Dynamic framework for edge-connectivity maintenance of simple graphs

TL;DR

The paper introduces a dynamic framework to actively maintain $k$-edge-connectivity in simple undirected graphs under single-edge updates, ensuring the invariant $\\lambda(G) \ge k$ at all times. It combines Nagamochi-Ibaraki sparse certificates with Link-Cut Trees to perform efficient redundancy elimination after edge insertions, and employs a localized augmentation plus Dinic-based max-flow on a sparsified graph to restore connectivity after deletions. The main contributions are a $O(k \log n)$ amortized update for additions, and a $O(k \cdot n^{5/3})$-time restoration pathway after removals, alongside formal correctness proofs and complexity analyses. This framework enables safe, resource-efficient self-healing in networks, with potential applications in SDN, data centers, and critical infrastructure.

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 $λ(G) \ge k$. 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.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2: Nagamochi92
• Corollary 1
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof