Extremal incomplete sets in finite abelian groups

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ 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 ${\rm cr}(G)-1$ which fail to span $G$ has also been characterized except for the case that $|G|$ is an even number and the case that $|G|=pq$ with $p,q$ are primes. In this paper, we characterize these extremal subsets for $|G|\geq 36$ is an even number, or $|G|=pq$ with $p,q$ are primes and $q\geq 2p+3$.Comment: Swith some notations into ones which are more popular and put the materials of Appendix into the body. The present version is to appear in Ars Combinatori