Skip to main content

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

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Data Availibility

Enquiries about data availability should be directed to the authors.

References

  • Alvaro JD, Dantas S, Rautenbach D (2015) Perfectly relating the domination, total domination, and paired domination numbers of a graph. Discr. Math 338:1424–1431

    MathSciNet  Article  Google Scholar 

  • Brešar B, Henning MA, Rall DF (2007) Paired-domination of Cartesian product of graphs. Util. Math. 73:255–265

    MathSciNet  MATH  Google Scholar 

  • Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of Domination in Graphs. Marcel Dekker, New York

    MATH  Google Scholar 

  • Haynes TW, Slater PJ (1998) Paired-domination in graphs. Networks 32:199–206

    MathSciNet  Article  Google Scholar 

  • Henning MA, Vestergaard PD (2006) Trees with paired–domination number twice their domination number, Aalborg University

  • Hou X (2008) A characterization of \((2\gamma , \gamma _{\rm p})\)-trees. Discrere Math. 308:3420–3426

    Article  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco

    MATH  Google Scholar 

  • Meir A, Moon JW (1975) Relations between packing and covering numbers of a tree. Pacific J. Math. 61:225–233

    MathSciNet  Article  Google Scholar 

  • Rall DF (2008) Packing and Domination Invariants on Cartesian Products and Direct Products

  • Shang E, Kang L, Henning MA (2004) A characterization of trees with equal total domination and paired-domination numbers. Australas. J. of Comb. 30:31–39

    MathSciNet  MATH  Google Scholar 

Download references

Funding

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Didem Gözüpek.

Ethics declarations

Competing Interests

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dettlaff, M., Gözüpek, D. & Raczek, J. Paired domination versus domination and packing number in graphs. J Comb Optim 44, 921–933 (2022). https://doi.org/10.1007/s10878-022-00873-y

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-022-00873-y

Keywords

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