Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank $2$. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class $C$ interpretable in a efficiently bounded graph class of tree rank at most $2$, there is a polynomial time algorithm that to any $G \in C$ computes a (sparse) graph $H$ from a fixed graph class of tree rank at most $2$ such that $G = I(H)$ for a fixed interpretation $I$. To the best of our knowledge, this is the first efficient "interpretation reversal" result that generalizes the result of Gajarsk\'y et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.

翻译：有界树秩的图类最近在一阶逻辑的图模型检验问题背景下被引入。这些图类是有界度图类和有界树深度图类的共同推广，并且是有界扩展图类的特例。我们为这些图类引入了一种分解概念，并证明了这些分解可以高效计算。此外，我们的分解的一种自然扩展为有界扩展图类提供了新的刻画和分解（以及计算该分解的高效算法）。然后，我们关注有界树秩图类的解释。我们给出了树秩至多为$2$的图类中可解释图类的刻画。重要的是，我们的刻画导致了一种高效稀疏化过程：对于任何在树秩至多为$2$的高效有界图类中可解释的图类$C$，存在一个多项式时间算法，对于任意$G \in C$，从固定树秩至多为$2$的图类中计算一个（稀疏的）图$H$，使得$G = I(H)$，其中$I$是固定解释。据我们所知，这是首个推广了Gajarsk\'y等[LICS 2016]结果的高效“解释逆变换”结果，该文献对在有界度图类中可解释的图类证明了类似结果。