Discrete Mathematics
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
Fixed-Parameter Tractability and Completeness I: Basic Results
SIAM Journal on Computing
On incidence coloring and star arboricity of graphs
Discrete Mathematics
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
Wdm and directed star arboricity
Combinatorics, Probability and Computing
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In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynomial-time solvable when restricted to strong digraphs. In fact, we prove that restricted to this class, the Spanning Galaxy problem is equivalent to the problem of deciding whether a strong digraph has a strong subdigraph with an even number of vertices. We then show a polynomial-time algorithm to solve this problem. We also consider some parameterized version of the Spanning Galaxy problem. Finally, we improve some results concerning the notion of directed star arboricity of a digraph D, which is the minimum number of galaxies needed to cover all the arcs of D. We show in particular that dst(D)@?@D(D)+1 for every digraph D and that dst(D)@?@D(D) for every acyclic digraph D.