My lurchers are fine with cats that my missus mum has IN the house but wouldn't trust them unsupervised with them, my old bitch would run them but that's about it, my white bitch has had one incident with a cat outside, she ended up with the old cat hat and by the time I got there (less than 10 seconds) it was dead, I was dubious after that with the mother in laws cats BUT touch wood she's not showed any interest, outside however it would be a different story.