Maker-breaker percolation games II: Escaping to infinity

Abstract

Let Λ be an infinite connected graph, and let v0 be a vertex of Λ. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of Λ are marked as unsafe. On each of her turns, Maker marks p unsafe edges as safe, while on each of his turns Breaker takes q unsafe edges and deletes them from the graph. Breaker wins if at any time in the game the component containing v0 becomes finite. Otherwise if Maker is able to ensure that v0 remains in an infinite component indefinitely, then we say she has a winning strategy. This game can be thought of as a variant of the celebrated Shannon switching game. Given (p,q) and (Λ,v0), we would like to know: which of the two players has a winning strategy? Our main result in this paper establishes that when Λ=Z2 and v0 is any vertex, Maker has a winning strategy whenever p≥2q, while Breaker has a winning strategy whenever 2p≤q. In addition, we completely determine which of the two players has a winning strategy for every pair (p,q) when Λ is an infinite d-regular tree. Finally, we give some results for general graphs and lattices and pose some open problems.