Hindrance from a wasteful partial linkage

TL;DR

This paper investigates conditions for the existence of hindrances in infinite digraphs, showing that a wasteful partial linkage implies the presence of a hindrance, which is significant for Menger's theorem and related open problems.

Contribution

It establishes that a wasteful partial linkage in a digraph guarantees the existence of a hindrance, advancing understanding of structural obstructions in infinite graph theory.

Findings

Wasteful partial linkages imply hindrances in digraphs.

Hindrances are linked to configurations with fewer unused vertices.

Results contribute to the proof of the infinite Menger's theorem.

Abstract

Let $ D=(V,E) $ be a (possibly infinite) digraph and $ A,B\subseteq V $. A hindrance consists of an $ AB $-separator $ S $ together with a set of disjoint $ AS $-paths linking a proper subset of $ A $ onto $ S $. Hindrances and configurations guaranteeing the existence of hindrances play an essential role in the proof of the infinite version of Menger's theorem and are important in the context of certain open problems as well. This motivates the investigation of circumstances under which hindrances appear. In this paper we show that if there is a ``wasteful partial linkage'', i.e. a set $ \mathcal{P} $ of disjoint $ AB $-paths with fewer unused vertices in $ B $ than in $ A $, then there exists a hindrance.