Hanani–Tutte, Monotone Drawings, and Level-Planarity

  • Radoslav Fulek
  • Michael J. Pelsmajer
  • Marcus Schaefer
  • Daniel ŠtefankovičEmail author


A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even number of times, then the graph has an x-monotone embedding in which the x-coordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly. This answers a question posed by Pach and Tóth. We show that a further strengthening to a “removing even crossings” lemma is impossible by separating monotone versions of the crossing and the odd crossing number.

Our results extend to level-planarity, which is a well-studied generalization of x-monotonicity. We obtain a new and simple algorithm to test level-planarity in quadratic time, and we show that x-monotonicity of edges in the definition of level-planarity can be relaxed.



We thank the anonymous referee for careful proofreading.

Radoslav Fulek gratefully acknowledges support from the Swiss National Science Foundation Grant No. 200021-125287/1.

Michael J. Pelsmajer gratefully acknowledges the support from NSA Grant H98230-08-1-0043 and the Swiss National Science Foundation Grant No. 200021-125287/1.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Radoslav Fulek
    • 1
  • Michael J. Pelsmajer
    • 2
  • Marcus Schaefer
    • 3
  • Daniel Štefankovič
    • 4
    Email author
  1. 1.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  3. 3.Department of Computer ScienceDePaul UniversityChicagoUSA
  4. 4.Computer Science DepartmentUniversity of RochesterRochesterUSA

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