Incremental $2$-Edge-Connectivity in Directed Graphs

TL;DR

This paper introduces the first dynamic algorithm for maintaining 2-edge connectivity in directed graphs, enabling efficient updates and queries after edge insertions, with optimal or near-optimal time complexities.

Contribution

It presents a novel incremental algorithm for 2-edge connectivity in directed graphs with efficient update and query times, filling a gap in dynamic graph connectivity research.

Findings

Supports constant-time connectivity queries with witnesses.

Updates 2-edge-connected blocks in O(mn) total time.

First dynamic solution for 2-edge connectivity in directed graphs.

Abstract

In this paper, we initiate the study of the dynamic maintenance of $2$-edge-connectivity relationships in directed graphs. We present an algorithm that can update the $2$-edge-connected blocks of a directed graph with $n$ vertices through a sequence of $m$ edge insertions in a total of $O(mn)$ time. After each insertion, we can answer the following queries in asymptotically optimal time: (i) Test in constant time if two query vertices $v$ and $w$ are $2$-edge-connected. Moreover, if $v$ and $w$ are not $2$-edge-connected, we can produce in constant time a "witness" of this property, by exhibiting an edge that is contained in all paths from $v$ to $w$ or in all paths from $w$ to $v$. (ii) Report in $O(n)$ time all the $2$-edge-connected blocks of $G$. To the best of our knowledge, this is the first dynamic algorithm for $2$-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.