How do I discourage my neighbor's cat from entering my house?

I have a large dog that really dislikes (in other words, wants to kill) cats. I also have a cat that we adopted when our neighbors moved away and abandoned it. Our cat stays in a bedroom or outside during the day. In the evenings we lock our dog in our master bedroom and let our cat out to roam. We also keep the dog door to our back yard open so our cat can come and go. Our 'lock in, lock out' process is not perfect, but it works, especially since our cat knows to avoid our dog. However, our neighbors have a cat that's fearless, has learned to use our dog door, and really likes to spend time in our house. Last week he snuck in through the dog door and pooped in a spare bedroom. Last night he snuck in and hid in our master bedroom. A few minutes after putting our dog Lanikai in the master for the night she discovered the neighbors cat hiding in the room. Luckily the neighbor's cat made it under the bed and wasn't killed.

Anyone have any recommendations on how to scare off the neighbor's cat without scaring off our own? Pepper spray near the dog door isn't an option since that would affect my cat as well. I need something that will target the neighbor's cat without doing any real harm, but deter him from coming in the house or even our yard. Otherwise he'll eventually get caught and killed by the dog.

Don't bother recommending talking to the neighbors. We've done that 3 times already.

Any help would be appreciated.

Thanks.

There are collars that work with dog/cat doors that will only allow the wearer throughthe door. Maybe you can outfit your cat and dog? It will keep the neighbor's cat out.

The fact that he's sneaking in and pooping tells me he thinks he's the dominant cat. If he isn't fixed, that might cause this behavior.

If the cat gets in and your dog attacks it, the neighbors may cause problems. It sounds like you may want to alert animal control to the situation, just to cover yourselves. The neighbors don't sound very responsible and that could cause more issues in future.

Thank you for being the responsible parties in this situation!