Maker-Breaker domination number

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 γ 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 γ MB (G). Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant γ MB (G) is also compared with the domination number. Using the Erd˝ os-Selfridge Criterion a large class of graphs G is found for which γ MB (G) > γ(G) holds. Residual graphs are introduced and used to bound/determine γ MB (G) and γ MB (G). Using residual graphs, γ MB (T) and γ 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.
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https://hal.archives-ouvertes.fr/hal-01935244
Contributor : Valentin Gledel <>
Submitted on : Monday, November 26, 2018 - 3:25:49 PM
Last modification on : Thursday, November 21, 2019 - 2:31:12 AM
Long-term archiving on : Wednesday, February 27, 2019 - 3:17:41 PM

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Valentin Gledel, Vesna Iršič, Sandi Klavžar. Maker-Breaker domination number. Bulletin of the Malaysian Mathematical Sciences Society, 2019, ⟨10.1007/s40840-019-00757-1⟩. ⟨hal-01935244⟩

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