Polytopes Close to Being Simple
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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.
KeywordsReconstruction Simple polytope k-Skeleton
Mathematics Subject ClassificationPrimary: 52B05 Secondary: 52B12
Guillermo Pineda would like to thank Michael Joswig for the hospitality at the Technical University of Berlin and for suggesting looking at the reconstruction problem for polytopes with small number of vertices.
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