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arXiv · 1704.00854

Polytopes close to being simple

Guillermo Pineda-Villavicencio·Julien Ugon·David Yost
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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.

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Guillermo Pineda-Villavicencio, Julien Ugon, David Yost. 2017-04-04. Polytopes close to being simple. https://arxiv.org/abs/1704.00854

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