There was a rule that the Imperial Prince who did not ascend to the throne was given Ingo as the father of the Emperor when his son was enthroned. (See Daijo Tenno)
その他、皇位についていない親王の皇子が天皇の後継に立てられ即位した場合にも天皇の実父として院号が贈られた制度がある(太上天皇参照)。