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 $\mathrm{\Delta }=3$.

A dynamic geometry system, as an important application in the field of geometric constraint solving, is widely used in elementary mathematics education; moreover, the dynamic geometry system is also a fundamental environment for automated theorem proving in geometry. In a geometric constraint solving process, a situation involving a critical point is often encountered, and geometric element degeneracy may occur at this point. Usually, the degeneracy situation must be substantively focused on during the learning and exploration process. However, many degeneracy situations cannot be completely presented even by the well-known dynamic geometry software. In this paper, the mechanisms causing the degeneracy of a geometric element are analyzed, and relevant definitions and formalized descriptions for the problem are provided according to the relevant modern Euclidean geometry theories. To solve the problem, the data structure is optimized, and a domain model design for the geometric element and the constraint relationships thereof in the dynamic geometry system are formed; furthermore, an update algorithm for the element is proposed based on the novel domain model. In addition, instances show that the proposed domain model and the update algorithm can effectively cope with the geometric element degeneracy situations in the geometric constraint solving process, thereby achieving unification of the dynamic geometry drawing and the geometric intuition of the user.