Free Edge Lengths in Plane Graphs
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Author(s)
Connelly, Robert
Fulek, Radoslav
Morić, Filip
Okamoto, Yoshio
Szabó, Tibor
Tóth, Csaba D.
Abel, Zachary R
Eisenstat, Sarah Charmian
Date Issued
May 2015
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Abel, Zachary et al. “Free Edge Lengths in Plane Graphs.” Discrete & Computational Geometry 54.1 (2015): 259–289.
Version
Author's final manuscript
Abstract
We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the “host” of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all constraints on the edge lengths of G depend on G only, irrespective of additional edges of the host H. We characterize the planar graphs G that are free in every host H, G⊆H, and all the planar graphs G that are extrinsically free in every host H, G⊆H. The case of cycles G=C[subscript k] provides a new version of the celebrated carpenter’s rule problem. Even though cycles C[subscript k], k≥4, are not extrinsically free in all triangulations, it turns out that “nondegenerate” edge lengths are always realizable, where the edge lengths are considered degenerate if the cycle can be flattened (into a line) in two different ways. Separating triangles, and separating cycles in general, play an important role in our arguments. We show that every star is free in a 4-connected triangulation (which has no separating triangle).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00454-015-9704-z
