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Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs

Olivier Fischer, Yonggang Jiang, Sagnik Mukhopadhyay, Sorrachai Yingchareonthawornchai

TL;DR

This paper presents an extremely simple reduction from directed to undirected vertex connectivity for dense graphs, and obtains a subcubic-time algorithm for weighted directed vertex connectivity, improving upon a three-decade-old bound for dense graphs.

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 $n^{ω+o(1)}$ 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].

Directed and Undirected Vertex Connectivity Problems are Equivalent for Dense Graphs

TL;DR

This paper presents an extremely simple reduction from directed to undirected vertex connectivity for dense graphs, and obtains a subcubic-time algorithm for weighted directed vertex connectivity, improving upon a three-decade-old bound for dense graphs.

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 time [LNPSY25], and obtain a parallel vertex connectivity algorithm for directed graphs with work and 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].

Paper Structure

This paper contains 22 sections, 14 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. All-pairs minimum vertex cuts.
  3. Weighted vertex mincuts.
  4. Remark.
  5. Beyond the sequential model.
  6. Comparison to Edge Connectivity.
  7. Independent Work.
  8. Preliminaries
  9. Vertex cut.
  10. Minimum vertex cut.
  11. Single-source minimum vertex cut.
  12. All-pairs minimum vertex cut.
  13. Steiner minimum vertex cut.
  14. A Reduction from Directed to Undirected Vertex Connectivity
  15. Construction.
  16. ...and 7 more sections

Key Result

theorem 1.1

The directed vertex connectivity problem on an $n$-vertex directed graph reduces to the undirected vertex connectivity problem on a graph with $2n$ vertices and $O(n^2)$ edges.

Figures (1)

  • Figure 1: (Left) An input graph $G = (V,E,w)$. (Middle) The output graph $G' = (V',E',w')$. (Right) The split graph construction (ignoring weights) $SG$ given $G$. The split graph usually has the edge weight as capacity in one direction; our graph has vertex weights and no edge weights.

Theorems & Definitions (22)

  • theorem 1.1: Informal version of \ref{['thm:reduction']}
  • corollary 1.2
  • corollary 1.3
  • theorem 1.4: Informal version of \ref{['thm:reduction']}
  • theorem 1.5: Informal version of \ref{['thm:reduction']}
  • corollary 1.6
  • theorem 3.1
  • proposition 3.2
  • proof
  • proposition 3.3
  • ...and 12 more