  We give the rectangle condition for strong irreducibility of Heegaard
splittings of $3$-manifolds with non-empty boundary. We apply this to a
generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a
link. The condition implies that any thin meridional level surface in the link
complement is incompressible. We also show that the additivity of knot width
holds for a composite knot satisfying the condition.
