We present a dynamic framework for maintaining $k$-edge-connectivity of undirected, simple graphs subject to structural updates, specifically single edge additions and removals. The required edge-connectivity $k$ is a chosen, constant parameter. Unlike standard dynamic graph problems, such as dynamic minimum-cut, which focus solely on reporting the value of the minimum cut, our approach actively modifies the graph $G$ to maintain the edge-connectivity invariant $λ(G) \ge k$. We address two fundamental maintenance tasks: redundancy elimination, which identifies and removes an existing edge rendered redundant for $k$-edge-connectivity by new edge insertion, and connectivity restoration, which computes and inserts a minimal set of augmenting edges to restore graph's $k$-edge-connectivity following an old edge deletion. To preclude trivial reversals, we strictly enforce that the eliminated edge is distinct from the inserted edge and that restoration excludes the already deleted edge. Our solution of the first problem integrates Nagamochi-Ibaraki sparse certificates [Nagamochi and Ibaraki 1992] with Link-Cut Trees [Sleator and Tarjan 1983] to remove redundant edges in $O(k \log n)$ amortized time. For restoration, we propose a localized augmentation strategy that exploits the residual graph structure to bridge the minimum cut. By executing Dinic's [Dinic 1970] algorithm on the sparsified input graph, we identify the minimal edge set required to reconnect the graph in $O(k \cdot n^{5/3})$ time.

翻译：我们提出了一种动态框架，用于维护无向简单图在结构更新（特别是单边添加与删除）下的 $k$-边连通性。所需的边连通度 $k$ 是一个选定的常数参数。与标准的动态图问题（如动态最小割）仅关注报告最小割值不同，我们的方法主动修改图 $G$ 以维持边连通性不变量 $λ(G) \ge k$。我们处理两个基本维护任务：冗余消除——识别并移除因新边插入而变得对 $k$-边连通性冗余的现有边；连通性恢复——在旧边删除后计算并插入最小增广边集以恢复图的 $k$-边连通性。为避免平凡逆转，我们严格规定被消除的边必须不同于插入的边，且恢复过程排除已删除的边。针对第一个问题，我们将 Nagamochi-Ibaraki 稀疏证书 [Nagamochi and Ibaraki 1992] 与 Link-Cut Trees [Sleator and Tarjan 1983] 相结合，以 $O(k \log n)$ 的摊还时间移除冗余边。对于恢复任务，我们提出一种局部化增广策略，利用残差图结构桥接最小割。通过在稀疏化输入图上执行 Dinic [Dinic 1970] 算法，我们以 $O(k \cdot n^{5/3})$ 的时间识别出重连图所需的最小边集。