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36
Vertex decomposable graphs and obstructions to shellability
, 2009
"... Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMac ..."
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Cited by 29 (2 self)
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Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMacaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.
RESOLUTIONS OF SQUAREFREE MONOMIAL IDEALS VIA FACET IDEALS: A SURVEY
, 2006
"... We survey some recent results on the minimal graded free resolution of a squarefree monomial ideal. The theme uniting these results is the pointofview that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a ne ..."
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Cited by 22 (3 self)
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We survey some recent results on the minimal graded free resolution of a squarefree monomial ideal. The theme uniting these results is the pointofview that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the StanleyReisner correspondence, to associate to a squarefree monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current stateoftheart with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part
Chordal and sequentially cohenmacaulay clutters
"... We 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 CohenMacaulay; ..."
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Cited by 21 (1 self)
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We 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 CohenMacaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal nonchordal 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. 1
BOUNDS FOR THE REGULARITY OF EDGE IDEAL OF VERTEX DECOMPOSABLE AND SHELLABLE GRAPHS
, 2009
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On the structure of the hvector of a paving matroid
 European J. Combin
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COHENMACAULAY EDGE IDEAL WHOSE HEIGHT IS HALF OF THE NUMBER OF VERTICES
, 2009
"... We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number ♯V (G) of the vertices. We give CohenMacaulay criteria for such graphs. ..."
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Cited by 9 (6 self)
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We consider a class of graphs G such that the height of the edge ideal I(G) is half of the number ♯V (G) of the vertices. We give CohenMacaulay criteria for such graphs.
Vertex decomposability and regularity of very wellcovered graphs
 J. PURE AND APPL. ALG
, 2010
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