License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.40
URN: urn:nbn:de:0030-drops-109840
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10984/
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Einarson, Carl ; Reidl, Felix

Domination Above r-Independence: Does Sparseness Help?

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LIPIcs-MFCS-2019-40.pdf (0.6 MB)

Abstract

Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r+1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the "residual" set R := V(G) \ N[X].
Our work aims to answer two questions: How does the constant r affect the tractability of the problem and does the restriction to sparse graph classes help here? For the base case r = 2, we find that the problem is paraNP-complete even in apex- and bounded-degree graphs. For r = 3, the problem is W[2]-hard for general graphs but in FPT for nowhere dense classes and it admits a linear kernel for bounded expansion classes. For r >= 4, the parametrisation becomes essentially equivalent to the natural parameter, the size of the dominating set.

BibTeX - Entry

@InProceedings{einarson_et_al:LIPIcs:2019:10984,
  author =	{Carl Einarson and Felix Reidl},
  title =	{{Domination Above r-Independence: Does Sparseness Help\l{}}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{40:1--40:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10984},
  URN =		{urn:nbn:de:0030-drops-109840},
  doi =		{10.4230/LIPIcs.MFCS.2019.40},
  annote =	{Keywords: Dominating Set, Above Guarantee, Kernel, Bounded Expansion, Nowhere Dense}
}

Keywords: Dominating Set, Above Guarantee, Kernel, Bounded Expansion, Nowhere Dense
Collection: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 20.08.2019

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