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.ISAAC.2017.13
URN: urn:nbn:de:0030-drops-82570
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8257/
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Biedl, Therese ; Chimani, Markus ; Derka, Martin ; Mutzel, Petra

Crossing Number for Graphs with Bounded~Pathwidth

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LIPIcs-ISAAC-2017-13.pdf (0.6 MB)

Abstract

The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth.

In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an O(n)xO(n)-grid to achieve such a drawing.

Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a 4w^3-approximation for maximal graphs of pathwidth w. This is a constant approximation for bounded pathwidth graphs.

BibTeX - Entry

@InProceedings{biedl_et_al:LIPIcs:2017:8257,
  author =	{Therese Biedl and Markus Chimani and Martin Derka and Petra Mutzel},
  title =	{{Crossing Number for Graphs with Bounded~Pathwidth}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{13:1--13:13},
  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},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8257},
  URN =		{urn:nbn:de:0030-drops-82570},
  doi =		{10.4230/LIPIcs.ISAAC.2017.13},
  annote =	{Keywords: Crossing Number, Graphs with Bounded Pathwidth}
}

Keywords: Crossing Number, Graphs with Bounded Pathwidth
Collection: 28th International Symposium on Algorithms and Computation (ISAAC 2017)
Issue Date: 2017
Date of publication: 07.12.2017

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