(First-order) transductions are a basic notion capturing graph modifications that can be described in first-order logic. In this work, we propose an efficient algorithmic method to approximately reverse the application of a transduction, assuming the source graph is sparse. Precisely, for any graph class $\mathcal{C}$ that has structurally bounded expansion (i.e., can be transduced from a class of bounded expansion), we give an $O(n^4)$-time algorithm that given a graph $G\in \mathcal{C}$, computes a vertex-colored graph $H$ such that $G$ can be recovered from $H$ using a first-order interpretation and $H$ belongs to a graph class $\mathcal{D}$ of bounded expansion. This answers an open problem raised by Gajarský et al. In fact, for our procedure to work we only need to assume that $\mathcal{C}$ is monadically stable (i.e., does not transduce the class of all half-graphs) and has inherently linear neighborhood complexity (i.e., the neighborhood complexity is linear in all graph classes transducible from $\mathcal{C}$). This renders the conclusion that the graph classes satisfying these two properties coincide with classes of structurally bounded expansion.

翻译：（一阶）转换是描述可由一阶逻辑表达的图修改操作的基本概念。本研究提出一种高效算法方法，可在源图为稀疏图的假设下近似逆推转换过程。具体而言，对于任何具有结构有界扩张性（即可从有界扩张类转换得到）的图类$\mathcal{C}$，我们给出一个$O(n^4)$时间算法：当输入图$G\in \mathcal{C}$时，该算法输出顶点着色图$H$，使得$G$可通过一阶解释从$H$还原，且$H$属于具有有界扩张性的图类$\mathcal{D}$。这解决了Gajarský等人提出的公开问题。实际上，我们的算法仅需假设$\mathcal{C}$是单元稳定的（即不会转换出所有半图类）且具有固有线性邻域复杂度（即从$\mathcal{C}$可转换的所有图类中邻域复杂度均为线性）。由此可得出结论：满足这两个性质的图类恰好与具有结构有界扩张性的图类相一致。