International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 87-98 | Cite as

Genus, Treewidth, and Local Crossing Number

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.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vida Dujmović
    • 1
  • David Eppstein
    • 2
  • 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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