Spanning galaxies in digraphs

Daniel Gonçalves (1) , Frédéric Havet (2) , Alexandre Pinlou (1) , Stéphan Thomassé (1)

1 ALGCO - Algorithmes, Graphes et Combinatoire
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications

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.

Domaines: Mathématique discrète [cs.DM]

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⟩