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On the Reconstruction of Polytopes

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Abstract

Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most \(d-2\) nonsimple vertices is determined by its 2-skeleton; and (3) for any \(d>3\) there are two d-polytopes with \(d-1\) nonsimple vertices, isomorphic \((d-3)\)-skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes.

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Notes

  1. Every subset of the d-sphere which is a homeomorphic image of the \((d-1)\)-sphere divides the d-sphere into two connected components.

  2. A sink is a vertex with no directed edges going out.

  3. A source is a vertex with no directed edges coming in.

References

  1. Blind, G., Blind, R.: The almost simple cubical polytopes. Discrete Math. 184(1–3), 25–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Friedman, E.J.: Finding a simple polytope from its graph in polynomial time. Discrete Comput. Geom. 41(2), 249–256 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fukuda, K., Miyata, H., Moriyama, S.: Complete enumeration of small realizable oriented matroids. Discrete Comput. Geom. 49(2), 359–381 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

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

  8. http://guillermo.com.au/pdfs/ConstructPolytope-A.txt

  9. 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)

    Chapter  Google Scholar 

  10. Joswig, M., Kaibel, V., Körner, F.: On the $k$-systems of a simple polytope. Isr. J. Math. 129, 109–117 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaibel, V.: Reconstructing a simple polytope from its graph. In: Jünger, M., et al. (eds.) Combinatorial Optimization-Eureka, You Shrink! Lecture Notes in Computer Science, vol. 2570, pp. 105–118. Springer, Berlin (2003)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Pineda-Villavicencio, G., Ugon, J., Yost, D.: The excess degree of a polytope. arXiv:1703.10702 (2017)

  14. Pineda-Villavicencio, G., Ugon, J., Yost, D.: Polytopes close to being simple. arXiv:1704.00854 (2017)

  15. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

    Book  Google Scholar 

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Acknowledgements

We thank Micha Perles for helpful discussions and the referees for many valuable comments and suggestions. Guillermo Pineda would like to thank Michael Joswig for the hospitality at the Technical University of Berlin and for many fruitful discussions on the topics of this research. Joseph Doolittle would like to thank Margaret Bayer for pushing for more results and keeping the direction of exploration straight.

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Authors and Affiliations

  1. College of Liberal Arts and Sciences, University of Kansas, Lawrence, KS, 66045, USA

    Joseph Doolittle

  2. Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Eran Nevo

  3. Centre for Informatics and Applied Optimisation, Federation University Australia, Mount Helen, Victoria, 3353, Australia

    Guillermo Pineda-Villavicencio & David Yost

  4. School of Information Technology, Deakin University, Melbourne, Burwood VIC, 3125, Australia

    Julien Ugon

Authors
  1. Joseph Doolittle
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  2. Eran Nevo
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  3. Guillermo Pineda-Villavicencio
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  4. Julien Ugon
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  5. David Yost
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Corresponding author

Correspondence to Guillermo Pineda-Villavicencio.

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Editor in Charge: Kenneth Clarkson

Research of Nevo was partially supported by Israel Science Foundation Grant ISF-1695/15. Research of Pineda-Villavicencio was partly supported by the Indonesian government Scheme P3MI, Grant No. 1016/I1.C01/PL/2017. Research of Ugon was supported by ARC discovery Project DP180100602.

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Doolittle, J., Nevo, E., Pineda-Villavicencio, G. et al. On the Reconstruction of Polytopes. Discrete Comput Geom 61, 285–302 (2019). https://doi.org/10.1007/s00454-018-9997-9

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  • DOI: https://doi.org/10.1007/s00454-018-9997-9

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