OK, forget what I said, because I can't seem to win unless the cat is dumb and heads for a section already littered with blackened dots.
But here's one hint I can offer. When presented with the following:
It's tempting to play A here, but don't do it. The dumb cat will waste a turn moving towards A, if you don't sew it up prematurely. Put a fence post at B, instead, then close A.
Again, if those two dark spaces between A and B didn't happen to be there already, I would've been screwed.