Free Edge Lengths in Plane Graphs
Authors: 
Zachary Abel, Robert Connelly, Sarah Eisenstat, Radoslav Fulek, Filip Morić, Yoshio Okamoto, Tibor Szabó, Csaba D. TóthAuthors Info & Claims
Zachary Abel, Robert Connelly, Sarah Eisenstat, Radoslav Fulek, Filip Morić, Yoshio Okamoto, Tibor Szabó, Csaba D. TóthAuthors Info & ClaimsPages 426 - 435
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 all 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 = Ck provides a new version of the celebrated carpenter's rule problem. Even though cycles Ck, 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).
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Published: 08 June 2014
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