Paired domination versus domination and packing number in graphs

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 subset of 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 kpacking and is denoted by rho(k) (G). It iswell known that gamma(pr) (G) <= 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.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).