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Monotone Paths in Planar Convex Subdivisions and Polytopes

Chapter
Part of the Fields Institute Communications book series (FIC, volume 69)

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 \(\Omega (\log _{\Delta }n)\) 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 \(\Omega (\log (n/k)/\log \log (n/k))\) edges that is monotone in some direction. This bound is also the best possible. Our methods are constructive and lead to efficient algorithms for computing monotone paths of lengths specified above. 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 \(O({\log }^{2}n)\) 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.

Key words

Monotone path Convex subdivision Geometric graph Convex polytope 

Subject Classifications

52B10 52C45 

Notes

Acknowledgements

The work of Adrian Dumitrescu was supported in part by NSF grant DMS-1001667. The work of Günter Rote was supported in part by CIB (Centre Interfacultaire Bernoulli) in Lausanne, by the NSF (National Science Foundation), and by the ESF EUROCORES programme EuroGIGA-VORONOI, Deutsche Forschungsgemeinschaft (DFG): RO 2338/5-1. This work was initiated while Günter Rote was a guest of CIB in Fall 2010. The work of Csaba Tóth was supported in part by NSERC grant RGPIN 35586. Research by Csaba Tóth was conducted at the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada M5T 3J1.

The authors thank János Pach for insistently asking the question to which Theorem 1 gives the answer. A recursive construction involving a hierarchy of squeezed zigzags, similar to the upper-bound construction for Theorem 2 in Sect. 3.2, has been suggested by Boris Bukh.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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