Edmonds' Branching Theorem in Digraphs Without Forward-Infinite Paths

Let D be a finite digraph, and let V0, . . . , Vk−1 be nonempty subsets of V (D). The (strong form of) Edmonds’ branching theorem states that there are pairwise edge-disjoint spanning branchings B0, . . . ,Bk−1 in D such that the root set of Bi is Vi (i = 0, . . . , k − 1) if and only if for all ∅ 6= X ⊆ V (D) the number of ingoing edges of X is greater than or equal to the number of sets Vi disjoint from X. As was shown by R. Aharoni and C. Thomassen in [1], this theorem does not remain true for infinite digraphs. Thomassen also proved that for the class of digraphs without backward-infinite paths, the above theorem of Edmonds remains true. Our main result is that for digraphs without forward-infinite paths, Edmonds’ branching theorem remains true as well. 1 Notions and notation The digraphs D = (V,A) considered here may have multiple edges and arbitrary size. Loops are also allowed but are irrelevant to our subject. If B ⊆ V , then we write D[B] for the subgraph of D spanned by B. For X ⊆ V let inD(X) and outD(X) be the set of ingoing and outgoing edges respectively of X in D, and let ̺D(X), δD(X) be their respective cardinalities. By a path, we mean a directed, possibly infinite, simple path (the repetition of vertices is not allowed). We denote by start(P ) and end(P ) the first and last vertex of the path P , if they exist. For an edge e from x to y, let start(e) = x and end(e) = y. For X,Y ⊆ V , let eD(X,Y ) = {e ∈ A : start(e) ∈ X, end(e) ∈ Y }; for singletons we write e(x, y) instead of e({x}, {y}). We say that the path P goes from X to Y if V (P ) ∩ X = {start(P )} and V (P ) ∩ Y = {end(P )} (start(P ) = end(P ) is allowed). We call min{̺D(X) : ∅ 6= X ⊆ V \ {r}} the edge-connectivity of D from r, and D is κ-edge-connected from r if this cardinal is at least κ. A digraph is an arborescence with root vertex r if it is a directed tree such that all vertices are reachable from r. A digraph is a branching with root set W if its weakly connected components are arborescences and the vertex set W consists of the roots of these arborescences. B is a k-branching ∗MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary. E-mail: [email protected].