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 \(k\geqslant 5\). 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.

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  1. 1.

    Balinski, M.L.: On the graph structure of convex polyhedra in \(n\)-space. Pac. J. Math. 11, 431–434 (1961)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Blind, R., Mani-Levitska, P.: Puzzles and polytope isomorphisms. Aequ. Math. 34(2–3), 287–297 (1987)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Britton, D., Dunitz, J.D.: A complete catalogue of polyhedra with eight or fewer vertices. Acta Crystallogr. Sect. A 29(4), 362–371 (1973)

    Google Scholar 

  4. 4.

    Brøndsted, A.: An Introduction to Convex Polytopes. Graduate Texts in Mathematics, vol. 90. Springer, New York (1983)

  5. 5.

    Doolittle, J., Nevo, E., Pineda-Villavicencio, G., Ugon, J., Yost, D.: On the reconstruction of polytopes. Discrete Comput. Geom. https://doi.org/10.1007/s00454-018-9997-9. arXiv: 1702.08739

  6. 6.

    Fukuda, K., Miyata, H., Moriyama, S.: Classification of oriented matroids. http://www-imai.is.s.u-tokyo.ac.jp/~hmiyata/oriented_matroids/ (2013)

  7. 7.

    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry, 2nd edn. Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton (2004)

  8. 8.

    Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes—Combinatorics and Computation (Oberwolfach, 1997). DMV Seminar, vol. 29, pp. 43–73. Birkhäuser, Basel (2000)

  9. 9.

    Grünbaum, B.: Convex Polytopes. 2nd edn. Graduate Texts in Mathematics, vol. 221. Springer, New York (2003). Prepared and with a preface by V. Kaibel, V. Klee and G.M. Ziegler

  10. 10.

    Joswig, M.: Reconstructing a non-simple polytope from its graph. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes-Combinatorics and Computation (Oberwolfach, 1997). DMV Seminar, vol. 29, pp. 167–176. Birkhäuser, Basel (2000)

  11. 11.

    Kalai, G.: A simple way to tell a simple polytope from its graph. J. Comb. Theory Ser. A 49(2), 381–383 (1988)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Pineda-Villavicencio, G., Ugon, J., Yost, D.: Lower bound theorems for general polytopes. Eur. J. Comb. (to appear) (arXiv: 1509.08218)

  13. 13.

    Pineda-Villavicencio, G., Ugon, J., Yost, D.: The excess degree of a polytope. SIAM J. Discret. Math. 32(3), 2011–2046 (2018)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Przesławski, K., Yost, D.: More indecomposable polyhedra. Extr. Math. 31(2), 169–188 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Shephard, G.C.: Decomposable convex polyhedra. Mathematika 10, 89–95 (1963)

    MathSciNet  MATH  Google Scholar 

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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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Correspondence to Guillermo Pineda-Villavicencio.

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Pineda-Villavicencio, G., Ugon, J. & Yost, D. Polytopes Close to Being Simple. Discrete Comput Geom 64, 200–215 (2020). https://doi.org/10.1007/s00454-018-00053-y

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  • Reconstruction
  • Simple polytope
  • k-Skeleton

Mathematics Subject Classification

  • Primary: 52B05
  • Secondary: 52B12