Zero.
I have three dogs and we used to have a cat years ago (she got really sick and we had to put her to sleep while I was in high school, freshman or junior year...we believe it was due to that Swiffer WetJet stuff they said was safe for pets, but she was also really old). The big dogs had no problem with the cat, they just ignored her. Our mini Dachshund always tried to play with the cat though.