  We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$
satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty
or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf
K}; {\mathbb Z}_2)$ facets, where $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$
denotes the $d$th reduced Betti number.
  In this paper, we show that if the link of each face of a pure simplicial
complex ${\mathbf K}$ (including the link of the empty face which is the whole
${\mathbf K}$) satisfy the removal-collapsibility condition, then the second
barycentric subdivision of ${\mathbf K}$ is vertex decomposable and in
particular shellable. This is a higher dimensional generalization of a result
of Hachimori, who proved that that if the link of each vertex of a pure
2-dimensional simplicial complex ${\mathbf K}$ is connected, and ${\mathbf K}$
becomes simplicially collapsible after removing $\tilde{\chi}({\mathbf K})$
facets, where $\tilde \chi ({\mathbf K})$ denotes the reduced Euler
characteristic, then the second barycentric subdivision of ${\mathbf K}$ is
shellable.
  For the proof, we introduce a new variant of decomposability of a simplicial
complex, stronger than vertex decomposability, which we call star
decomposability. This notion may be of independent interest.
