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 ${\cal O}_{\varphi,k}(\log n)$. 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：$\ varphie$ 在图形的签名上, 值 $\ mathsf{ CMSO%% 2 的值是 $\ mathsf{ CMSO% 2 的值 。 我们展示了一个动态数据结构, 对于某个通过边缘插入和边缘删除更新的图形$G$, 维持对美元是否满意的G$。 数据结构只有在反馈的顶点数不超过固定的不变值$G$时, 才会正确报告结果, 否则它报告反馈的顶点数太大 。 根据这个假设, 我们保证对更新时间进行摊销 $\ cal O ⁇ varphi, k} (\log n) 。 通过将这个结果与典型的Erd\ H{ o}s和 P\\ osa 的理论结合, 我们给出一个完全动态的数据结构, 以维持一个图表是否包含 $k$k verex- dission 的包装, 和 $\\\\ log n 更新时间的 。 我们的数据结构还将在更大的普通结构中工作。