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41
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.
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 on the regularity and projective dimension of ideals associated to graphs
 JOURNAL OF ALGEBRAIC COMBINATORICS
, 2013
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Regularity of edge ideals of C4free graphs via the topology of the lcmlattice
, 2010
"... We study the topology of the lcmlattice of edge ideals and derive upper bounds on the CastelnuovoMumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4cycle in their complement. Using the above method we obtain sharp upper bounds on t ..."
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Cited by 11 (0 self)
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We study the topology of the lcmlattice of edge ideals and derive upper bounds on the CastelnuovoMumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the original graph is claw free with no induced 4cycle in its complement. For the last family we show that the second power of the edge ideal has a linear resolution.
BOUNDS FOR THE REGULARITY OF EDGE IDEAL OF VERTEX DECOMPOSABLE AND SHELLABLE GRAPHS
, 2009
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Vertex decomposability and regularity of very wellcovered graphs
 J. PURE AND APPL. ALG
, 2010
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Cellular resolutions of cointerval ideals
, 2010
"... Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d–uniform hypergraphs coincides with the complements of interval graphs (for the case d = 2), and strictly contains the class of ‘strongly stable ’ hypergraphs corresponding to pure shifted simpl ..."
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Cited by 6 (1 self)
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Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d–uniform hypergraphs coincides with the complements of interval graphs (for the case d = 2), and strictly contains the class of ‘strongly stable ’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.