Maintaining ${\mathsf{C}\mathsf{M}\mathsf{S}\mathsf{O}}_2$ properties on dynamic structures with bounded feedback vertex number

Let $\phi$ be a sentence of ${\mathsf{C}\mathsf{M}\mathsf{S}\mathsf{O}}_2$ (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 $\phi$ 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.