We give new decomposition theorems for classes of graphs that can be transduced in first-order logic from classes of sparse graphs -- more precisely, from classes of bounded expansion and from nowhere dense classes. In both cases, the decomposition takes the form of a single colored rooted tree of bounded depth where, in addition, there can be links between nodes that are not related in the tree. The constraint is that the structure formed by the tree and the links has to be sparse. Using the decomposition theorem for transductions of nowhere dense classes, we show that they admit low-shrubdepth covers of size $O(n^\varepsilon)$, where $n$ is the vertex count and $\varepsilon>0$ is any fixed~real. This solves an open problem posed by Gajarsk\'y et al. (ACM TOCL '20) and also by Bria\'nski et al. (SIDMA '21).

翻译：我们给出了可以从稀有图表类别 -- -- 更精确地说,从捆绑扩张类别和不稠密类别中 -- -- 以第一阶逻辑从稀薄图表类别 -- -- 推导出来的图表类别分解论。在这两种情况下,分解方式是一棵有色、有色、深层的根树,此外,树上没有关联的结点之间可能存在联系。 制约是树和链接所形成的结构必须稀疏。 使用分解式图解图解,用于不高密度类的转导,我们显示它们接受低层灌木深度覆盖值$O(n ⁇ varepsilon),其中, 美元是脊椎计数, 美元是固定的 ~现实值。 这解决了Gairsk\'y等人(ACM TOCL'20)和Bria\nski等人(SIDMA'21)提出的一个未解决的问题。