The variety of domination games
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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.
KeywordsDomination game Total domination game L-domination game Z-domination game Grundy domination number
Mathematics Subject Classification05C69 05C57 91A43
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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