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Let  D  be a finite digraph, and letV  0   ,  ⋯  ,   V   k  −  1be nonempty subsets of    V  (  D  )   . The (strong form of) Edmonds' branching theorem states that there are pairwise edge‐disjoint spanning branchingsB  0   ,  ⋯  ,   B   k  −  1in  D  such that the root set of    B  i    isV  i(  i  =  0  ,  ⋯  ,  k  −  1  )if and only if for all    ⌀  ≠  X  ⊆  V  (  D  )    the number of ingoing edges of  X  is greater than or equal to the number of sets    V  i    disjoint from  X . As was shown by R. Aharoni and C. Thomassen (J Graph Theory 13 (1989), 71–74), 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.

Edmonds' Branching Theorem in Digraphs Without Forward‐Infinite Paths