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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Vertex decomposable graphs and obstructions to shellability


Author: Russ Woodroofe
Journal: Proc. Amer. Math. Soc. 137 (2009), 3235-3246
MSC (2000): Primary 13F55, 05C38, 05E99
Published electronically: June 4, 2009
MathSciNet review: 2515394
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Abstract: 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 $ 5$-chordal graphs with no chordless $ 4$-cycles are shellable and sequentially Cohen-Macaulay. 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.


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Additional Information

Russ Woodroofe
Affiliation: Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130
Email: russw@math.wustl.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09981-X
PII: S 0002-9939(09)09981-X
Keywords: Sequentially Cohen-Macaulay, independence complex, edge ideal, chordal graphs
Received by editor(s): January 8, 2009
Published electronically: June 4, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.