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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 \(S \subseteq V(G)\) 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 \(\gamma _{pr}(G)=2\rho _{3}(G)\).

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This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant No. 118E799.

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Correspondence to Didem Gözüpek.

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Dettlaff, M., Gözüpek, D. & Raczek, J. Paired domination versus domination and packing number in graphs. J Comb Optim 44, 921–933 (2022).

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  • Graph theory
  • Domination
  • Paired Domination
  • Total domination
  • Packing number