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Covering Directed Graphs by In-Trees

  • Naoyuki Kamiyama
  • Naoki Katoh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)

Abstract

Given a directed graph D = (V,A) with a set of d specified vertices S = {s 1,...,s d } ⊆ V and a function \(f\colon S \to \mathbb{Z}_+\) where ℤ +  denotes the set of non-negative integers, we consider the problem which asks whether there exist \(\sum_{i=1}^d f(s_i)\) in-trees denoted by \(T_{i,1},T_{i,2},\ldots, T_{i,f(s_i)}\) for every i = 1,...,d such that \(T_{i,1},\ldots,T_{i,f(s_i)}\) are rooted at s i , each T i,j spans vertices from which s i is reachable and the union of all arc sets of T i,j for i = 1,...,d and j = 1,...,f(s i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in \(\sum_{i=1}^df(s_i)\) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Naoyuki Kamiyama
    • 1
  • Naoki Katoh
    • 1
  1. 1.Department of Architecture and Architectural EngineeringKyoto University, Kyotodaigaku-Katsura, Nishikyo-kuKyotoJapan

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