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Maker–Breaker Domination Number

  • Valentin Gledel
  • Vesna IršičEmail author
  • Sandi Klavžar
Article
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Abstract

The Maker–Breaker domination game is played on a graph G by Dominator and Staller. The players alternatively select a vertex of G that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper, we introduce the Maker–Breaker domination number \(\gamma _{\mathrm{MB}}(G)\) of G as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted \(\gamma _{\mathrm{MB}}'(G)\). Comparing the two invariants, it turns out that they behave much differently than the related game domination numbers. The invariant \(\gamma _{\mathrm{MB}}(G)\) is also compared with the domination number. Using the Erdős-Selfridge criterion, a large class of graphs G is found for which \(\gamma _{\mathrm{MB}}(G) > \gamma (G)\) holds. Residual graphs are introduced and used to bound/determine \(\gamma _{\mathrm{MB}}(G)\) and \(\gamma _{\mathrm{MB}}'(G)\). Using residual graphs, \(\gamma _{\mathrm{MB}}(T)\) and \(\gamma _{\mathrm{MB}}'(T)\) are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given.

Keywords

Maker–Breaker domination game Maker–Breaker domination number Domination game Perfect matching Tree Cycle Union of graphs 

Mathematics Subject Classification

05C57 05C69 91A43 

Notes

Acknowledgements

We acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-9109, N1-0095).

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205LyonFrance
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  3. 3.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  4. 4.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia

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