Free Edge Lengths in Plane Graphs
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 th...
| Main Authors: | , , , , , , , |
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| Format: | Article |
| Language: | English |
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Springer US
2017
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| Online Access: | http://hdl.handle.net/1721.1/106900 https://orcid.org/0000-0002-4295-1117 https://orcid.org/0000-0002-3182-1675 |
| _version_ | 1826192577301315584 |
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| author | Connelly, Robert Fulek, Radoslav Morić, Filip Okamoto, Yoshio Szabó, Tibor Tóth, Csaba D. Abel, Zachary R Eisenstat, Sarah Charmian |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Connelly, Robert Fulek, Radoslav Morić, Filip Okamoto, Yoshio Szabó, Tibor Tóth, Csaba D. Abel, Zachary R Eisenstat, Sarah Charmian |
| author_sort | Connelly, Robert |
| collection | MIT |
| description | 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). |
| first_indexed | 2024-09-23T09:22:31Z |
| format | Article |
| id | mit-1721.1/106900 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:22:31Z |
| publishDate | 2017 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1069002022-09-30T14:19:49Z Free Edge Lengths in Plane Graphs Connelly, Robert Fulek, Radoslav Morić, Filip Okamoto, Yoshio Szabó, Tibor Tóth, Csaba D. Abel, Zachary R Eisenstat, Sarah Charmian 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 Abel, Zachary R Eisenstat, Sarah Charmian 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). 2017-02-10T18:56:05Z 2017-02-10T18:56:05Z 2015-05 2015-04 2016-05-23T12:14:19Z Article http://purl.org/eprint/type/JournalArticle 0179-5376 1432-0444 http://hdl.handle.net/1721.1/106900 Abel, Zachary et al. “Free Edge Lengths in Plane Graphs.” Discrete & Computational Geometry 54.1 (2015): 259–289. https://orcid.org/0000-0002-4295-1117 https://orcid.org/0000-0002-3182-1675 en http://dx.doi.org/10.1007/s00454-015-9704-z Discrete & Computational Geometry Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Connelly, Robert Fulek, Radoslav Morić, Filip Okamoto, Yoshio Szabó, Tibor Tóth, Csaba D. Abel, Zachary R Eisenstat, Sarah Charmian Free Edge Lengths in Plane Graphs |
| title | Free Edge Lengths in Plane Graphs |
| title_full | Free Edge Lengths in Plane Graphs |
| title_fullStr | Free Edge Lengths in Plane Graphs |
| title_full_unstemmed | Free Edge Lengths in Plane Graphs |
| title_short | Free Edge Lengths in Plane Graphs |
| title_sort | free edge lengths in plane graphs |
| url | http://hdl.handle.net/1721.1/106900 https://orcid.org/0000-0002-4295-1117 https://orcid.org/0000-0002-3182-1675 |
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