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Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

  • Clinton Bowen
  • Stephane Durocher
  • Maarten Löffler
  • Anika Rounds
  • André Schulz
  • Csaba D. Tóth
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane.

Keywords

Unit Disk Boolean Formula Regular Hexagon Fixed Orientation Contact Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

Our results in Sect. 4 were developed at the First International Workshop on Drawing Algorithms for Networks of Changing Entities (DANCE 2014), held in Langbroek, the Netherlands, and supported by the NWO project 639.023.208. Research by Rounds and Tóth was supported in part by the NSF awards CCF-1422311 and CCF-1423615. Research by Durocher was supported in part by NSERC.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Clinton Bowen
    • 1
  • Stephane Durocher
    • 2
  • Maarten Löffler
    • 3
  • Anika Rounds
    • 4
  • André Schulz
    • 5
  • Csaba D. Tóth
    • 1
    • 4
  1. 1.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  2. 2.Department of Computer ScienceUniversity of ManitobaWinnipegCanada
  3. 3.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  4. 4.Department of Computer ScienceTufts UniversityMedfordUSA
  5. 5.Theoretical Computer ScienceUniversity of HagenHagenGermany

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