Monotone Paths in Planar Convex Subdivisions

  • Adrian Dumitrescu
  • Günter Rote
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

Consider a connected subdivision of the plane into n convex faces where every vertex is incident to at most Δ edges. Then, starting from every vertex there is a path with at least Ω(logΔn) edges that is monotone in some direction. This bound is the best possible. Consider now a connected subdivision of the plane into n convex faces where exactly k faces are unbounded. Then, there is a path with at least Ω(log(n/k)/loglog(n/k)) edges that is monotone in some direction. This bound is also the best possible.

In 3-space, we show that for every n ≥ 4, there exists a polytope P with n vertices, bounded vertex degrees, and triangular faces such that every monotone path on the 1-skeleton of P has at most O(log2n) edges. We also construct a polytope Q with n vertices, and triangular faces, (with unbounded degree however), such that every monotone path on the 1-skeleton of Q has at most O(logn) edges.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Günter Rote
    • 2
  • Csaba D. Tóth
    • 3
  1. 1.Department of Computer ScienceUniversity of WisconsinMilwaukeeUSA
  2. 2.Institut für InformatikFreie Universität BerlinGermany
  3. 3.Department of MathematicsUniversity of CalgaryCanada

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