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Let $\varphi$ be a sentence of $\mathsf{CMSO}_2$ (monadic second-order logicwith quantification over edge subsets and counting modular predicates) over thesignature 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 correctlyreport the outcome only when the feedback vertex number of $G$ does not exceeda fixed constant $k$, otherwise it reports that the feedback vertex number istoo large. With this assumption, we guarantee amortized update time ${\calO}_{\varphi,k}(\log n)$.  By combining this result with a classic theorem of Erd\H{o}s and P\'osa, wegive a fully dynamic data structure that maintains whether a graph contains apacking of $k$ vertex-disjoint cycles with amortized update time ${\calO}_{k}(\log n)$. Our data structure also works in a larger generality ofrelational structures over binary signatures.

Maintaining $\mathsf{CMSO}_2$ properties on dynamic structures with bounded feedback vertex number