To whom is it permissible to designate the Udhiyah if the father is deceased? Is it permissible for the children or the wife?

It is prescribed to mention the name of the owner of the sacrifice at the time of slaughter. The one slaughtering says, if it is for himself, "for myself," and if he is slaughtering for someone else, he says, "This is for so-and-so." This is not obligatory. The proof for this is what the Prophet, peace and blessings be upon him, did when he slaughtered and said, "In the name of Allah, Allah is the Greatest. This is on my behalf and on behalf of those of my Ummah who have not offered a sacrifice." If the father passes away, the children offer the sacrifice on their own behalf, because the sacrifice is prescribed for the living, not for the dead. The one offering the sacrifice says, "on my behalf and on behalf of my household."