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Inserting Multiple Edges into a Planar Graph

Authors Markus Chimani, Petr Hlinený

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Keywords
  • crossing number
  • edge insertion
  • parameterized complexity
  • path homotopy
  • funnel algorithm

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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.

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Markus Chimani and Petr Hlinený. Inserting Multiple Edges into a Planar Graph. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.30

Author Details

Markus Chimani
Petr Hlinený

References

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