Paired domination versus domination and packing number in graphs

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