The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes of graphs with structurally bounded expansion , defined as first-order transductions of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth covers (or colorings), replacing treedepth by its dense analogue called shrubdepth.

中文翻译：

---

有界扩展类的一阶解释

有界扩展的概念捕获了图类的均匀稀疏性，并呈现了通常难以处理的各种算法问题。特别是，一阶逻辑的模型检查问题在此类图类上是固定参数可处理的。为了将这些结果推广到密集图，我们引入了图类结构有界的扩展，定义为有界扩展类的一阶转换。作为算法处理的第一步，我们提供了类似于通过低树深覆盖（或着色）来描述有界扩展类的特征，将树深度替换为称为灌木深度的密集类似物。