S.D. Codified Laws § 3-4-10
If any position on a board or commission whose members are appointed by the Governor without the consent of the Senate becomes vacant pursuant to § 3-4-1, the Governor must appoint a person to fill the vacancy within one hundred twenty days of the occurrence of the vacancy. If the vacancy is the result of a resignation, the person who has resigned remains on the board or commission until the Governor appoints a person to fill the vacancy or until a date set in writing by the person who resigned, whichever is sooner.
Except as provided below, if a person's term on the board or commission has expired, the person remains on the board or commission until the Governor appoints a person to the new term, and if the Governor has not appointed a person to the new term within one hundred twenty days after the term expired, the person is deemed to be reappointed for another term. If a person is unable to serve an additional term due to a term limit, the person may not remain on the board or commission after the person's term has expired and may not be deemed reappointed under this section.
This section applies to any board or commission whose members are appointed by the Governor without the consent of the Senate unless the board or commission is specifically exempted from this section by law.
SDCL 3-4-10