  Inspired by the works of Adiprasito, Babson, Nevo, and Murai on the
$g$-conjecture, we consider different classes of PL-spheres and the relations
between them. We focus on a certain class of spheres that is in the
intersection of vertex-decomposable spheres (a concept due to Provan and
Billera) and strongly edge-decomposable spheres (a concept due to Eran Nevo).
The spheres in this class are exactly those vertex-decomposable ones for which
Adiprasito's recent proof method works.
