Game-perfect Graphs with Clique Number 2

We consider two variants of Bodlaender's graph coloring games: one (A) in which Alice has the right to have the first move and to miss one or several turns, the other (B) in which Bob has the first move and is allowed to miss one or several turns. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. We determine those graphs for which these numbers are 2 and those graphs with clique number 2 which are game-perfect in the respective sense.