  We define two new families of invariants for (3-manifold, graph) pairs which
detect the unknot and are additive under connected sum of pairs and
(-1/2)-additive under trivalent vertex sum of pairs. The first of these
families is closely related to both bridge number and tunnel number. The second
of these families is a variation and generalization of Gabai's width for knots
in the 3-sphere. We give applications to the tunnel number and higher genus
bridge number of connected sums of knots.
