Advertisement

Geometric Realization of Simplicial Complexes

  • Patrice Ossona de Mendez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)

Abstract

We show that an abstract simplicial complex Δ may be realized on a grid of IRd-1, where d = dim P(Δ) is the order dimension (Dushnik-Miller dimension) of the face poset of Δ.

Keywords

Planar Graph Linear Order Simplicial Complex Maximal Element Total Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    L. Barany, R. Howe, and H. Scarf, The complex of maximal lattice free simplices, Mathematical Programming 66 (Ser. A) (1994), 273–281.CrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Bayer, I. Peeva, and B. Sturmfels, Monomial resolutions, AMS electronic preprint #1996-10-14-012, 1996, (a.k.a. alg-geom/9610012).Google Scholar
  3. 3.
    G. Brightwell and W.T. Trotter, The order dimension of convex polytopes, SIAM Journal of Discrete Mathematics 6 (1993), 230–245.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    —, The order dimension of planar maps, SIAM jouranl on Discrete Mathematics 10 (1997), no. 4, 515–528.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    H. de Fraysseix and P. Ossona de Mendez, On topological aspects of orientations, Proc. of the Fifth Czech-Slovak Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Discrete Math., (to appear).Google Scholar
  6. 6.
    —, Regular orientations, arboricity and augmentation, DIMACS International Workshop, Graph Drawing 94, Lecture notes in Computer Science, vol. 894, 1995, pp. 111–118.Google Scholar
  7. 7.
    H. de Fraysseix, P. Ossona de Mendez, and P. Rosenstiehl, On triangle contact graphs, Combinatorics, Probability and Computing 3 (1994), 233–246.zbMATHGoogle Scholar
  8. 8.
    H. de Fraysseix, J. Pach, and R. Pollack, Small sets supporting Fary embeddings of planar graphs, Twentieth Annual ACM Symposium on Theory of Computing, 1988, pp. 426–433.Google Scholar
  9. 9.
    —, How to draw a planar graph on a grid, Combinatorica 10 (1990), 41–51.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    B. Dushnik, Concerning a certain set of arrangements, Proceedings of the AMS, vol. 1, 1950, pp. 788–796.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    B. Dushnik and E.W. Miller, Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Hibi, Algebraic combinatorics on convex polytopes, Carlslaw Publications, 1992.Google Scholar
  13. 13.
    G. Kant, Algorithms for drawing planar graphs, Ph.D. thesis, Utrecht University, Utrecht, 1993.Google Scholar
  14. 14.
    H. Scarf, The computation of economic equilibria, Cowles foundation monograph, vol. 24, Yale University Press, 1973.Google Scholar
  15. 15.
    W. Schnyder, Planar graphs and poset dimension, Order 5 (1989), 323–343.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Patrice Ossona de Mendez
    • 1
  1. 1.CNRS UMR 8557 — E.H.E.S.S.ParisFrance

Personalised recommendations