Dynamic framework for edge-connectivity maintenance of simple graphs

TL;DR

This paper introduces a dynamic framework for maintaining k-edge connectivity in simple graphs efficiently during edge insertions and deletions, combining advanced data structures and flow computations.

Contribution

It presents a novel dynamic algorithm that efficiently updates k-edge connectivity with amortized time bounds, integrating sparse certificates, link-cut trees, and flow algorithms.

Findings

Maintains O(kn) edges throughout updates.

Handles edge insertions in O(k log n) amortized time.

Restores k-edge connectivity after deletions in O(k^{3/2} n^{3/2}) time.

Abstract

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.