Divisibility of great webs and reducible Dehn surgery

We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a manifold with more than two connected summands, we show that r is bounded in absolute value by the bridge number of K. As a consequence, this possibility is ruled out for knots with bridge number at most five and for positive braid closures.