述，该描述以一种稠密遗传的方式推广了乘积结构的概念——后一个概念是 Hlineny 和 Jedelsky 最近在 H-团宽 [MFCS 2024] 的名称下引入的。利用这一刻画，我们证明了三维网格图类，以及一类特定的二维网格变体图类，都不是承认乘积结构的图类的一阶转导，特别地，也不是平面图类的一阶转导。