Arc-disjoint in-trees in directed graphs

AbstractGiven a directed graph D = (V,A) with a set of d specified vertices S = {s1,…, sd} ⊆ V and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition such that there exist Σi=1df(si) arc-disjoint in-trees denoted by Ti,1,Ti,2,…, $$ T_{i,f(s_0 )} $$ for every i = 1,…,d such that Ti,1,…,$$ T_{i,f(s_0 )} $$ are rooted at si and each Ti,j spans the vertices from which si 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.