Some algebraic properties of hypergraphs

Emtander, Eric; Mohammadi, Fatemeh; Moradi, Somayeh

doi:10.1007/s10587-011-0031-0

Cited by 9 publications

(6 citation statements)

References 26 publications

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“…As mentioned in Example 4.8, there are clutters such that every subclutter contains a complete-neighborhood vertex, but that are not chordal. We can however use a similar technique to show that clutters with a complete-neighborhood vertex in every induced subclutter are vertex decomposable, improving the previous result [14,Theorem 4.3] that such clutters are shellable: Proposition 6.11. Let C be a d-uniform clutter such that every induced subclutter has a complete-neighborhood vertex.…”

Section: Alexander Duals Of Complements To Chordal Clutterssupporting

confidence: 55%

Chordal and Sequentially Cohen-Macaulay Clutters

Woodroofe¹

2011

Electron. J. Combin.

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Mathematics Subject Classifications: 05E45, 13F55, 13C14, 05C65. AbstractWe extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices.

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Section: Alexander Duals Of Complements To Chordal Clutterssupporting

confidence: 55%

Chordal and Sequentially Cohen-Macaulay Clutters

Woodroofe¹

2011

Electron. J. Combin.

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“…In particular, if C is a chordal clutter in Emtander's sense, then I C has a linear resolution over any field. In his later work, he showed that for a chordal clutter C, in fact, the ideal I C has linear quotients [8] while it is still an open question whether this is true for generalized chordal clutters as well.…”

Section: Claimmentioning

confidence: 99%

“…Let ∆ be a triangulation of the dunce hat with 8 vertices as shown in Figure 4, which is originally introduced by Zeeman [25]. Let C be the 5-uniform clutter C = C 8,5 \ F ∆ , where ∆ = [8] \ F : F ∈ F (∆) .…”

Section: Other Known Chordalities Another Notion Of Chordality Has Be...mentioning

confidence: 99%

Stability of Betti numbers under reduction processes: Towards chordality of clutters

Bigdeli

Pour

Zaare-Nahandi

2017

Journal of Combinatorial Theory, Series A

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For a given clutter C, let I := I C be the circuit ideal in the polynomial ring S. In this paper, we show that the Betti numbers of I and I + (xF ) are the same in their non-linear strands, for some suitable F ∈ C. Motivated by this result, we introduce a class of clutters that we call chordal. This class, is a natural extension of the class of chordal graphs and has the nice property that the circuit ideal associated to any member of this class has a linear resolution over any field. Finally we compare this class with all known families of clutters which generalize the notion of chordality, and show that our class contains several important previously defined classes of chordal clutters. We also show that in comparison with others, this class is possibly the best approximation to the class of d-uniform clutters with linear resolution over any field.2010 Mathematics Subject Classification. Primary 13D02, 13F55; Secondary 05E45, 05C65.

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“…What spoils things is the connectedness property for graphs. This property is central in the definition of graphical building set whereas for hypergraphs the notion of being connected may be defined in many different but equally natural ways (see [6] and [10] for examples). It is thus not clear how to approach the problem.…”

Section: About Being Chordal Triangulated Et Ceteramentioning

confidence: 99%

A class of hypergraphs that generalizes chordal graphs

Emtander¹

2010

MATH. SCAND.

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In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given by H. T. Hà and A. Van Tuyl, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. R. Fröberg has showed that the chordal graphs corresponds to graph algebras, R/I (G), with linear resolutions. We extend Fröberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call d-flag complexes.

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Some algebraic properties of hypergraphs

Cited by 9 publications

References 26 publications

Chordal and Sequentially Cohen-Macaulay Clutters

Chordal and Sequentially Cohen-Macaulay Clutters

Stability of Betti numbers under reduction processes: Towards chordality of clutters

A class of hypergraphs that generalizes chordal graphs

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