Paired domination versus domination and packing number in graphs

Abstract

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 $\gamma _{pr}(G)=2\rho _{3}(G)$.