Covering directed graphs by in-trees

Naoyuki Kamiyama, Naoki Katoh

Research output: Contribution to journalArticlepeer-review


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 positive integers, we consider the problem which asks whether there exist Σi=1 d f(s i ) in-trees denoted by Ti,1, Ti,2,...,Ti,f(si) for every i=1,...,d such that Ti,1,...,Ti,f(si) 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 Σ i=1 d f(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.

Original languageEnglish
Pages (from-to)2-18
Number of pages17
JournalJournal of Combinatorial Optimization
Issue number1
Publication statusPublished - Jan 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics


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