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SimplicialDecomposability -- various decomposability routines for simplicial complexes.

Description

This package includes routines for vertex decomposability and shellability for arbitrary simplicial complexes as well as routines for k-decomposability. Moreover, it can find a shelling order for a shellable simplicial complex.

References:

[BW-1] A. Bjoerner and M. Wachs, "Shellable nonpure complexes and posets, I," Trans. of the AMS 348 (1996), 1299–1327.

[BW-2] A. Bjoerner and M. Wachs, "Shellable nonpure complexes and posets, II," Trans. of the AMS 349 (1997), 3945–3975.

[MT] S. Moriyama and F. Takeuchi, "Incremental construction properties in dimension two: shellability, extendable shellability and vertex decomposability," Discrete Math. 263 (2003), 295–296.

[PB] J. S. Provan and L. J. Billera, "Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra," Math. of Operations Research 5 (1980), 576–594.

[St] R. Stanley, "Combinatorics and Commutative Algebra," 2nd edition. Progress in Mathematics, 41. Birkhaeuser Boston, Inc. Boston, MA, 1996.

[Wo] R. Woodroofe, "Chordal and sequentially Cohen-Macaulay clutters," arXiv:0911.4697v1.

Author

Certification a gold star

Version 1.0.5 of this package was accepted for publication in volume 2 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2010-08-03, in the article Simplicial Decomposability (DOI: 10.2140/jsag.2010.2.20). That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.0.6 of SimplicialDecomposability.

Source code

The source code from which this documentation is derived is in the file SimplicialDecomposability.m2.

Exports

  • Functions and commands
    • allFaces -- returns all faces of a simplicial complex, up to a given dimension
    • faceDelete -- computes the face deletion for a simplicial complex
    • fTriangle -- determines the f-Triangle of a simplicial complex
    • hTriangle -- determines the h-Triangle of a simplicial complex
    • hVector -- the h-vector of a simplicial complex
    • isDecomposable -- determines whether a simplicial complex is k-decomposable
    • isSheddingFace -- determines whether a face of a simplicial complex is a shedding face
    • isSheddingVertex -- determines whether a vertex of a simplicial complex is a shedding vertex
    • isShellable -- determines whether a simplicial complex is shellable
    • isShelling -- determines whether a list of faces is a shelling
    • isSimplex -- determines whether a simplicial complex is simplex
    • isVertexDecomposable -- determines whether a simplicial complex is vertex-decomposable
    • shellingOrder -- finds a shelling of a simplicial complex, if one exists
  • Methods
    • allFaces(SimplicialComplex) -- see allFaces -- returns all faces of a simplicial complex, up to a given dimension
    • allFaces(ZZ,SimplicialComplex) -- see allFaces -- returns all faces of a simplicial complex, up to a given dimension
    • faceDelete(RingElement,SimplicialComplex) -- see faceDelete -- computes the face deletion for a simplicial complex
    • fTriangle(SimplicialComplex) -- see fTriangle -- determines the f-Triangle of a simplicial complex
    • hTriangle(SimplicialComplex) -- see hTriangle -- determines the h-Triangle of a simplicial complex
    • hVector(SimplicialComplex) -- see hVector -- the h-vector of a simplicial complex
    • isDecomposable(ZZ,SimplicialComplex) -- see isDecomposable -- determines whether a simplicial complex is k-decomposable
    • isSheddingFace(RingElement,SimplicialComplex) -- see isSheddingFace -- determines whether a face of a simplicial complex is a shedding face
    • isSheddingVertex(RingElement,SimplicialComplex) -- see isSheddingVertex -- determines whether a vertex of a simplicial complex is a shedding vertex
    • isShellable(SimplicialComplex) -- see isShellable -- determines whether a simplicial complex is shellable
    • isShelling(List) -- see isShelling -- determines whether a list of faces is a shelling
    • isSimplex(SimplicialComplex) -- see isSimplex -- determines whether a simplicial complex is simplex
    • isVertexDecomposable(SimplicialComplex) -- see isVertexDecomposable -- determines whether a simplicial complex is vertex-decomposable
    • shellingOrder(SimplicialComplex) -- see shellingOrder -- finds a shelling of a simplicial complex, if one exists
  • Symbols

For the programmer

The object SimplicialDecomposability is a package.