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Hanani–Tutte, Monotone Drawings, and Level-Planarity

In: Thirty Essays on Geometric Graph Theory

Author

Listed:
  • Radoslav Fulek

    (Ecole Polytechnique Fédérale de Lausanne)

  • Michael J. Pelsmajer

    (Illinois Institute of Technology, Department of Applied Mathematics)

  • Marcus Schaefer

    (DePaul University, Department of Computer Science)

  • Daniel Štefankovič

    (University of Rochester, Computer Science Department)

Abstract

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.

Suggested Citation

  • Radoslav Fulek & Michael J. Pelsmajer & Marcus Schaefer & Daniel Štefankovič, 2013. "Hanani–Tutte, Monotone Drawings, and Level-Planarity," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 263-287, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_14
    DOI: 10.1007/978-1-4614-0110-0_14
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