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Genus, Treewidth, and Local Crossing Number

  • Vida Dujmović
  • David EppsteinEmail author
  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth \(O(\sqrt{(g+1)(k+1)n})\) and layered treewidth \(O((g+1)k)\), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth \(O(\sqrt{(k+1)n})\) and layered treewidth \(O(k+1)\), which are tight bounds that improve a previously known \(O((k+1)^{3/4}n^{1/2})\) treewidth bound. Additionally, we show that for \(g<m\), every m-edge graph can be embedded on a surface of genus g with \(O((m/(g+1))\log ^2 g)\) crossings per edge, which is tight to a polylogarithmic factor.

Keywords

Planar Graph Tree Decomposition Grid Graph Graph Minor Cyclomatic Number 
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.

Notes

Acknowledgement

This research was initiated at the Workshop on Graphs and Geometry held at the Bellairs Research Institute in 2015. Vida Dujmović was supported by NSERC. David Eppstein was supported in part by NSF grant CCF-1228639. David Wood was supported by the Australian Research Council.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vida Dujmović
    • 1
  • David Eppstein
    • 2
    Email author
  • David R. Wood
    • 3
  1. 1.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  2. 2.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  3. 3.School of Mathematical SciencesMonash UniversityMelbourneAustralia

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