Arc-disjoint in-trees in directed graphs

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 d i=1 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 sV, 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.

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Correspondence to Naoyuki Kamiyama.

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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.

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Kamiyama, N., Katoh, N. & Takizawa, A. Arc-disjoint in-trees in directed graphs. Combinatorica 29, 197–214 (2009). https://doi.org/10.1007/s00493-009-2428-z

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Mathematics Subject Classification (2000)

  • 05C70
  • 05C40