Extremal incomplete sets in finite abelian groups

Let $G$ be a finite abelian group. The critical number $\mathrm{c}\mathrm{r}(G)$ of $G$ is the least positive integer $\mathrm{\ell }$ such that every subset $A\subseteq G\setminus{0}$ofcardinalityatleast$\ell$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 ${\rm cr}(G)-1$whichfailtospan$G$ has also been characterized except for the case that $\mathrm{\mid }G\mathrm{\mid }$ is an even number and the case that $\mathrm{\mid }G\mathrm{\mid }=pq$ with $p,q$ are primes. In this paper, we characterize these extremal subsets for $\mathrm{\mid }G\mathrm{\mid }\ge 36$ is an even number, or $\mathrm{\mid }G\mathrm{\mid }=pq$ with $p,q$ are primes and $q\ge 2p+3$.