Perfect Graphs for Domination Games

Annals of Combinatorics (2021)Cite this article

Abstract

Let \(\gamma (G)\) and \(\gamma _{t}(G)\) be the domination number and the total domination number of a graph G, respectively, and let \(\gamma _g(G)\) and \(\gamma _{tg}(G)\) be the game domination number and the game total domination number of G, respectively. Then, G is \(\gamma _g\)-perfect (resp. \(\gamma _{tg}\)-perfect) if every induced subgraph F of G satisfies \(\gamma _g(F)=\gamma (F)\) (resp. \(\gamma _{tg}(F)=\gamma _t(F)\)). A recursive characterization of \(\gamma _g\)-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for \(\gamma _g\)-perfect graphs. It is proved that every minimally \(\gamma _g\)-imperfect graph has domination number 2. All minimally \(\gamma _g\)-imperfect triangle-free graphs are determined. It is also proved that \(\gamma _{tg}\)-perfect graphs are precisely \(\overline{2P_3}\)-free cographs.

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

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    J. D. Alvarado, S. Dantas, D. Rautenbach, Perfectly relating the domination, total domination, and paired domination numbers of a graph, Discrete Math. 338 (2015) 1424–1431.

    MathSciNet  Article  Google Scholar 

  2. 2.

    S. Arumugam, Bibin K. Jose, Cs. Bujtás, Zs. Tuza, Equality of domination and transversal numbers in hypergraphs, Discrete Appl. Math. 161 (2013) 1859–1867.

    MathSciNet  Article  Google Scholar 

  3. 3.

    G. Boruzanlı Ekinci, Cs. Bujtás, Bipartite graphs with close domination and \(k\)-domination numbers, Open Math. 18 (2020) 873–885.

    MathSciNet  Article  Google Scholar 

  4. 4.

    B. Brešar, S. Klavžar, D.F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010) 979–991.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cs. Bujtás, On the game total domination number, Graphs Combin. 34 (2018) 415–425.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cs. Bujtás, M.A. Henning, Zs. Tuza, Transversal game on hypergraphs and the \(\frac{3}{4}\)-conjecture on the total domination game, SIAM J. Discrete Math. 30 (2016) 1830–1847.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cs. Bujtás, General upper bounds on the game domination number, Discrete Appl. Math. 285 (2020) 530–538.

    MathSciNet  Article  Google Scholar 

  8. 8.

    E. Camby, F. Plein, A note on an induced subgraph characterization of domination perfect graphs, Discrete Appl. Math. 217 (2017) 711–717.

    MathSciNet  Article  Google Scholar 

  9. 9.

    M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. of Math. 164 (2006) 51–229.

    MathSciNet  Article  Google Scholar 

  10. 10.

    D. G. Corneil, H. Lerchs, L. Stewart Burlingham, Complement reducible graphs, Discrete Appl. Math. 3 (1981) 163–174.

    MathSciNet  Article  Google Scholar 

  11. 11.

    P. Dorbec, G. Košmrlj, G. Renault, The domination game played on unions of graphs, Discrete Math. 338 (2015) 71–79.

    MathSciNet  Article  Google Scholar 

  12. 12.

    M.A. Henning, S. Jäger, D. Rautenbach, Hereditary equality of domination and exponential domination, Discuss. Math. Graph Theory 38 (2018) 275–285.

    MathSciNet  Article  Google Scholar 

  13. 13.

    M.A. Henning, S. Klavžar, D.F. Rall, Total version of the domination game, Graphs Combin. 31 (2015) 1453–1462.

    MathSciNet  Article  Google Scholar 

  14. 14.

    M.A. Henning, S. Klavžar, D.F. Rall, The \(4/5\) upper bound on the game total domination number, Combinatorica 37 (2017) 223–251.

    MathSciNet  Article  Google Scholar 

  15. 15.

    M. A. Henning, S. Klavžar, D. F. Rall, Game total domination critical graphs, Discrete Appl. Math. 250 (2018) 28–37.

    MathSciNet  Article  Google Scholar 

  16. 16.

    M. A. Henning, A. Yeo, Total Domination in Graphs, Springer, New York, 2013.

    Google Scholar 

  17. 17.

    V. Iršič, Effect of predomination and vertex removal on the game total domination number of a graph, Discrete Appl. Math. 257 (2019) 216–225.

    MathSciNet  Article  Google Scholar 

  18. 18.

    T. James, S. Klavžar, A. Vijayakumar, The domination game on split graphs, Bull. Aust. Math. Soc. 99 (2019) 327–337.

    MathSciNet  Article  Google Scholar 

  19. 19.

    W.B. Kinnersley, D.B. West, R. Zamani, Extremal problems for game domination number, SIAM J. Discrete Math. 27 (2013) 2090–2107.

    MathSciNet  Article  Google Scholar 

  20. 20.

    S. Klavžar, G. Košmrlj, S. Schmidt, On graphs with small game domination number, Appl. Anal. Discrete Math. 10 (2016) 30–45.

    MathSciNet  Article  Google Scholar 

  21. 21.

    S. Klavžar, D.F. Rall, Domination game and minimal edge cuts, Discrete Math. 342 (2019) 951–958.

    MathSciNet  Article  Google Scholar 

  22. 22.

    L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253–267.

    MathSciNet  Article  Google Scholar 

  23. 23.

    A.J. Nadjafi-Arani, M. Siggers, H. Soltani, Charactertisation of forests with trivial game domination numbers, J. Comb. Optim. 32 (2016) 800–811.

    MathSciNet  Article  Google Scholar 

  24. 24.

    W. Ruksasakchai, K. Onphaeng, C. Worawannotai, Game domination numbers of a disjoint union of paths and cycles, Quaest. Math. 42 (2019) 1357–1372.

    MathSciNet  Article  Google Scholar 

  25. 25.

    D. P. Sumner, J. I. Moore, Domination perfect graphs, Notices Amer. Math. Soc. 26 (1979) A-569.

    Google Scholar 

  26. 26.

    K. Xu, X. Li, On domination game stable graphs and domination game edge-critical graphs, Discrete Appl. Math. 250 (2018) 47–56.

    MathSciNet  Article  Google Scholar 

  27. 27.

    I. E. Zverovich, V. E. Zverovich, An induced subgraph characterization of domination perfect graphs, J. Graph Theory 20 (1995) 375–395.

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We are grateful to Gašper Košmrlj for providing us with his software that computes game domination invariants. We acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-9109, J1-1693, N1-0095, N1-0108).

Author information

Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Csilla Bujtás, Vesna Iršič & Sandi Klavžar

  2. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Vesna Iršič & Sandi Klavžar

  3. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

Authors
  1. Csilla Bujtás
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vesna Iršič
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sandi Klavžar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sandi Klavžar.

Additional information

Publisher's Note

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

Communicated by Torsten Ueckerdt

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bujtás, C., Iršič, V. & Klavžar, S. Perfect Graphs for Domination Games. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00523-w

Download citation

Keywords

  • Domination game
  • Total domination game
  • Perfect graph for domination game
  • Triangle-free graph
  • Cograph

Mathematics Subject Classification

  • 05C57
  • 05C69
  • 68Q25