Hindrance from a wasteful partial linkage

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.