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A generalisation of Menger's theorem in bidirected graphs

Ebrahim Ghorbani, Jana Katharina Nickel, Florian Reich

TL;DR

The paper addresses extending Menger's theorem to bidirected graphs by incorporating nontrivial $X$-$X$ and $Y$-$Y$ paths. It introduces $X$-$Y$ turnarounds and $X$-$Y$ links, and proves a max-min relation: the maximum number of vertex-disjoint $X$-$Y$ links, with turnarounds counted twice, is at least the minimum size of an $X$-$Y$-link cut; this subsumes previous sufficient conditions for the classic Menger setting when turnarounds are constrained. The approach relies on linear programming and the integrality properties of the incidence matrix of bidirected graphs via $k$-regularity, yielding a concise, self-contained proof and several corollaries and remarks on optimality. The results strengthen prior work and offer potential variants, including edge-based questions and extensions when no $X$-$X$ turnarounds are allowed. The work contributes to a deeper understanding of path-packings in bidirected graphs and links to matching theory.

Abstract

Menger's theorem - the maximum number of vertex-disjoint $X$-$Y$ paths is equal to the minimum size of an $X$-$Y$ separator - is generally not true in bidirected graphs. We prove that Menger's theorem holds true if we take the nontrivial $X$-$X$ paths and the nontrivial $Y$-$Y$ paths into account.

A generalisation of Menger's theorem in bidirected graphs

TL;DR

The paper addresses extending Menger's theorem to bidirected graphs by incorporating nontrivial - and - paths. It introduces - turnarounds and - links, and proves a max-min relation: the maximum number of vertex-disjoint - links, with turnarounds counted twice, is at least the minimum size of an --link cut; this subsumes previous sufficient conditions for the classic Menger setting when turnarounds are constrained. The approach relies on linear programming and the integrality properties of the incidence matrix of bidirected graphs via -regularity, yielding a concise, self-contained proof and several corollaries and remarks on optimality. The results strengthen prior work and offer potential variants, including edge-based questions and extensions when no - turnarounds are allowed. The work contributes to a deeper understanding of path-packings in bidirected graphs and links to matching theory.

Abstract

Menger's theorem - the maximum number of vertex-disjoint - paths is equal to the minimum size of an - separator - is generally not true in bidirected graphs. We prove that Menger's theorem holds true if we take the nontrivial - paths and the nontrivial - paths into account.

Paper Structure

This paper contains 3 sections, 8 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. From vertices to sets of vertices
  3. Proof of \ref{['thm:main_vtx']}

Key Result

Theorem 1

Let $B$ be a bidirected graph and let $X$ and $Y$ be sets of vertices of $B$ such that there is either no nontrivial $X$--$X$ path or no nontrivial $Y$--$Y$ path. Then the maximum number of vertex-disjoint $X$--$Y$ paths is equal to the minimum size of a set $S$ of vertices such that there is no $X$

Figures (1)

  • Figure 1: Two bidirected graphs in which every $X$--$Y$ link is an $X$--$Y$ turnaround. The perpendicular bar at each half-edge indicates that it has sign $+$.

Theorems & Definitions (12)

  • Theorem 1: bowler2023mengertheorembidirected*Theorem 1.2
  • Theorem 2
  • Corollary 3
  • Corollary 4: nickel2025disjoint*Theorem 4.1
  • Theorem 6
  • proof : Proof of \ref{['thm:main']} from \ref{['thm:main_vtx']}
  • proof : Proof of \ref{['cor:ep']} from \ref{['thm:main']}
  • Lemma 7: APPA2004k-regular*Theorem 23
  • Lemma 8: APPA2004k-regular*Theorem 16
  • Lemma 9
  • ...and 2 more