2019/11/11 by Magda Dettlaff, Dettlaff, Magda, Didem Gözüpek +3
Computer Science · #Advanced Graph Theory Research #Interconnection Networks and Systems #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.1911.04098
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\rm pr(G), and γt(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 ρk(G). It is well known that γ\rm pr(G) ≤ 2γ(G). In this paper, we prove that it is NP-hard to determine whether γ\rm pr(G) = 2γ(G) even for bipartite graphs. We provide a simple characterization of trees with γ\rm 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\rm pr(G)=γt(G). We finally prove that it is both NP-hard to determine whether γ\rm pr(G)=2ρ4(G) and whether γ\rm pr(G)=2ρ3(G).