We study various aspects of the first-order transduction quasi-order on graph classes, which provides a way of measuring the relative complexity of graph classes based on whether one can encode the other using a formula of first-order (FO) logic. In contrast with the conjectured simplicity of the transduction quasi-order for monadic second-order logic, the FO-transduction quasi-order is very complex, and many standard properties from structural graph theory and model theory naturally appear in it. We prove a local normal form for transductions among other general results and constructions, which we illustrate via several examples and via the characterizations of the transductions of some simple classes. We then turn to various aspects of the quasi-order, including the (non-)existence of minimum and maximum classes for certain properties, the strictness of the pathwidth hierarchy, the fact that the quasi-order is not a lattice, and the role of weakly sparse classes in the quasi-order.

翻译：我们研究图类上一阶转换拟序的各个方面，该拟序提供了一种基于一阶逻辑公式编码能力来衡量图类相对复杂度的方式。与单子二阶逻辑转换拟序所推测的简单性相反，一阶转换拟序具有高度复杂性，结构图论和模型论中的许多标准性质自然呈现其中。我们证明了转换的局部范式及其他一般性结果与构造，并通过若干示例及对某些简单图类转换的特征刻画加以阐释。随后我们探讨该拟序的多个层面，包括特定性质下最小类与最大类的（非）存在性、路径宽度层级的严格性、拟序非格结构的事实，以及弱稀疏类在拟序中的作用。