Reconstructing Nearly Simple Polytopes from their Graph

TL;DR

This paper extends the understanding of polytope reconstruction from graphs by identifying classes of nearly simple polytopes that are reconstructible, and providing examples and partial classifications beyond simple polytopes.

Contribution

It introduces the concept of $h$-nearly simple polytopes and proves reconstructibility for 1- and 2-nearly simple cases, expanding prior results.

Findings

1-nearly simple and 2-nearly simple polytopes are reconstructible from their graphs

A 3-nearly simple polytope may not be reconstructible from its graph

Partial list of reconstructible polytopes provided in a non-constructive manner

Abstract

We present a partial description of which polytopes are reconstructible from their graphs. This is an extension of work by Blind and Mani (1987) and Kalai (1988), which showed that simple polytopes can be reconstructed from their graphs. In particular, we introduce a notion of $h$-nearly simple and prove that 1-nearly simple and 2-nearly simple polytopes are reconstructible from their graphs. We also give an example of a 3-nearly simple polytope which is not reconstructible from its graph. Furthermore, we give a partial list of polytopes which are reconstructible from their graphs in an entirely non-constructive way.