Dynamic framework for edge-connectivity maintenance of simple graphs

We present a framework for dynamically maintaining $k$-edge-connectivity of an undirected simple graph $G$ under edge insertions and deletions, where $k$ is a fixed constant. After an edge insertion, the algorithm identifies and removes a distinct redundant edge to maintain sparsity, in $O(k \log n)$ amortized time. After an edge deletion that reduces $\lambda(G)$ below $k$, the algorithm restores $k$-edge-connectivity by adding at most two new edges (excluding the deleted edge), in $O(k^{3/2} n^{3/2})$ time. The insertion procedure combines Nagamochi-Ibaraki sparse certificates with Link-Cut Trees; the deletion procedure uses a single maximum-flow computation on the sparsified graph. Throughout all updates, the graph is maintained with $O(kn)$ edges.