Publications Repository - Gdańsk University of Technology

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), γpr(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 kpacking and is denoted by ρk (G). It is well known that γpr(G) ≤ 2γ (G). In this paper, we prove that it is NP-hard to determine whether γpr(G) = 2γ (G) even for bipartite graphs. We provide a simple characterization of trees with γpr(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 γpr(G) = γt(G). We finally prove that it is both NP-hard to determine whether γpr(G) = 2ρ4(G) and whether γpr(G) = 2ρ3(G).

Authors

• dr inż. Magda Dettlaff link open in new tab ,
• Didem Gözüpek,
• dr inż. Joanna Raczek link open in new tab