Given an undirected multigraph G = (V, E) and requirement functions r(lambda) : ((V)(2)) --> Z(+) and r(kappa) : ((V)(2)) --> Z(+) (where Z(+) is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the edge-connectivity and vertex-connectivity between every pair x,y is an element of V become at least r(lambda)(x,y) and r(kappa)(x,y), respectively, in the resulting graph G '. In this paper, we show that the problem can be solved in polynomial time if r(kappa) is given by r(kappa)(x,y) = 2 for all x,y is an element of V.