As a simple approximation, debris of an explosion would need to reach the escape velocity of that asteroids gravity field to permanentely be separated from the asteroid. To "blow up" an asteroid, one would need a sufficiently strong explosion to reach the escape velocity for at least a large part of the debris. The recombination of the asteroid would take quite a lot of time, considering the weak gravity. So if it is blown up at the right moment, recombination does not have enough time. Just reducing the density of the asteroid or rather its cloud of debris beyond a certain point might be sufficient to prevent a large disaster on earth. It then doesn't matter if the asteroid would recombine after crossing the earths path.