Disjoint non-balanced A-paths in biased graphs

Let A be a set of vertices in a graph G. An A-path is a non-trivial path in G that has both of its ends in A. In 1961, Gallai showed that, for any integer k, either G has k disjoint A-paths or there is a set of at most 2(k−1) vertices that hits all of the A-paths. There have been a number of extensions of this result; in each of these extensions we want to find a maximum collection of disjoint “allowable” A-paths, where the collection of allowed A-paths varies according to the application. We prove a new extension of this type, in the context of biased graphs, unifying many of the others.