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Graphs, skeleta and reconstruction of polytopes

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Abstract

A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grünbaum’s 1967 book. This survey paper includes varied results and open questions by many researchers on simplicial polytopes, nearly simple polytopes, cubical polytopes, zonotopes, crosspolytopes, and Eulerian posets.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, USA

    M. M. Bayer

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  1. M. M. Bayer
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Correspondence to M. M. Bayer.

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Dedicated to Tibor Bisztriczky, Gábor Fejes Tóth and Endre Makai, on the occasion of their birthdays

This article is based in part on work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, and while supported by a University of Kansas sabbatical, during the Fall 2017 semester.

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Bayer, M.M. Graphs, skeleta and reconstruction of polytopes. Acta Math. Hungar. 155, 61–73 (2018). https://doi.org/10.1007/s10474-018-0804-0

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