Weak and strong k-connectivity games

For a positive integer k, we consider the k-vertex-connectivity game, played on the edge set of _{K n}, the graph on n vertices. We first study the Maker-Breaker version of this game and prove that, for any integer k ≥ 2 and sufficiently large n, Maker has a strategy to win this game within ⌊k n / 2 ⌋ + 1 moves, which is easily seen to be best possible. This answers a question fromHefetz etal. (2009) [6]. We then consider the strong k-vertex-connectivity game. For every positive integer k and sufficiently large n, we describe an explicit first player's winning strategy for this game.