Signable posets and partitionable simplicial complexes
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The notion of apartitionable simplicial complex is extended to that of asignable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termedtotal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.
- L. J. Billera and J. S. Provan. Decomposition of simplicial complexes related to diameters of convex polyhedra.Math. Oper. Res., 5:576–594, 1980. CrossRef
- A. Björner. Homology and shellability of matroids and geometric lattices. In N. White, editor,Matroid Applications, chapter 7, pages 226–283. Cambridge University Press, Cambridge, 1992.
- A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler.Oriented Matroids. Cambridge University Press, Cambridge, 1993.
- A. Björner and M. Wachs. On lexicographically shellable posets.Trans. Amer. Math. Soc., 277:323–341, 1983. CrossRef
- G. Danaraj and V. Klee. A representation of 2-dimensional pseudomanifolds and its use in the design of a linear-time shelling algorithm.Ann. Discrete Math., 2:53–63, 1977.
- G. Danaraj and V. Klee. Which spheres are shellable?Annals of Discrete Mathematics, 2:33–52, 1978.
- M. R. Garey and D. S. Johnson.Computers and Intractability. Freeman, San Francisco, CA, 1979.
- B. Kind and P. Kleinschmidt. Schälbare Cohen-Macaulay Komplexe und ihre Parametrisierung.Math. Z., 167:173–179, 1979. CrossRef
- P. Kleinschmidt and Z. Smilansky. New results for simplicial spherical polytopes. In J. E. Goodman, R. Pollack, and W. Steiger, editors,Special Year on Discrete and Computational Geometry, pages 187–197. DIMACS Series, vol. 6. American Mathematical Society, Providence, RI, 1991.
- M. Las Vergnas. Convexity in oriented matroids.J. Combin. Theory Ser. B, 29:231–243, 1980. CrossRef
- N. Linial. Hard enumeration problems in geometry and combinatorics.SIAM J. Algebraic Discrete Methods, 7:331–335, 1986. CrossRef
- P. Mani-Levitska. Convex polytopes and smooth structures on manifolds. Euroconference on Combinatorial Geometry, Crete, July 1994.
- P. McMullen. The maximum numbers of faces of a convex polytope.Mathematika, 17:179–184, 1970. CrossRef
- S. D. Noble. Recognizing a partitionable simplicial complex is inNP. Discr. Math., to appear.
- S. Onn. Partitionable Posets. Manuscript, 20 pages, 1995.
- R. Seidel. Constructing higher dimensional convex hulls at logarithmic cost per face.Proc. 18th Ann. ACM Symp. on Theory of Computing, pages 404–413, 1986.
- R. P. Stanley.Combinatorics and Commutative Algebra. Birkhäuser, Boston, 1983.
- R. P. Stanley.Enumerative Combinatorics vol. 1, Wadsworth and Brooks/Cole, Pacific Grove, CA, 1986.
- I. A. Volodin, V. E. Kuznetsov, and A. T. Fomenko. The problem of discriminating algorithmically the standard three-dimensional sphere.Russian Math. Surveys, 29:71–172, 1974. CrossRef
- Signable posets and partitionable simplicial complexes
Discrete & Computational Geometry
Volume 15, Issue 4 , pp 443-466
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