This is but one of several similar ancient Roman mosaics. It has been suggested that the signs were not to warn trespassers about a guard dog, but rather to warn visitors to not accidentally step on the wee house dog! (These were probably small Italian Greyhounds.)
Here’s another cute example:
For more about animals in ancient Roman animals, check out the book Animals in Roman Life & Art:
I also have a curated list of books about dogs in art from around the world available as an Amazon Idea List here.