Arc-disjoint in-trees in directed graphs

Authors:
Naoyuki Kamiyama;Naoki Katoh;Atsushi Takizawa
Affiliations:
Kyoto University, Department of Architecture and Architectural Engineering, Kyotodaigaku-Katsura, Nishikyo-ku, 615-8540, Kyoto, Japan;Kyoto University, Department of Architecture and Architectural Engineering, Kyotodaigaku-Katsura, Nishikyo-ku, 615-8540, Kyoto, Japan;Kyoto University, Department of Architecture and Architectural Engineering, Kyotodaigaku-Katsura, Nishikyo-ku, 615-8540, Kyoto, Japan
Venue:
Combinatorica
Year:
2009

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Abstract

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,…, $$T_{i,f(s_0 )}$$ for every i = 1,…,d such that T i,1,…, $$T_{i,f(s_0 )}$$ 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.