Morrow [9] classified all weighted dual graphs of the boundary of the minimal normal compactifications of the affine plane $\mathbf{A}^{2}$ by using a result of Ramanujam [10] that any minimal normal compactification of $\mathbf{A}^{2}$ has a linear chain as the graph of the boundary divisor. In this article, we give a new proof of the above-mentioned results of Ramanujam-Morrow [9] from a different point of view and by the purely algebro-geometric arguments. Moreover, we show that the affine plane $\mathbf{A}^{2}$ is characterized by the weighted dual graph of the boundary divisor.