2-Vertex Connectivity in Directed Graphs

TL;DR

This paper presents linear-time algorithms for computing 2-vertex-connectivity and vertex-resilience relations in directed graphs, along with a sparse certificate that preserves these relations, enabling efficient queries and analysis.

Contribution

It introduces efficient algorithms for 2-vertex-connectivity and vertex-resilience in directed graphs, including a linear-time sparse certificate construction.

Findings

Linear-time computation of 2-vertex-connected relations

Linear-time construction of sparse certificates

Constant-time queries for vertex connectivity relations

Abstract

We complement our study of 2-connectivity in directed graphs, by considering the computation of the following 2-vertex-connectivity relations: We say that two vertices v and w are 2-vertex-connected if there are two internally vertex-disjoint paths from v to w and two internally vertex-disjoint paths from w to v. We also say that v and w are vertex-resilient if the removal of any vertex different from v and w leaves v and w in the same strongly connected component. We show how to compute the above relations in linear time so that we can report in constant time if two vertices are 2-vertex-connected or if they are vertex-resilient. We also show how to compute in linear time a sparse certificate for these relations, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-vertex-connectivity and vertex-resilience relations as the input graph, where n is the number of vertices.