| Summary: | Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X=\left[\kern-0.15em\left[ { - 1,1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m,n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X)=2(n+m).$
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