2-Edge Connectivity in Directed Graphs

TL;DR

This paper investigates 2-edge connectivity in directed graphs, defining a relation that partitions vertices into blocks, and provides linear-time algorithms to compute this relation and a sparse certificate preserving these blocks.

Contribution

It introduces a linear-time method to compute 2-edge connectivity relations and constructs a sparse subgraph maintaining these connectivity blocks in directed graphs.

Findings

Linear-time algorithm for computing 2-edge connectivity relation

Constant-time query for 2-edge connectivity between vertices

Construction of a sparse certificate with O(n) edges

Abstract

Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study $2$-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices $v$ and $w$ are $2$-edge-connected if there are two edge-disjoint paths from $v$ to $w$ and two edge-disjoint paths from $w$ to $v$. This relation partitions the vertices into blocks such that all vertices in the same block are $2$-edge-connected. Differently from the undirected case, those blocks do not correspond to the $2$-edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are $2$-edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has $O(n)$ edges and maintains the same $2$-edge-connected blocks as the input graph.