Vertex-Flames in Countable Rooted Digraphs Preserving an Erdős-Menger Separation for Each Vertex

Abstract

It follows from a theorem of Lovasz that if D is a finite digraph with r ∈ V(D), then there is a spanning subdigraph E of D such that for every vertex v ≠ r the following quantities are equal: the local connectivity from r to v in D, the local connectivity from r to v in E and the indegree of v in E. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a result is the generalization of Menger's theorem to infinite digraphs. We generalize the result of Lovasz above in this spirit. Our main result is that every countable r-rooted digraph D has a spanning subdigraph E with the following property. For every v ≠ r, E contains a system Rv of internally disjoint r → v paths such that the ingoing edges of v in E are exactly the last edges of the paths in Rv. Furthermore, the path-system Rv is “big” in D in the Erdős-Menger sense, i.e., one can choose from each path in Rv either an edge or an internal vertex in such a way that a resulting set separates v from r in D.