It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d -polytopes with at most $$d-2$$ d - 2 nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and $$d-2$$ 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 $$d+k$$ d + k vertices and at most $$d-k+3$$ d - k + 3 nonsimple vertices, provided $$k\geqslant 5$$ k ⩾ 5 . For $$k\leqslant 4$$ 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 $$\xi (P):=2f_1-df_0$$ ξ ( P ) : = 2 f 1 - d f 0 , where $$f_k$$ 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 $$d-1$$ d - 1 are reconstructible from their graphs, and this is best possible. An interesting intermediate result is that d -polytopes with less than 2 d vertices, and at most $$d-1$$ d - 1 nonsimple vertices, are necessarily pyramids.

中文翻译：

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接近简单的多面体

已知最多具有两个非简单顶点的多面体可以从它们的图中重构，并且最多具有 $$d-2$$ d-2 个非简单顶点的 d-多面体可以从它们的 2-骨架中重构。在这里，我们缩小了 2 和 $$d-2$$ d - 2 之间的差距，表明具有两个以上非简单顶点的某些多胞体可以从它们的图中重建。特别是，我们证明了图的可重构性也适用于具有 $$d+k$$ d + k 个顶点和至多 $$d-k+3$$ d - k + 3 个非简单顶点的 d -polytope，前提是 $$ k\geqslant 5$$ k ⩾ 5 。对于 $$k\leqslant 4$$ k ⩽ 4 ，同样的结论在稍微强一点的假设下成立。另一个偏离简单性的度量是多胞体的过度度，定义为 $$\xi (P):=2f_1-df_0$$ ξ ( P ) : = 2 f 1 - df 0 ，其中 $$f_k$$ fk 表示多胞体的 k 维面数。简单的多面体是那些具有多余零的多面体。我们证明了最多 $$d-1$$ d - 1 的多胞体可以从它们的图中重构，这是最好的可能。一个有趣的中间结果是具有少于 2 d 个顶点且至多 $$d-1$$ d - 1 个非简单顶点的 d -polytopes 必然是金字塔。