333 research outputs found
Obstructions to weak decomposability for simplicial polytopes
Provan and Billera introduced notions of (weak) decomposability of simplicial
complexes as a means of attempting to prove polynomial upper bounds on the
diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera
and Klee provided the first examples of simplicial polytopes that are not
weakly vertex-decomposable. These polytopes are polar to certain simple
transportation polytopes. In this paper, we refine their analysis to prove that
these -dimensional polytopes are not even weakly -decomposable.
As a consequence, (weak) decomposability cannot be used to prove a polynomial
version of the Hirsch conjecture
More indecomposable polyhedra
We apply combinatorial methods to a geometric problem: the classification of
polytopes, in terms of Minkowski decomposability. Various properties of
skeletons of polytopes are exhibited, each sufficient to guarantee
indecomposability of a significant class of polytopes. We illustrate further
the power of these techniques, compared with the traditional method of
examining triangular faces, with several applications. In any dimension , we show that of all the polytopes with or fewer edges,
only one is decomposable. In 3 dimensions, we complete the classification, in
terms of decomposability, of the 260 combinatorial types of polyhedra with 15
or fewer edges.Comment: PDFLaTeX, 21 pages, 6 figure
Not all simplicial polytopes are weakly vertex-decomposable
In 1980 Provan and Billera defined the notion of weak -decomposability for
pure simplicial complexes. They showed the diameter of a weakly
-decomposable simplicial complex is bounded above by a polynomial
function of the number of -faces in and its dimension. For weakly
0-decomposable complexes, this bound is linear in the number of vertices and
the dimension. In this paper we exhibit the first examples of non-weakly
0-decomposable simplicial polytopes
More indecomposable polyhedra
We apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of Minkowski decomposability. Various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. We illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. In any dimension dâ 2, we show that of all the polytopes with d^2 + ½d or fewer edges, only one is decomposable. In 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges.peerReviewe
Obstructions to weak decomposability for simplicial polytopes
International audienceProvan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these -dimensional polytopes are not even weakly -decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial version of the Hirsch Conjecture
Combinatorial 3-manifolds with 10 vertices
We give a complete enumeration of all combinatorial 3-manifolds with 10
vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as
well as 518 vertex-minimal triangulations of the sphere product
and 615 triangulations of the twisted sphere product S^2_\times_S^1.
All the 3-spheres with up to 10 vertices are shellable, but there are 29
vertex-minimal non-shellable 3-balls with 9 vertices.Comment: 9 pages, minor revisions, to appear in Beitr. Algebra Geo
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