Is a Persian greyhound an Afghan?
No, "Persian Greyhound" is another name for the Saluki, which is a different breed from the Afghan Hound (known as "Afghan" for short).
There is only one breed called Greyhound. However, there are multiple breeds that are also known as a type of Greyhound -- for example, the Saluki is also known as the Persian Greyhound. Italian Greyhounds are another breed that looks similar to a Greyhound but smaller, though they are not considered sighthounds and are instead a toy breed.
There are many sighthound breeds other than the Greyhound, including: Mudhol Hound Saluki Afghan Hound Whippet Galgo/Spanish Greyhound Tazi Taigan Dog Sloughi Silken Windhound Borzoi Rampur Hound Poligar Hound Irish Wolfhound Scottish Deerhound Cirneco dell'Etna Lurcher (cross-breed) Brazilian Greyhound Balkh Hound