The neighbors across the street have a new cat and it keeps coming in our cat door and I keep finding it in our house. I don't want their cat in our house. It is "marking" territory in our house, and eating my cats' food. What can I do? I don't want to close our cat door because then my cats can't get in and out during the day. (They are kept inside at night with the cat door closed.)

Any ideas for how to keep this cat out of my house? Is there a special cat door that will let in only my cats? Is there something I can do to the invading cat to repel it? Do I need to give my cats assertiveness training so they do more than just hiss at this cat?