A generalisation of Menger's theorem in bidirected graphs

TL;DR

This paper extends Menger's theorem to bidirected graphs by considering nontrivial paths within the same vertex sets, providing conditions under which the theorem holds in this broader context.

Contribution

It introduces a generalized version of Menger's theorem applicable to bidirected graphs, focusing on nontrivial paths within the same vertex sets.

Findings

Menger's theorem does not hold in bidirected graphs generally.

The theorem is valid when considering nontrivial $X$-$X$ and $Y$-$Y$ paths.

Provides a new characterization of connectivity in bidirected graphs.

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.