Arc-disjoint in-trees in directed graphs

Authors:
Naoyuki Kamiyama;Naoki Katoh;Atsushi Takizawa
Affiliations:
Kyoto University, Kyoto, Japan;Kyoto University, Kyoto, Japan;Kyoto University, Kyoto, Japan
Venue:
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2008

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Abstract

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