Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-Line Drawings

  • Luca Castelli Aleardi
  • Olivier Devillers
  • Éric Fusy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

We extend the notion of canonical orderings to cylindric triangulations. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack to this setting. Our algorithm yields in linear time a crossing-free straight-line drawing of a cylindric triangulation G with n vertices on a regular grid ℤ/wℤ×[0..h], with w ≤ 2n and h ≤ n(2d + 1), where d is the (graph-) distance between the two boundaries. As a by-product, we can also obtain in linear time a crossing-free straight-line drawing of a toroidal triangulation with n vertices on a periodic regular grid ℤ/wℤ×ℤ/hℤ, with w ≤ 2n and h ≤ 1 + n(2c + 1), where c is the length of a shortest non-contractible cycle. Since \(c\leq\sqrt{2n}\), the grid area is O(n5/2). Our algorithms apply to any triangulation (whether on the cylinder or on the torus) that have no loops nor multiple edges in the periodic representation.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luca Castelli Aleardi
    • 1
  • Olivier Devillers
    • 2
  • Éric Fusy
    • 1
  1. 1.LIX - École PolytechniqueFrance
  2. 2.INRIA Sophia AntipolisMéditerranéeFrance

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