Planar and Quasi Planar Simultaneous Geometric Embedding

  • Emilio Di Giacomo
  • Walter Didimo
  • Giuseppe Liotta
  • Henk Meijer
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

A simultaneous geometric embedding (SGE) of two planar graphs G1 and G2 with the same vertex set is a pair of straight-line planar drawings Γ1 of G1 and Γ2 of G2 such that each vertex is drawn at the same point in Γ1 and Γ2. Many papers have been devoted to the study of which pairs of graphs admit a SGE, and both positive and negative results have been proved. We extend the study of SGE, by introducing and characterizing a new class of planar graphs that makes it possible to immediately extend several positive results that rely on the property of strictly monotone paths. Moreover, we introduce a relaxation of the SGE setting where Γ1 and Γ2 are required to be quasi planar (i.e., they can have crossings provided that there are no three mutually crossing edges). This relaxation allows for the simultaneous embedding of pairs of planar graphs that are not simultaneously embeddable in the classical SGE setting and opens up to several new interesting research questions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • Walter Didimo
    • 1
  • Giuseppe Liotta
    • 1
  • Henk Meijer
    • 2
  • Stephen Wismath
    • 3
  1. 1.Dipartimento di IngegneriaUniversità degli Studi di PerugiaItaly
  2. 2.University College RooseveltThe Netherlands
  3. 3.Department of Mathematics and Computer ScienceUniversity of LethbridgeCanada

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