Upper paired domination in graphs

A set $PD\subseteq V(G)$ in a graph $G$ is a paired dominating set if every vertex $v\notin PD$ is adjacent to a vertex in $PD$ and the subgraph induced by $PD$ contains a perfect matching. A paired dominating set $PD$ of $G$ is minimal if there is no proper subset $P{D}^{\prime }\subset PD$ which is a paired dominating set of $G$. A minimal paired dominating set of maximum cardinality is called an upper paired dominating set, denoted by ${\mathrm{\Gamma }}_{pr}(G)$-set. Denote by $Upper$- $PDS$ the problem of computing a ${\mathrm{\Gamma }}_{pr}(G)$-set for a given graph $G$. Michael et al. showed the APX-completeness of $Upper$- $PDS$ for bipartite graphs with $\mathrm{\Delta }=4$ ^[11]. In this paper, we show that $Upper$- $PDS$ is APX-complete for bipartite graphs with $\mathrm{\Delta }=3$.