On the Reconstruction of Polytopes.
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| Title: | On the Reconstruction of Polytopes. |
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| Authors: | Doolittle, Joseph1 jdoolitt@ku.edu, Nevo, Eran2 nevo@math.huji.ac.il, Pineda-Villavicencio, Guillermo3 work@guillermo.com.au, Ugon, Julien4 julien.ugon@deakin.edu.au, Yost, David3 d.yost@federation.edu.au |
| Source: | Discrete & Computational Geometry. Mar2019, Vol. 61 Issue 2, p285-302. 18p. |
| Subjects: | Polytopes, Isomorphous structures, Lattice constants, Generalization, Reconstruction (Graph theory) |
| Abstract: | Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d-2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d>3 there are two d-polytopes with d-1 nonsimple vertices, isomorphic (d-3)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d-2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d>3 there are two d-polytopes with d-1 nonsimple vertices, isomorphic (d-3)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-018-9997-9 |
