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Crossing Number and Weighted Crossing Number of Near-Planar Graphs

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Abstract

A nonplanar graph G is near-planar if it contains an edge e such that G−e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněný and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs.

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Authors and Affiliations

  1. Department of Mathematics, FMF, University of Ljubljana, Ljubljana, Slovenia

    Sergio Cabello

  2. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Bojan Mohar

Authors
  1. Sergio Cabello
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  2. Bojan Mohar
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Corresponding author

Correspondence to Sergio Cabello.

Additional information

S. Cabello is supported in part by the Slovenian Research Agency, project J1-7218 and program P1-0297.

B. Mohar is supported in part by the ARRS, Research Program P1-0297, by an NSERC Discovery Grant, and by the Canada Research Chair Program.

B. Mohar is on leave from IMFM & FMF, Department of Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia.

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Cabello, S., Mohar, B. Crossing Number and Weighted Crossing Number of Near-Planar Graphs. Algorithmica 60, 484–504 (2011). https://doi.org/10.1007/s00453-009-9357-5

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