稀疏图变换的树型分解
Treelike decompositions for transductions of sparse graphs
摘要：
我们给出了新的分解定理的类图，可以转换在一阶逻辑类的稀疏图-更准确地说，从类的有界扩张和无处密集类。在这两种情况下，分解都采用了一个有界深度的单色根树的形式，此外，树中不相关的节点之间可能存在链接。约束是树和链接形成的结构必须是稀疏的。使用无处密集类的转换的分解定理，我们表明，他们承认低shrubdepth覆盖的大小$O(n^\varepsilon)$，其中$n$是顶点计数和$\varepsilon>0$是任何固定的~实数。这解决了由Gajarskeleton等人提出的一个公开问题。（ACM TOCL '20），也由Briaeletski等人提出。（SIMMA'21）。
英文摘要：
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 Gajarský et al. (ACM TOCL '20) and also by Briański et al. (SIDMA '21).
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