In a graph $G$, a vertex dominates itself and its neighbors. A subset $S$ of vertices of $G$ is a double dominating set of $G$ if every vertex is dominated by at least two vertices in $S$. The double domination number $γ_{\times 2}(G)$ of $G$ is the minimum cardinality of a double dominating set of $G$. In this paper, we prove that, for a maximal outerplanar graph $G$, the double domination number $γ_{\times 2}(G)$ is at most $(n+k)/2$, where $k$ is the number of pairs of consecutive vertices on the outer cycle but at distance at least 3. Although this bound was previously proposed by Abd Aziz, Rad and Kamarulhaili (A note on the double domination number in maximal outerplanar and planar graphs, RAIRO Operations Research, 56 (2022) 3367--3371), their proof was found to be incomplete. In this paper we establish the validity of this result by providing a complete proof.

翻译：在图中，一个顶点控制自身及其所有邻点。图$G$的顶点子集$S$称为$G$的双控制集，若每个顶点至少被$S$中的两个顶点所控制。图$G$的双控制数$γ_{\times 2}(G)$定义为$G$的最小双控制集的基数。本文证明，对于极大外平面图$G$，双控制数$γ_{\times 2}(G)$不超过$(n+k)/2$，其中$k$为外圈上连续顶点对（且距离至少为3）的数目。尽管Abd Aziz、Rad和Kamarulhaili先前曾提出这一界（关于极大外平面图与平面图中双控制数的注释，RAIRO运筹研究，56 (2022) 3367--3371），但他们的证明被发现存在不完整性。本文通过提供完整证明，确立了该结果的有效性。