Question: Are the Bullenbeisser and Mexican Hairless Dog both dogs?

Context paragraphs:
Elmendorf Beast: The Elmendorf Beast was the name given to a coyote blamed for several attacks on livestock in Elmendorf, Texas.  Various opinions have been offered as to the identity of the creature, including that it was a Mexican Hairless Dog whose appearance had been altered by sickness and/or congenital ailments, and that it was a wolf–coyote cross.  Some local people have linked it to the legend of the Chupacabra, while others believe that it was the product of a lab experiment that escaped, or that it was a previously unknown form of canid that was forced into contact with humans after its natural habitat was destroyed.

Chiribaya Dog: The Chiribaya Dog (Spanish: "perro Chiribaya" ) or Peruvian shepherd dog ("perro pastor Peruano ") was a pre-Columbian breed of dog from the southwest of Peru, identified by the 42 mummies discovered by anthropologist Sonia Guillén Oneglio in the Ilo District, Moquegua Region, on the south coast of Peru.  It has been established that it was a llama herding dog.  The dogs were not only an important part of the social structure of the ancient Peruvians, but they received special treatment after death as well.  The dog variety has been referred to in various Spanish-language documentaries under different terms, such as "el perro pastor Chribaya" ('the Chiribaya shepherd dog') and "pastor Peruano " ('Peruvian shepherd'), though the ancient Peruvians did not keep sheep.  Its original name is unknown.  (It has been referred to more ambiguously by the term "perro Peruano " or "perro del Perú " ('Peruvian dog', 'dog of Peru'), but this has also been applied to an extant but ancient hairless variety, referred to in more detail as "perro sin pelo del Perú", 'hairless dog of Peru', or the Peruvian hairless dog, a favorite in South American dog shows.)

Ecuadorian Hairless Dog: The Ecuadorian Hairless Dog (in Spanish: Perro calvo dorado ecuatoriano) is a breed of hairless dog originated from Santa Elena Peninsula in Ecuador.  It is now considered the rarest breed between the hairless dogs.

Peruvian Hairless Dog: The Peruvian Hairless Dog is a breed of dog with its origins in Peruvian pre-Inca cultures.  It is one of several breeds of hairless dog.

Canine follicular dysplasia: Follicular dysplasia is a genetic disease of dogs causing alopecia, or hair loss.  It is caused by hair follicles that are misfunctioning due to structural abnormality.  There are several types, some affecting only certain breeds.  Diagnosis is achieved through a biopsy, and treatment is rarely successful.  Certain breeds, such as the Mexican Hairless Dog and Chinese Crested Dog, are bred specifically for alopecia.

Mexican Hairless Dog: The Xoloitzcuintli ( ; Nahuatl pronunciation: ), or Xolo for short, is a hairless breed of dog, found in toy, miniature, and standard sizes.  The Xolo also comes in a coated variety and coated and hairless can be born in the same litter.  It is also known as Mexican hairless dog in English-speaking countries, and is one of several breeds of hairless dog.

Chinese Crested Dog: The Chinese crested dog is a hairless breed of dog.  Like most hairless dog breeds, the Chinese crested comes in two varieties, with and without fur, which are born in the same litter: the Powderpuff and the Hairless.

Abyssinian Sand Terrier: The Abyssinian Sand Terrier or African Hairless Dog is a breed of hairless dog.  It can grow 15.5 to tall and weigh 21 to .  Abyssinian Sand Terriers are hairless except for the skull and towards the end of the tail.  However, some dogs are totally hairless.  They come in bronze, elephant gray, grayish black, black, pale sandy, and mottled.  Their ears are rose shaped.

Bullenbeisser: The Bullenbeisser (also known as the German Bulldog) was a breed of dog known for its strength and agility.  The breed was closely related to the Bärenbeisser (some believe that the two breeds were the same; the names mean "bull-biter" and "bear-biter," respectively), and the Boxer.  It was, in all its aspects, similar to the present Alano Español (Spanish Bulldog) and very alike to the Dogo Argentino, not only in aspect, but also in usage.  There were two regional varieties, the Brabanter Bullenbeisser and the Danziger Bullenbeisser.  The breed is now extinct.

Mexican Hairless Cat: The Mexican Hairless cat, Aztec cat or New Mexican Hairless was a purported breed of cat first documented in 1902 by Mr. E.J.Shinick who owned a pair of hairless cats.  The cats were novelties at this time because of their lack of a coat (fur), although they grew light fur along the back and tail in winter.  These cats also had long Whiskers and eyebrows.  Because they had no coat, they lost more body heat than coated cats.  This made them warm to the touch as well as heat-seeking.  They are well documented in media reports in the early 20th Century.

Supporting paragraphs: Bullenbeisser Mexican Hairless Dog
Answer: yes