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
$\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that
$\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We
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.
An upper bound on the paired-domination number in terms of the number of edges in the graph
