Does my cat's obesity contribute to her pooping on the floor?

My one year old cat is occasionally having accidents around the house. (I can only conclude they're accidents, as it's only a little bit of poop at a time, not what she usually leaves in the litter box.) She's also obese. (Yes, we're beginning serious attempts to handle this. We also have her brother, a very thin cat, and we don't want to starve him in the process.)

Does her weight contribute to this or are there other factors I should worry about? They eat a high quality, grain free diet with both wet and dry food.