Final answer:
The vessels transected during heart valve surgery are the Left Coronary Artery, Brachiocephalic, and Left Subclavian. The Left Common Carotid artery is not transected during heart valve recovery.
Step-by-step explanation:
The vessels transected during heart valve surgery are:
- Left Coronary Artery
 - Brachiocephalic
 - Left Subclavian
 
Therefore, the vessel that is not transected during heart valve recovery is the Left Common Carotid artery.