Spanning galaxies in digraphs

Daniel Gonçalves 1 Frédéric Havet 2 Alexandre Pinlou 1 Stéphan Thomassé 1
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D, denoted by dst(D), is the minimum number of galaxies needed to cover A(D). In this paper, we show that dst(D)less-than-or-equals, slantΔ(D)+1 and that if D is acyclic then dst(D)less-than-or-equals, slantΔ(D). These results are proved by considering the existence of spanning galaxies in digraphs. Thus, we study the problem of deciding whether a digraph D has a spanning galaxy or not. We show that it is NP-complete (even when restricted to acyclic digraphs) but that it becomes polynomial-time solvable when restricted to strongly connected digraphs.
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Conference papers
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00433050
Contributor : Alexandre Pinlou <>
Submitted on : Wednesday, November 18, 2009 - 8:27:42 AM
Last modification on : Tuesday, January 22, 2019 - 7:10:05 PM

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Daniel Gonçalves, Frédéric Havet, Alexandre Pinlou, Stéphan Thomassé. Spanning galaxies in digraphs. EuroComb: European Conference on Combinatorics, Graph Theory and Applications, Sep 2009, Bordeaux, France. pp.139-143, ⟨10.1016/j.endm.2009.07.023⟩. ⟨lirmm-00433050⟩

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