Upper paired domination versus upper domination

A paired dominating set $P$ is a with the additional property that has perfect matching. While maximum cardainality of minimal in graph $G$ called upper domination number $G$, denoted by $\Gamma(G)$, cardinality $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any without isolated vertices. We focus on graphs satisfying equality $\Gamma_{pr}(G)= 2\Gamma(G)$. give characterizations two special classes: bipartite unicyclic using results Ulatowski (2015). Besides, study restricted girth. In this context, provide characterizations: one girth at least 6 other $C_3$-free cactus also pose characterization general case as an open question.