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
Abstract
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.