Incremental Strong Connectivity and 2-Connectivity in Directed Graphs

TL;DR

This paper introduces new incremental algorithms for maintaining connectivity structures in directed graphs, specifically focusing on strong connectivity and 2-vertex connectivity, with bounds close to theoretical limits.

Contribution

It presents novel incremental algorithms for all connectivity cuts of size one and 2-vertex-connected components in directed graphs, with tight bounds supported by lower bounds.

Findings

Algorithms maintain connectivity cuts efficiently.

Maintains 2-vertex-connected components in O(mn) time.

Results are near optimal based on lower bounds.

Abstract

In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of those cuts. We give a conditional lower bound that provides evidence that our algorithms may be tight up to sub-polynomial factors. As an additional result, with our approach we can also maintain dynamically the $2$-vertex-connected components of a digraph during any sequence of edge insertions in a total of $O(mn)$ time. This matches the bounds for the incremental maintenance of the $2$-edge-connected components of a digraph.