Abstract: Let $G$ be a finite abelian group and $k$ be a positive integer. The Davenport constant is a central invariant in zero-sum thoery. The invariant $D_k(G)$ generalizes the Davenport constant $D(G)$ and is defined as the maximum length $l$ such that there exists a sequence $B$ of length $l$ over $G$ containing $k$ disjoint non-empty zero-sum subsequences. This paper studies the inverse problem associated with this invariant for the elementary abelian 2-groups $C_2^r$. For $r \in [2,4]$, we characterize the structures of zero-sum sequences of length $ D_2(C_2^r)$ and $D_2(C_2^r) - 1$ in $C_2^r$ that can be decomposed into at most two minimal zero-sum subsequences. For $r \in [2,5]$, we characterize the structures of sequences of length $D_2(C_2^r) - 1$.

Key words: Elementary abelian 2-group, Davenport constant, Inverse problem, Zero-sum sequence