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 → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition such that there exist Σ _i=1 ^d f(s _i ) arc-disjoint in-trees denoted by T _i,1,T _i,2,…, for every i = 1,…,d such that T _i,1,…, 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. Supported by JSPS Research Fellowships for Young Scientists. Supported by the project New Horizons in Computing, Grand-in-Aid for Scientific Research on Priority Areas, MEXT Japan.