Planar Embeddings of Graphs with Specified Edge Lengths

We consider the problem of finding a planar embedding of a (planar) graph with a prescribed Euclidean length on every edge. There has been substantial previous work on the problem without the planarity restrictions, which has close connections to rigidity theory, and where it is easy to see that the problem is NP-hard. In contrast, we show that the problem is tractable—indeed, solvable in linear time on a real RAM—for planar embeddings of planar 3-connected triangulations, even if the outer face is not a triangle. This result is essentially tight: the problem becomes NP-hard if we consider instead planar embeddings of planar 3-connected infinitesimally rigid graphs with unit edge lengths, a natural relaxation of triangulations in this context. A preliminary version of this paper has appear in the Proceedings of Graph Drawing 2003, Lecture Notes in Computer Science 2912, Springer Verlag. The research by the first author has been done as PhD student at the Institute of Information and Computing Sciences, Universiteit Utrecht, The Netherlands, supported by Cornelis Lely Stichting and by a Marie Curie Fellowship of the European Community programme “Combinatorics, Geometry, and Computation” under contract number HPMT-CT2001-00282. The third author is partially supported by the National Science Foundation under grant number ITR ANI-0205445.