Paired Domination versus Domination and Packing Number in Graphs
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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 γ(G), γ_ pr(G), and γ_t(G), respectively. For a positive integer k, a k-packing in G is a set S ⊆ 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 ρ_k(G). It is well known that γ_ pr(G) < 2γ(G). In this paper, we prove that it is NP-hard to determine whether γ_ pr(G) = 2γ(G) even for bipartite graphs. We provide a simple characterization of trees with γ_ pr(G) = 2γ(G), implying a polynomial-time recognition algorithm. We also prove that even for a bipartite graph, it is NP-hard to determine whether γ_ pr(G)=γ_t(G). We finally prove that it is both NP-hard to determine whether γ_ pr(G)=2ρ_4(G) and whether γ_ pr(G)=2ρ_3(G).
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