The sweetpotato vine (Ipomoea batatas) has many varieties grown for either its tuberous roots or for its decorative foliage. The ASPCA Animal Poison Control Center lists the sweetpotato vine as toxic to dogs, cats, and horses with clinical signs of diarrhea and hallucinations. However, the sweetpotato root and its sprouts are safe for your dog to consume as long as they are clean, free of mold, and cooked. Do not add sugar or butter to a sweetpotato if you cook one for your dog as that would add excess calories.
In addition, regular potatoes (Solanum tuberosum) are from the Nightshade family and their shoots and leaves are poisonous to humans and animals. The sweetpotato is only distantly related to the potato.
Other varieties of sweetpotato exist, though they are rare and only grown regionally, and their greens are poisonous. It's best to make sure you know which type of sweetpotato you have growing. If you notice any changes in your dog's physical behavior (vomiting, diarrhea, etc.), it's a good idea to check with your veterinarian or contact the ASPCA Animal Poison Control Center at 888-426-4435.