# Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs

## TL;DR

This paper introduces a simple reduction from directed to undirected vertex connectivity in dense graphs, simplifying proofs and enabling new efficient algorithms for various variants of the problem.

## Contribution

The paper presents a straightforward reduction from directed to undirected vertex connectivity in dense graphs, simplifying existing proofs and enabling new algorithms for weighted, all-pairs, and Steiner variants.

## Key findings

- Simplifies proof for directed vertex connectivity in $n^{2+o(1)}$ time.
- Develops a parallel algorithm for directed graphs with near-linear work and depth.
- Provides a subcubic-time algorithm for weighted directed vertex connectivity.

## Abstract

Vertex connectivity and its variants are among the most fundamental problems in graph theory, with decades of extensive study and numerous algorithmic advances. The directed variants of vertex connectivity are usually solved by manually extending fast algorithms for undirected graphs, which has required considerable effort. In this paper, we present an extremely simple reduction from directed to undirected vertex connectivity for dense graphs. As immediate corollaries, we vastly simplify the proof for directed vertex connectivity in $n^{2+o(1)}$ time [LNPSY25], and obtain a parallel vertex connectivity algorithm for directed graphs with $n^{\omega+o(1)}$ work and $n^{o(1)}$ depth, via the undirected vertex connectivity algorithm of [BJMY25]. Our reduction further extends to the weighted, all-pairs and Steiner versions of the problem. By combining our reduction with the recent subcubic-time algorithm for undirected weighted vertex cuts [CT25], we obtain a subcubic-time algorithm for weighted directed vertex connectivity, improving upon a three-decade-old bound [HRG00] for dense graphs. For the all-pairs version, by combining the conditional lower bounds on the all-pairs vertex connectivity problem for directed graphs [AGIKPTUW19], we obtain an alternate proof of the conditional lower bound for the all-pairs vertex connectivity problem on undirected graphs, vastly simplifying the proof by [HLSW23].