The top of the Etrog is referred to as the “Pitom.” It is comprised of the neck and the bulbous wood-like “Shoshanta,” literally meaning rose. If the Pitom fell off in its entirety, Maran rules that the Etrog is Pasul (invalid) on the first day. It should be noted that there is a minority opinion of Rishonim that is lenient even if the Pitom fell off. Apparently they hold that since there are species of Etrogim that grow without a Pitom, e.g. Yemenite and Moroccan Etrogim, its removal does not impeach the “Hadar”-beauty of the Etrog. Nevertheless, the Halacha is strict, since the neck of the Pitom is made of the same texture as the body of the Etrog, its absence renders the Etrog lacking in wholeness. It is no different than if someone took a bite out of his Etrog. However, if only the Shoshanta fell off, but the neck remains, the Etrog is Kosher. The wood-like Shoshanta is not an integral part of the Etrog, and its absence does not render the Etrog missing from its wholeness or Hidur.
The bottom stem of the Etrog is called the “Okes.” If it falls off altogether, the Etrog is Pasul. However, as long as enough of it remains to fill the cavity from which it grows, the Etrog is Kosher. In fact, if the stem was very long, it is permissible to cut it. If the stem detached from the Etrog, Hacham Ovadia permits reattaching it to the cavity using an adhesive or pins. Since its absence is only an issue of Hadar, its beauty can be restored with “cosmetic surgery.” On the other hand, the lack of a Pitom is a defect in the wholeness of the Etrog; therefore, artificially reattaching it would not help. Of course, one must exercise caution that the pins don’t cause more damage to the Etrog.