Given a graph \(G=(V(G), E(G))\), the size of a minimum dominating set, minimum paired dominating set, and a minimum total dominating set of a graph G are denoted by \(\gamma (G)\), \(\gamma _{pr}(G)\), and \(\gamma _{t}(G)\), respectively. For a positive integer k, a k-packing in G is a set \(S \subseteq V(G)\) such that for every pair of distinct vertices u and v in S, the distance between u and v is at least \(k+1\). The k-packing number is the order of a largest k-packing and is denoted by \(\rho _{k}(G)\). It is well known that \(\gamma _{pr}(G) \le 2\gamma (G)\). In this paper, we prove that it is NP-hard to determine whether \(\gamma _{pr}(G) = 2\gamma (G)\) even for bipartite graphs. We provide a simple characterization of trees with \(\gamma _{pr}(G) = 2\gamma (G)\), implying a polynomial-time recognition algorithm. We also prove that even for a bipartite graph, it is NP-hard to determine whether \(\gamma _{pr}(G)=\gamma _{t}(G)\). We finally prove that it is both NP-hard to determine whether \(\gamma _{pr}(G)=2\rho _{4}(G)\) and whether \(\gamma _{pr}(G)=2\rho _{3}(G)\).

给定一个图\(G=(V(G), E(G))\) ，图G的最小支配集、最小配对支配集和最小总支配集的大小用\(\ gamma (G)\)，\(\gamma _{pr}(G)\)和\(\gamma _{t}(G)\)，分别。对于正整数k，G中的k-packing是一个集合\(S \subseteq V(G)\)，使得对于 S 中的每对不同的顶点u和v ， u和v之间的距离至少为\ (S \subseteq V(G)\) (k+1\)。k-包装数是最大k -packing 的阶数，用\(\rho _{k}(G)\)表示。众所周知\(\gamma _{pr}(G) \le 2\gamma (G)\)。在本文中，我们证明即使对于二分图，也很难确定\(\gamma _{pr}(G) = 2\gamma (G)\) 。我们用\(\gamma _{pr}(G) = 2\gamma (G)\)提供了树的简单表征，这意味着多项式时间识别算法。我们还证明，即使对于二分图，也很难确定\(\gamma _{pr}(G)=\gamma _{t}(G)\)是否。我们最终证明，确定是否\(\gamma _{pr}(G)=2\rho _{4}(G)\)和是否\(\gamma _{pr}(G) =2\rho _{3}(G)\).