Drawing Planar Graphs with Many Collinear Vertices

  • Giordano Da Lozzo
  • Vida Dujmović
  • Fabrizio FratiEmail author
  • Tamara Mchedlidze
  • Vincenzo Roselli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


Given a planar graph G, what is the maximum number of collinear vertices in a planar straight-line drawing of G? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known: Every n-vertex planar graph has a planar straight-line drawing with \(\varOmega (\sqrt{n})\) collinear vertices; for every n, there is an n-vertex planar graph whose every planar straight-line drawinghas \(O(n^{0.986})\) collinear vertices; every n-vertex planar graph of treewidth at most two has a planar straight-line drawingwith \(\varTheta (n)\) collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs, partially answering two problems posed by Ravsky and Verbitsky. Similar results are not possible for all bounded treewidth or bounded degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.



Research partially supported by MIUR Project “AMANDA” under PRIN 2012C4E3KT, by NSERC, and by Ontario Ministry of Research and Innovation. The authors thank Giuseppe Liotta, Sue Whitesides, and Stephen Wismath for stimulating discussions.


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Giordano Da Lozzo
    • 1
  • Vida Dujmović
    • 2
  • Fabrizio Frati
    • 1
    Email author
  • Tamara Mchedlidze
    • 3
  • Vincenzo Roselli
    • 1
  1. 1.University Roma TreRomeItaly
  2. 2.University of OttawaOttawaCanada
  3. 3.Karlsruhe Institute of TechnologyKarlsruheGermany

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