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.

翻译：配对 $P 设置为 配对 $P 设置为 附加属性 $P 具有完美的匹配值 。 虽然图形 $ G$ 中最小支配值的最大硬度为 $G$ 的上限支配数, 由$\ Gamma (G) 表示, 最小对称支配值 $G$ 的最大基本度为 $G$ 的上对称支配数 $G$, 由$\Gamma pr} (G) 表示 $(G) 。 我们首先显示 $G) (G)\ leq 2 gamma (G) $ 。 我们然后关注满足 $G$(G) = 2\ Gamma(G) = G$。 我们用 Ulatowkow $ (G) 表示两种特殊的图形类别: 双面和单面图形 $ gamma(G) $ (G) 以 Ulatow $ (G) $ (G) 美元 和两面 G gamma(G) 提供 $G= gurma as a gs gs g cas (O) as (O) as a g) a g= gs gs gs gs gs gs gs gs) sualmas gs gs gs gs gs a gs subs gs g) subs subus ex