It is a two-person game with complete information played on a hypergraph (V,H) where V is an arbitrary set (called the board of the game) and H is a family of subsets of V, called the winning sets. The two players alternately occupy previously unoccupied elements of V.
The ﬁrst player, Maker, has to occupy a winning set to win; and the second player, Breaker, has to stop Maker from doing so; if Breaker successfully prevents maker from occupying a winning set to the end of the game, then Breaker wins. Thus, in a Maker–Breaker positional game, Maker wins if he occupies all elements of some winning set and Breaker wins if he prevents Maker from doing so. There can be no draw in a Maker-Breaker positional game: one player always wins.
The definition of Maker-Breaker game has a subtlety when and . In this case we say that Breaker has a winning strategy if, for all j > 0, Breaker can prevent Maker from completely occupying a winning set by turn j.
Maker-Breaker games on graphs
There has been quite some research done on playing Maker-Breaker games when the board of the game is the edge-set of a graph (usually taken as the complete graph) and the family of winning sets is , where is some graph property (usually taken to be monotone increasing) such as connectivity (see, e.g.,).
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