Journal of Combinatorial Theory, Ser. A 118 Issue 5 (2011), 1608-1623.

Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness by Masahiro Hachimori and Kenji Kashiwabara


For a property $\cal P$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\cal P$ if $\Gamma$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. In this paper, we determine all obstructions to shellability of dimensions $\le 2$, refining the previous work by Wachs. This result derives that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes.