Polytopes close to being simple
Abstract
It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$polytopes with at most $d2$ nonsimple vertices are reconstructible from their 2skeletons. Here we close the gap between 2 and $d2$, 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 $dk+3$ 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 $\xi(P):=2f_1df_0$, 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 $d1$ 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 $d1$ nonsimple vertices, are necessarily pyramids.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.00854
 arXiv:
 arXiv:1704.00854
 Bibcode:
 2017arXiv170400854P
 Keywords:

 Mathematics  Combinatorics;
 52B05 (Primary) 52B12 (Secondary)
 EPrint:
 17 pages