Realizability of Graphs and Linkages

  • Marcus Schaefer


We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers, ETR; we introduce the class \(\exists \mathbb{R}\) that captures the computational complexity of ETR and many other problems. The graph realizability problem remains \(\exists \mathbb{R}\)-complete if all edges have unit length, which implies that recognizing unit distance graphs is \(\exists \mathbb{R}\)-complete. We also consider the problem for linkages: In a realization of a linkage, vertices are allowed to overlap and lie on the interior of edges. Linkage realizability is \(\exists \mathbb{R}\)-complete and remains so if all edges have unit length. A linkage is called rigid if any slight perturbation of its vertices that does not break the linkage (i.e., keeps edge lengths the same) is the result of a rigid motion of the plane. Testing whether a configuration is not rigid is \(\exists \mathbb{R}\)-complete.


Existential Theory Graph Drawing Graph Realizability Euclidean Dimension Nontrivial Zero 
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.



I would like to thank the anonymous referee for pointing out the universality papers by Jordan and Steiner [23] and Kapovich and Millson [25].


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceDePaul UniversityChicagoUSA

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