  The connected domination game was recently introduced by Borowiecki,
Fiedorowicz and Sidorowicz as another variation of the domination game. The
rules are essentially the same, except that the set of played vertices must be
connected at all stages of the game. We answer a problem from their paper
regarding the relation between the number of moves in a game where
Dominator/Staller starts the game. In this paper we also study the relation to
the diameter and present graphs with small connected game domination number. We
determine the values on the lexicographic product graphs, and consider the
effect of predomination of a vertex on the connected game domination number.
