Introductions to other dogs should occur in neutral territory outside the home. Let them sniff and get acquainted, keeping the situation calm and friendly. Enter the home while walking the dogs side-by-side.
Cat introductions, however, are a two-step process. Best to make an indoor/outdoor cat stay inside for a few days. Introduce kitty indoors, making sure to put the dog on leash and bring him to the cat, not the other way around. Don’t make a fuss; be as natural as possible. Every greyhound will show interest in the cat, but if his tail and ears go up, and his body becomes rigid with eyes fixed on kitty, give him a gentle, but firm pull on the leash and say in a very stern, deep voice “No!” When your dog breaks his concentration, tell him “Good boy!” in a gentle, higher-sounding voice. Dogs really do want to know the rules.
If your cat is allowed outside, take your greyhound outside on a leash and repeat the process above, using the appropriate commands and tone of voice. Greyhounds are extremely voice sensitive. Your tone tells them if they are good or bad. They also earn by repetition so be consistent in what you say and how you say it. Follow all these instructions for any neighborhood cat your dog meets while on leash. In fact, small dogs, bunnies, birds and assorted animals are similar to cats in the mind of an inquisitive greyhound.
Remember to never force any small animal to run away from your dog. A greyhound can misinterpret that act, and start chasing his new friend!