Extremal incomplete sets in finite abelian groups

Let G be a finite abelian group. The critical number cr(G) of G is the least positive integer l such that every subset A subset of G\ {0} of cardinality at least l spans G, i.e., every element of G can be written as a nonempty sum of distinct elements of A. The exact values of the critical number have been completely determined recently for all finite abelian groups. The structure of these sets of cardinality cr(G) - 1 which fail to span G has also been characterized except for the case that Cl I is an even number and the case that vertical bar G vertical bar = pq with p, q are primes. In this paper, we characterize these extremal subsets for vertical bar G vertical bar >= 36 is an even number, or vertical bar G vertical bar = pq with p, q are primes and q >= 2p + 3.