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级的过程中，某些问题尤为突出；例如，作为平面图（及类似图类）一阶转导的图类具有哪些性质？在探讨这个（迄今为止仍广泛开放）问题时，我们转向乘积结构的概念——即由Dujmovic等人[JACM 2020]引入的、作为路径和有界树宽图的强乘积子图的结构。具体而言，我们证明了任何作为承认此类乘积结构的图类的一阶转导的图类，在扰动意义下，也满足一种结构描述，该描述以一种稠密遗传的方式推广了乘积结构的概念——后一概念由Hlineny和Jedelsky最近以H-团宽[MFCS 2024]为名引入。利用这一特征，我们证明了3D网格图类，以及一类特定的2D网格变体图类，不能从承认乘积结构的图类一阶转导得到，尤其不能从平面图类转导得到。