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Aequationes mathematicae

, Volume 93, Issue 6, pp 1085–1109 | Cite as

The variety of domination games

  • Boštjan Brešar
  • Csilla Bujtás
  • Tanja Gologranc
  • Sandi KlavžarEmail author
  • Gašper Košmrlj
  • Tilen Marc
  • Balázs Patkós
  • Zsolt Tuza
  • Máté Vizer
Article
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Abstract

Domination game (Brešar et al. in SIAM J Discrete Math 24:979–991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453–1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6 / 7 of the order of a graph.

Keywords

Domination game Total domination game L-domination game Z-domination game Grundy domination number 

Mathematics Subject Classification

05C69 05C57 91A43 

Notes

Acknowledgements

Work supported by the Slovenian Research Agency (research core funding P1-0297, Projects J1-9109, N1-0095, N1-0108, J1-1693), by the National Research, Development and Innovation Office—NKFIH under the Grants SNN 129364, K 116769 and KH130371, by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the New National Excellence Program under the Grant Number ÚNKP-19-4-BME-287.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  3. 3.Faculty of Information TechnologyUniversity of PannoniaVeszprémHungary
  4. 4.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  5. 5.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  6. 6.Abelium R&DLjubljanaSlovenia

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