Spanning galaxies in digraphs.

*(English)*Zbl 1273.05087
Nešetřil, Jaroslav (ed.) et al., Extended abstracts of the 5th European conference on combinatorics, graph theory and applications, EuroComb’09, Bordeaux, France, September 7–11, 2009. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 34, 139-143 (2009).

Summary: 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 \(\mathsf{dst}(D)\), is the minimum number of galaxies needed to cover \(A(D)\). In this paper, we show that \(\mathsf{dst}(D)\leq \Delta(D)+1\) and that if \(D\) is acyclic then \(\mathsf{dst}(D)\leq \Delta(D)\). 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.

For the entire collection see [Zbl 1239.05008].

For the entire collection see [Zbl 1239.05008].

##### MSC:

05C20 | Directed graphs (digraphs), tournaments |

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\textit{D. Gonçalves} et al., Electron. Notes Discrete Math. 34, 139--143 (2009; Zbl 1273.05087)

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