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On the Reconstruction of Polytopes.

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Title: On the Reconstruction of Polytopes.
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]
Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Polytopes%22">Polytopes</searchLink><br /><searchLink fieldCode="DE" term="%22Isomorphous+structures%22">Isomorphous structures</searchLink><br /><searchLink fieldCode="DE" term="%22Lattice+constants%22">Lattice constants</searchLink><br /><searchLink fieldCode="DE" term="%22Generalization%22">Generalization</searchLink><br /><searchLink fieldCode="DE" term="%22Reconstruction+%28Graph+theory%29%22">Reconstruction (Graph theory)</searchLink>
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  Data: 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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  Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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        Type: general
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