On the Reconstruction of Polytopes

Joseph Doolittle, Eran Nevo, Guillermo Pineda-Villavicencio*, Julien Ugon, David Yost

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.

Original languageAmerican English
Pages (from-to)285-302
Number of pages18
JournalDiscrete and Computational Geometry
Volume61
Issue number2
DOIs
StatePublished - 15 Mar 2019

Bibliographical note

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Reconstruction
  • Simple polytope
  • k-Skeleton

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