Graphs, Skeleta and Reconstruction of Polytopes

TL;DR

This survey reviews key results and open questions on reconstructing face lattices of various polytopes from graphs and partial data, highlighting advances and challenges in polytope theory.

Contribution

It compiles and discusses diverse results and open problems related to reconstructing face lattices of different classes of polytopes from graphs and partial information.

Findings

Reconstruction of face lattices from graphs is possible for simple polytopes.

The survey highlights open questions in reconstructing face lattices of complex polytopes.

Various classes of polytopes have different reconstructibility properties.

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 Gr\"unbaum'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.