Fast Winning Strategies for the Maker-Breaker Domination Game.

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 M B ′ (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 Erdős-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 M B ′ (G). Using residual graphs, γ MB (T) and γ M B ′ (T) are determined for an arbitrary tree. The invariants are also obtained for cycles. A list of open problems and directions for further investigations is given. [ABSTRACT FROM AUTHOR]