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 $$d-2$$ 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 $$k\geqslant 5$$. For $$k\leqslant 4$$, the same conclusion holds under a slightly stronger assumption. Another measure of deviation from simplicity is the 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.