A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. We first show that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$. We then focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

中文翻译：

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上配对配对与上配对

配对控制集$ P $是具有$ P $具有完美匹配的附加属性的控制集。虽然将图形$ G $中的最小支配集的最大基数称为$ G $的较高支配数，用$ \ Gamma（G）$表示，但是$ G $中的最小配对支配集的最大基数为$ G $。称为$ G $的上对配对控制数，用$ \ Gamma_ {pr}（G）$表示。我们首先显示任何图$ G $的$ \ Gamma_ {pr}（G）\ leq 2 \ Gamma（G）$。然后，我们集中在满足等式$ \ Gamma_ {pr}（G）= 2 \ Gamma（G）$的图上。通过使用Ulatowski（2015）的结果，我们给出了两个特殊图类的表征：具有$ \ Gamma_ {pr}（G）= 2 \ Gamma（G）$的二部图和单环图。此外，我们用$ \ Gamma_ {pr}（G）= 2 \ Gamma（G）$和受限制的周长研究图。在这种情况下，我们提供了两个特征：一个用于$ \ Gamma_ {pr}（G）= 2 \ Gamma（G）$且周长至少为6的图，另一个用于$ \ Gamma_ {pr}（G）= 2 \的无$ C_3 $仙人掌图。伽玛（G）$。我们还用$ \ Gamma_ {pr}（G）= 2 \ Gamma（G）$作为一个开放性问题，来描述无$ C_3 $无图的一般情况。