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