Vertex decomposable graphs and obstructions to shellability

Details Vertex decomposable graphs and obstructions to shellability Vertex decomposable graphs and obstructions to shellability

2008-10-01

arxiv.org/abs/0810.0311v3

Proc. Amer. Math. Soc. 137 (2009), no. 10, 3235-3246

Mathematics

Combinatorics

13 pages, 3 figures. v2: Improved exposition, added Section 5.2 and additional references. v3: minor corrections for publicati

Scientific paper

10.1090/S0002-9939-09-09981-X

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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