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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Balinski, M.L.: On the graph structure of convex polyhedra in \(n\)-space. Pac. J. Math. 11, 431–434 (1961)
Blind, R., Mani-Levitska, P.: Puzzles and polytope isomorphisms. Aequ. Math. 34(2–3), 287–297 (1987)
Britton, D., Dunitz, J.D.: A complete catalogue of polyhedra with eight or fewer vertices. Acta Crystallogr. Sect. A 29(4), 362–371 (1973)
Brøndsted, A.: An Introduction to Convex Polytopes. Graduate Texts in Mathematics, vol. 90. Springer, New York (1983)
Fukuda, K., Miyata, H., Moriyama, S.: Classification of oriented matroids. http://www-imai.is.s.u-tokyo.ac.jp/~hmiyata/oriented_matroids/ (2013)
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)
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)
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
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)
Kalai, G.: A simple way to tell a simple polytope from its graph. J. Comb. Theory Ser. A 49(2), 381–383 (1988)
Pineda-Villavicencio, G., Ugon, J., Yost, D.: Lower bound theorems for general polytopes. Eur. J. Comb. (to appear) (arXiv: 1509.08218)
Pineda-Villavicencio, G., Ugon, J., Yost, D.: The excess degree of a polytope. SIAM J. Discret. Math. 32(3), 2011–2046 (2018)
Przesławski, K., Yost, D.: More indecomposable polyhedra. Extr. Math. 31(2), 169–188 (2016)
Shephard, G.C.: Decomposable convex polyhedra. Mathematika 10, 89–95 (1963)
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Editor in charge: Kenneth Clarkson
About this article
Cite this article
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
- Simple polytope
Mathematics Subject Classification
- Primary: 52B05
- Secondary: 52B12