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URN: urn:nbn:de:0030-drops-82460
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### On Directed Covering and Domination Problems

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### Abstract

In this paper, we study covering and domination problems on directed graphs.
Although undirected Vertex Cover and Edge Dominating Set are well-studied classical graph problems, the directed versions have not been studied much due to the lack of clear definitions.

We give natural definitions for Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set as directed generations of Vertex Cover and Edge Dominating Set.
For these problems, we show that
(1) Directed r-In (Out) Vertex Cover and Directed (p,q)-Edge Dominating Set are NP-complete on planar directed acyclic graphs except when r=1 or (p,q)=(0,0),
(2) if r>=2, Directed r-In (Out) Vertex Cover is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(3) if either p or q is greater than 1, Directed (p,q)-Edge Dominating Set is W[2]-hard and (c*ln k)-inapproximable on directed acyclic graphs,
(4) all problems can be solved in polynomial time on trees, and
(5) Directed (0,1),(1,0),(1,1)-Edge Dominating Set are fixed-parameter tractable in general graphs.

The first result implies that (directed) r-Dominating Set on directed line graphs is NP-complete even if r=1.

### BibTeX - Entry

```@InProceedings{hanaka_et_al:LIPIcs:2017:8246,
author =	{Tesshu Hanaka and Naomi Nishimura and Hirotaka Ono},
title =	{{On  Directed Covering and Domination Problems}},
booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages =	{45:1--45:12},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-054-5},
ISSN =	{1868-8969},
year =	{2017},
volume =	{92},
editor =	{Yoshio Okamoto and Takeshi Tokuyama},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},