Let $\varphi$ be a sentence of $\mathsf{CMSO}_2$ (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph $G$ that is updated by edge insertions and edge deletions, maintains whether $\varphi$ is satisfied in $G$. The data structure is required to correctly report the outcome only when the feedback vertex number of $G$ does not exceed a fixed constant $k$, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time ${\cal O}_{\varphi,k}(\log n)$. If we additionally assume that the feedback vertex number of $G$ never exceeds $k$, this update time guarantee is worst-case. By combining this result with a classic theorem of Erd\H{o}s and P\'osa, we give a fully dynamic data structure that maintains whether a graph contains a packing of $k$ vertex-disjoint cycles with amortized update time ${\cal O}_{k}(\log n)$. Our data structure also works in a larger generality of relational structures over binary signatures.

翻译：令 $\varphi$ 为定义在图的符号集上的一个 $\mathsf{CMSO}_2$ 句子（即带有对边子集的量化及模计数谓词的二元单子二阶逻辑）。我们提出一种动态数据结构，对于通过边插入和边删除更新的给定图 $G$，该结构能持续维护 $\varphi$ 是否在 $G$ 中被满足。该数据结构仅在 $G$ 的反馈顶点数不超过固定常数 $k$ 时才需要正确报告结果，否则它将报告反馈顶点数过大。在此假设下，我们保证摊销更新时间为 ${\cal O}_{\varphi,k}(\log n)$。若我们进一步假设 $G$ 的反馈顶点数始终不超过 $k$，则该更新时间保证为最坏情况。通过将此结果与 Erd\H{o}s 和 P\'osa 的经典定理相结合，我们给出一个全动态数据结构，该结构能以 ${\cal O}_{k}(\log n)$ 的摊销更新时间维护一个图是否包含 $k$ 个顶点不相交的圈构成的堆积。我们的数据结构也适用于更一般的二元符号集上的关系结构。