Constructions for positional games and applications to domination games

Abstract

We present constructions regarding the general behaviour of biased positional games, and amongst others show that the outcome of such a game can differ in an arbitrary way depending on which player starts the game, and that fair biased games can behave highly non-monotonic. We construct a gadget that helps to transfer such results to Maker-Breaker domination games, and by this we extend a recent result by Gledel, Ir\v{s}i\v{c}, and Klav\v{z}ar, regarding the length of such games. Additionally, we introduce Waiter-Client dominations games, give tight results when they are played on trees or cycles, and using our transference gadget we show that in general the length of such games can differ arbitrarily from the length of their Maker-Breaker analogue.