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：我们研究了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性的方法。与猜想中单子二阶逻辑转换拟序的简单性相反，一阶转换拟序非常复杂，结构图论和模型论中的许多标准性质自然出现在其中。我们证明了转换的局部范式以及其他一般性结果和构造，并通过若干示例及对一些简单类转换的特征刻画加以说明。随后我们转向该拟序的多个层面，包括特定性质下最小类和最大类的（不）存在性、路径宽度层次结构的严格性、拟序非格结构的事实，以及弱稀疏类在拟序中的作用。