$k$-shellable simplicial complexes and graphs

Rahmati-Asghar, Rahim

doi:10.7146/math.scand.a-102975

Cited by 3 publications

(4 citation statements)

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“…This gives us information about some homological invariants of I ∆ ∨ such as Betti numbers. Recently, the notion of a k-decomposable ideal was introduced in [19] and it was proved that it is the dual concept for k-decomposable simplicial complexes. In the case k = 0, 0-decomposable simplicial complexes are precisely vertex decomposable simplicial complexes.…”

Section: Vertex Decomposability and Vertex Splittable Idealsmentioning

confidence: 99%

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On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Moradi¹,

Khosh-Ahang²

2016

MATH. SCAND.

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Abstract. In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex ∆ is vertex decomposable if and only if I ∆ ∨ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of R/I ∆ , when ∆ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph G, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when G is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

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Section: Vertex Decomposability and Vertex Splittable Idealsmentioning

confidence: 99%

“…) and [u, v] is the greatest common divisor of u and v, while in Definition 2.1, it is not the case, i.e., the way that a monomial ideal splits in Definition 2.1 is different from one in [19,Definition 2.3]. For example let I = (xx 1 , .…”

Section: Vertex Decomposability and Vertex Splittable Idealsmentioning

confidence: 99%

On Vertex Decomposable Simplicial Complexes and Their Alexander Duals

Moradi¹,

Khosh-Ahang²

2016

MATH. SCAND.

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“…The following theorem was proved in [10]. In order to prove Theorem 2.4, one needs to know the construction of the order of linear quotients for a decomposable ideal.…”

Section: Regularity and Projective Dimension Of The Stanley-reisner R...mentioning

confidence: 99%

“…It is known that a d-dimensional simplicial complex is d-decomposable if and only if it is shellable (see [14,Theorem 3.6]). In [10], the concept of a kdecomposable monomial ideal was introduced and it was proved that a simplicial complex ∆ is k-decomposable if and only if I ∆ ∨ is a k-decomposable ideal.…”

Section: Introductionmentioning

confidence: 99%

Homological invariants of the Stanley–Reisner ring of a k-decomposable simplicial complex

Moradi

2017

Asian-European J. Math.

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We study the regularity and the projective dimension of the Stanley-Reisner ring of a k-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of kdecomposable monomial ideals which is the dual concept for k-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a chordal clutter H, an upper bound for reg(I(H)) is given in terms of the regularities of edge ideals of some chordal clutters which are minors of H.2010 Mathematics Subject Classification. Primary 13D02, 05E45, 05E40; Secondary 16E05.

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