We design a randomized data structure that, for a fully dynamic graph $G$ updated by edge insertions and deletions and integers $k, d$ fixed upon initialization, maintains the answer to the Split Completion problem: whether one can add $k$ edges to $G$ to obtain a split graph. The data structure can be initialized on an edgeless $n$-vertex graph in time $n \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$, and the amortized time complexity of an update is $5^k \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$. The answer provided by the data structure is correct with probability $1-\mathcal{O}(n^{-d})$.

翻译：我们设计了一种随机化数据结构，用于处理由边插入和删除操作更新的完全动态图$G$，其中整数$k, d$在初始化时固定。该数据结构维护了分裂完成问题的答案：是否可以通过向$G$添加$k$条边得到一个分裂图。该数据结构可在时间$n \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$内完成对初始无边的$n$顶点图的初始化，每次更新的分摊时间复杂度为$5^k \cdot (k d \cdot \log n)^{\mathcal{O}(1)}$。该数据结构提供的答案正确的概率为$1-\mathcal{O}(n^{-d})$。