To get rid of a pet cat in runescape just simply release it. Open the pet catalog and click the release button. And when it say "Do you really want to release" say YES!!
Or you can starve your cat to death.
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You can appeal it on the RuneScape site.
You can only pet your dog and cat if you have the Touch Screen Gloves which you can buy.
it is impossible to get rid of them
you cant get rid of a pet but you can sell items at the bottom of the w-shop
If you shoo away your cat, it will run away and you'll need to get another from Gertrude.