Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Koana, Tomohiro; Komusiewicz, Christian; Sommer, Frank https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-133646
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Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

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Abstract

A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩ N(v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class 𝒢 which is closed under taking subgraphs admits a kernel with at most γ k² vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

BibTeX - Entry

@InProceedings{koana_et_al:LIPIcs:2020:13364,
  author =	{Tomohiro Koana and Christian Komusiewicz and Frank Sommer},
  title =	{{Computing Dense and Sparse Subgraphs of Weakly Closed Graphs}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Yixin Cao and Siu-Wing Cheng and Minming Li},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/13364},
  URN =		{urn:nbn:de:0030-drops-133646},
  doi =		{10.4230/LIPIcs.ISAAC.2020.20},
  annote =	{Keywords: Fixed-parameter tractability, c-closure, degeneracy, clique relaxations, bicliques, dominating set}
}

Keywords: Fixed-parameter tractability, c-closure, degeneracy, clique relaxations, bicliques, dominating set
Seminar: 31st International Symposium on Algorithms and Computation (ISAAC 2020)
Issue date: 2020
Date of publication: 04.12.2020


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