Obstructions to shellability, partitionability, and sequential Cohen–Macaulayness

For a property P of simplicial complexes, a simplicial complex Γ is an obstruction to P if Γ itself does not satisfy P but all of its proper restrictions satisfy P. In this paper, we determine all obstructions to shellability of dimension ⩽2, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen–Macaulayness all coincide for dimensions ⩽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.