Polytopes close to being simple

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⩾ 5. For k⩽ 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 ξ(P) : = 2 f1- df0, where fk 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.