%0 Journal Article %T Weak and Strong k-connectivity games %A Asaf Ferber %A Dan Hefetz %J Mathematics %D 2012 %I arXiv %X For a positive integer $k$ we consider the $k$-vertex-connectivity game, played on the edge set of $K_n$, the complete graph on $n$ vertices. We first study the Maker-Breaker version of this game and prove that, for any integer $k \geq 2$ and sufficiently large $n$, Maker has a strategy for winning this game within $\lfloor k n/2 \rfloor + 1$ moves, which is clearly best possible. This answers a question of Hefetz, Krivelevich, Stojakovi\'c and Szab\'o. 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. %U http://arxiv.org/abs/1203.3447v1