Arc-disjoint in-trees in directed graphs

Given a directed graph D = (V,A) with a set of d specified vertices S = {s ₁, s _d } V and a function f: S → where denotes the set of natural numbers, we present a necessary and sufficient condition such that there exist ∑ _i=1 ^d f(s _i ) arc-disjoint in-trees denoted by T _i,1,T _i,2, T i,f(s0 ) for every i = 1, d such that T _i,1, T i,f (s0 ) are rooted at s _i and each T _i,j spans the vertices from which s _i is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D=(V,A) with a specified vertex s V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.