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.SoCG.2016.30
URN: urn:nbn:de:0030-drops-59223
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5922/
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Chimani, Markus ; Hlinený, Petr

Inserting Multiple Edges into a Planar Graph

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LIPIcs-SoCG-2016-30.pdf (0.6 MB)

Abstract

Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+F.

Finding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F|=1 [Gutwenger et al, SODA 2001/Algorithmica] and polynomial time solvable when all of F are incident to a new vertex [Chimani et al, SODA 2009]. The complexity for general F but with constant k=|F| was open, but algorithms both with relative and absolute approximation guarantees have been presented [Chuzhoy et al, SODA 2011], [Chimani-Hlineny, ICALP 2011]. We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V(G)|) time for any constant k.

BibTeX - Entry

@InProceedings{chimani_et_al:LIPIcs:2016:5922,
  author =	{Markus Chimani and Petr Hlinen{\'y}},
  title =	{{Inserting Multiple Edges into a Planar Graph}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{30:1--30:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5922},
  URN =		{urn:nbn:de:0030-drops-59223},
  doi =		{10.4230/LIPIcs.SoCG.2016.30},
  annote =	{Keywords: crossing number, edge insertion, parameterized complexity, path homotopy, funnel algorithm}
}

Keywords: crossing number, edge insertion, parameterized complexity, path homotopy, funnel algorithm
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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