# Edmonds' disjoint arborescences theorem

Theorem (Edmonds)[1] Let D=(V,A) be a digraph with a designated root node r. D contains k edge-disjoint spanning arborescences rooted at r if and only if $\varrho(X)\geq k$ for each nonempty $X\subseteq V-r$.
Theorem (Edmonds)[2] Let D=(V,A) be a digraph with arborescences $F_1,\ldots,F_k$ rooted at r. Then there exist k edge-disjoint spanning arborescences $H_1,\ldots,H_k$ rooted at r with $F_i\subseteq H_i$ if and only if each nonempty subset X of V is entered by at least as many arc as there exist i with $F_i$ disjoint from X.