🤖 AI Summary
This study addresses the problem of maintaining k-edge connectivity in dynamic undirected graphs under single-edge insertions or deletions. It is the first to model this task as a structural regulation problem, proposing an active maintenance framework that integrates redundant edge elimination and connectivity restoration. The approach combines Nagamochi–Ibaraki sparse certificates, Link-Cut Trees, and Dinic’s algorithm to achieve efficient updates on sparse graphs. A key innovation is the introduction of a local augmentation strategy that avoids trivial fallbacks. Theoretical analysis shows that redundant edge elimination can be performed in O(k log n) amortized time, while connectivity restoration completes in O(k·n^{5/3}) time, significantly improving the efficiency of dynamic k-edge connectivity maintenance.
📝 Abstract
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 $\lambda(G) \ge k$. 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.