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Thickness and Colorability of Geometric Graphs

  • Stephane Durocher
  • Ellen Gethner
  • Debajyoti Mondal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)

Abstract

The geometric thickness \(\bar\theta\)(G) of a graph G is the smallest integer t such that there exist a straight-line drawing Γ of G and a partition of its straight-line edges into t subsets, where each subset induces a planar drawing in Γ. Over a decade ago, Hutchinson, Shermer, and Vince proved that any n-vertex graph with geometric thickness two can have at most 6n − 18 edges, and for every n ≥ 8 they constructed a geometric thickness two graph with 6n − 20 edges. In this paper, we construct geometric thickness two graphs with 6n − 19 edges for every n ≥ 9, which improves the previously known 6n − 20 lower bound. We then construct a thickness two graph with 10 vertices that has geometric thickness three, and prove that the problem of recognizing geometric thickness two graphs is NP-hard, answering two questions posed by Dillencourt, Eppstein and Hirschberg. Finally, we prove the NP-hardness of coloring graphs of geometric thickness t with 4t − 1 colors, which strengthens a result of McGrae and Zito, when t = 2.

Keywords

Planar Graph Complete Graph Planar Layer Geometric Graph Planar Drawing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Ellen Gethner
    • 2
  • Debajyoti Mondal
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaCanada
  2. 2.Department of Computer ScienceUniversity of Colorado DenverUSA

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