Weak and strong k-connectivity games

For a positive integer k, we consider the k-vertex-connectivity game, played on the edge set of Kn, the complete 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 ⌊kn/2⌋+1 moves, which is easily seen to be best possible. This answers a question from Hefetz et al. (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.