The carrier suspended its PetSafe program in March, following a number of mishaps that resulted in pet deaths—dogs being put on the wrong flights; a puppy dying in-flight after a passenger was forced to put it into the overhead bin.

Now, United says it has started working with American Humane to improve its pet-carrying policies, revised versions of which will come into force from June 18. From that date, the airline will take reservations for pets travelling with their owners or guardians from July 9. And on July 16, United will start accepting reservations for pets flying on their own, as of July 30.

Under the new rules, only cats and dogs will be “accepted as PetSafe”—and not all breeds, either. Certain short- or snub-nosed dogs and cats, as well as strong-jawed dog breeds, will be banned. Here’s the list, which includes dog breeds such as boxers, pitbulls, pugs and chow chows, and cat breeds such as Persian, Burmese and Himalayans.

What’s more, during the hot summer months United won’t accept reservations for pets flying in and out of Las Vegas, Palm Springs, Phoenix and Tucson.

“As we continue our review process to ensure that we are always doing what’s right, we are committed to making significant improvements in our program and adhering to the best practices of animal comfort, well-being and travel on behalf of our customers and their pets,” said United cargo head Jan Krems.