We present a data structure that for a dynamic graph $G$ that is updated by edge insertions and deletions, maintains a tree decomposition of $G$ of width at most $6k+5$ under the promise that the treewidth of $G$ never grows above $k$. The amortized update time is ${\cal O}_k(2^{\sqrt{\log n}\log\log n})$, where $n$ is the vertex count of $G$ and the ${\cal O}_k(\cdot)$ notation hides factors depending on $k$. In addition, we also obtain the dynamic variant of Courcelle's Theorem: for any fixed property $\varphi$ expressible in the $\mathsf{CMSO}_2$ logic, the data structure can maintain whether $G$ satisfies $\varphi$ within the same time complexity bounds. To a large extent, this answers a question posed by Bodlaender [WG 1993].

翻译：我们提出一种数据结构，用于处理通过边插入和删除更新的动态图 $G$，在保证 $G$ 的树宽从不超过 $k$ 的前提下，维护 $G$ 的一个宽度至多为 $6k+5$ 的树分解。均摊更新时间为 ${\cal O}_k(2^{\sqrt{\log n}\log\log n})$，其中 $n$ 是 $G$ 的顶点数，且 ${\cal O}_k(\cdot)$ 记号隐藏了依赖于 $k$ 的因子。此外，我们还得到了库尔塞勒定理的动态变体：对于任何可在 $\mathsf{CMSO}_2$ 逻辑中表达的固定性质 $\varphi$，该数据结构能在相同的时间复杂度范围内维护 $G$ 是否满足 $\varphi$。这在很大程度上回答了 Bodlaender [WG 1993] 提出的一个问题。