Pet owners can start booking travel for their four-legged companions on United Airlines again this summer. Yet fewer animals will be flying thanks to new rules by the carrier prohibiting all but dogs and cats in cargo, and even then more than 20 breeds of felines and canines are banned.

After the death of a 10-month-old puppy in an overhead compartment on a United flight in March -- a month that also saw the airline loading three dogs onto the wrong planes -- the company stopped taking new cargo reservations for four-legged travelers pending a review.

Starting in June, United will resume flights for dogs and cats transported in planes' cargo holds. Notably, however, it will no longer accept 21 breeds of dogs and four breeds of cats (see below for a complete list). The ban on "brachycephalic" (or short- or snub-nosed) dogs and cats and strong-jawed breeds comes "out of concern for higher adverse health risks," the airline said.

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Short-nosed breeds including boxers, pugs and Boston terriers are more susceptible to respiratory issues. As a result, American, Delta and other airlines already to refuse to ship them.

Travelers may still bring small pets, including dogs and cats, into United cabins so long as they are in carriers that fit under the seat. That means fully grown dogs including Mastiffs and some bulldogs are effectively banned, as they are too big to ride in the cabin and are not allowed in the cargo hold.

The restrictions also mean United won't fly pets between May 1 and Sept. 30 to four cities -- Phoenix Sky Harbor International Airport, McCarran International Airport, Palm Springs International Airport and Tucson International Airport -- because of the heat.

United said it is working with American Humane, an animal welfare group, to improve animal safety on its flights.

United had the worst record of all U.S. airlines last year in the number of animals who died in flight. Beyond previously accepting higher-risk breeds that other airlines prohibited, United in the past flew animals including geese, leopards and foxes.

Of the nearly 507,000 animals that flew on U.S. commercial carriers in 2017, 24 died, with 18 of those deaths happening on United, according to the Department of Transportation. United also led airlines in the number of animals transported, carrying more than 138,000 animals.

Following are the dog and cat breeds that United will no longer transport:

Dog breeds

Affenpinscher

American Bully

American Pit Bull Terrier/Pit Bull

American Staffordshire Terrier/"Amstaff"

Belgian Malinois

Boston Terrier

Boxer

Brussels Griffon

Bulldog

- American Bulldog

- English Bulldog

- French Bulldog

- Old English Bulldogges

- Shorty Bulldogs

- Spanish Alano/Spanish Bulldog/Alano Espanol

- American Bulldog - English Bulldog - French Bulldog - Old English Bulldogges - Shorty Bulldogs - Spanish Alano/Spanish Bulldog/Alano Espanol Cavalier King Charles Spaniel

Chow Chow

English Toy Spaniel/Prince Charles Spaniel

Japanese Chin/Japanese Spaniel

Lhasa Apso

Mastiff

- American Mastiff

- Boerboel/South African Mastiff

- Bullmastiff

- Ca de Bou/Mallorquin Mastiff

- Cane Corso/Italian Mastiff

- Dogo Argentino/Argentinian Mastiff

- Dogue de Bordeaux/French Mastiff

- English Mastiff

- Fila Brasileiro/Brazilian Mastiff/Cao de Fila

- Indian Mastiff/Alangu

- Kangal/Turkish Kangal

- Neapolitan Mastiff/Mastino Napoletano

- Pakastani Mastiff/Bully Kutta

- Pyrenean Mastiff

- Presa Canario/Perro de Presa Canario/Dogo Canario/Canary Mastiff

- Spanish Mastiff / Mastin Espanol

- Tibetan Mastiff

- Tosa/Tosa Ken/Tosa Inu/Japanese Mastiff/Japanese Tosa

- American Mastiff - Boerboel/South African Mastiff - Bullmastiff - Ca de Bou/Mallorquin Mastiff - Cane Corso/Italian Mastiff - Dogo Argentino/Argentinian Mastiff - Dogue de Bordeaux/French Mastiff - English Mastiff - Fila Brasileiro/Brazilian Mastiff/Cao de Fila - Indian Mastiff/Alangu - Kangal/Turkish Kangal - Neapolitan Mastiff/Mastino Napoletano - Pakastani Mastiff/Bully Kutta - Pyrenean Mastiff - Presa Canario/Perro de Presa Canario/Dogo Canario/Canary Mastiff - Spanish Mastiff / Mastin Espanol - Tibetan Mastiff - Tosa/Tosa Ken/Tosa Inu/Japanese Mastiff/Japanese Tosa Pekingese

Pug

- Dutch Pug

- Japanese Pug

- Dutch Pug - Japanese Pug Shar-Pei/Chinese Shar-Pei

Shih-Tzu

Staffordshire Bull Terrier/"Staffys"

Tibetan Spaniel

Cat Breeds