Perfect Graphs for Domination Games

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.

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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).

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Correspondence to Sandi Klavžar.

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

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Keywords

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

Mathematics Subject Classification

  • 05C57
  • 05C69
  • 68Q25