Is it permissible to verbalize the intention when slaughtering the Udhiyah for the deceased, such as by saying, "O Allah, this is the Udhiyah of my father, so-and-so," or is it sufficient to perform the act without verbalizing it?

The intention resides in the heart. What one intends in their heart is sufficient, and there is no need to utter it verbally. One must pronounce the Basmalah and Takbir when slaughtering, based on the narration by Anas: "The Prophet (peace and blessings be upon him) sacrificed two rams; he slaughtered them with his own hand, and he pronounced the Basmalah and Takbir." There is no objection to saying, "O Allah, this is a sacrifice on behalf of my parents," as this is not considered verbalizing the intention.