Polytopes close to being simple

TL;DR

This paper extends the class of polytopes known to be reconstructible from their graphs, including those with more nonsimple vertices and bounded excess degree, advancing understanding of polytope reconstructibility.

Contribution

It proves that polytopes with certain bounds on nonsimple vertices and excess degree are reconstructible from their graphs, closing previous gaps in the theory.

Findings

Polytopes with at most $d-k+3$ nonsimple vertices are reconstructible for $k extgreater=5$.

Polytopes with excess degree at most $d-1$ are reconstructible, and this bound is optimal.

Polytopes with fewer than $2d$ vertices and at most $d-1$ nonsimple vertices are pyramids.

Abstract

It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and $d-2$, showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for $d$-polytopes with $d+k$ vertices and at most $d-k+3$ nonsimple vertices, provided $k\ge 5$. For $k\le4$, the same conclusion holds under a slightly stronger assumption. Another measure of deviation from simplicity is the {\it excess degree} of a polytope, defined as $\xi(P):=2f_1-df_0$, where $f_k$ denotes the number of $k$-dimensional faces of the polytope. Simple polytopes are those with excess zero. We prove that polytopes with excess at most $d-1$ are reconstructible from their graphs, and this is best possible. An interesting intermediate result is that $d$-polytopes with less than $2d$ vertices, and at most $d-1$ nonsimple vertices, are necessarily pyramids.