2026-08-20T19:38:28Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5922 2024-03-06T10:37:17Z ddc:004 open_access

Inserting Multiple Edges into a Planar Graph Chimani, Markus Hlinený, Petr crossing number edge insertion parameterized complexity path homotopy funnel algorithm 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Markus Chimani and Petr Hlinený 2016 Is Part Of LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2016.30 urn:nbn:de:0030-drops-59223 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.30 eng https://creativecommons.org/licenses/by/3.0/legalcode