Incremental construction properties in dimension two—shellability, extendable shellability and vertex decomposability

Moriyama, Sonoko; Takeuchi, Fumihiko

doi:10.1016/s0012-365x(02)00771-9

Cited by 13 publications

(12 citation statements)

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“…(Provan and Billera proved that any vertex decomposable simplicial complex satisfies Hirsh's conjecture, while Walkup's example was a counter-example to this example.) The short note of Moriyama and Takeuchi [21] contains a list of the minimal two dimensional simplicial complexes that are vertex decomposable, but not shellable. We checked these complexes, but none of them are the independence complex of a graph.…”

Section: Minimal Examplesmentioning

confidence: 99%

Independence Complexes of Well-Covered Circulant Graphs

Earl

Meulen

Tuyl

2016

Experimental Mathematics

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We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-Milanič, Brown-Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g., vertex decomposable, shellable) or topological (e.g., Cohen-Macaulay, Buchsbaum) properties. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen-Macaulay, and/or Buchsbaum. A highlight of this search is an example of a graph whose independence complex is shellable but not vertex decomposable.

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Section: Minimal Examplesmentioning

confidence: 99%

Independence Complexes of Well-Covered Circulant Graphs

Earl

Meulen

Tuyl

2016

Experimental Mathematics

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“…Theorem B] constructed a simplicial complex which is extendably shellable, but not vertex-decomposable. One can construct also examples of shellable, but not extendably shellable simplicial complexes (see, e.g.,[26, Theorem A]).…”

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confidence: 99%

Glicci simplicial complexes

Nagel

Römer

2008

Journal of Pure and Applied Algebra

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a b s t r a c tThis note is a case study for the potential of liaison-theoretic methods to applications in Combinatorics. One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertexdecomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and Stanley decompositions.

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“…This of course implies the k = n − 3 case of Simon's conjecture and also provides a generalization of Kleinschmidt's results. Our result is also best possible in the sense that there exists 2-dimensional shellable complexes on 6 vertices that are not extendably shellable (see [13], [3]).…”

Section: Introductionmentioning

confidence: 80%