A Puzzling Mystery!

Riddle: Two players, A and B, play an unequal game. There are n points on the game board. In each turn, Player A targets a pair of dots and Player B says whether those two dots are connected or not. A can only target each pair once and the game ends when all pairs are targeted. Player B wins if there is a point connected to all other points in the last turn, and player A wins if any point is connected to all other points in any turn except the last point or if no point is connected to all other points after that. The last turn. What n values do any of the players have for a winning strategy?