  The connected domination game is played just as the domination game, with an
additional requirement that at each stage of the game the vertices played
induce a connected subgraph. The number of moves in a D-game (an S-game, resp.)
on a graph $G$ when both players play optimally is denoted by $\gamma_{\rm
cg}(G)$ ($\gamma_{\rm cg}'(G)$, resp.). Connected Game Continuation Principle
is established as a substitute for the classical Continuation Principle which
does not hold for the connected domination game. Let $G|x$ denote the graph $G$
together with a declaration that the vertex $x$ is already dominated. The first
main result asserts that if $G$ is a graph with $\gamma_{\rm cg}(G) \geq 3$ and
$x \in V(G)$, then $\gamma_{\rm cg}(G|x) \leq 2 \gamma_{\rm cg}(G) - 3$ and the
bound is sharp. The second main theorem states that if $G$ is a graph with
$n(G) \geq 2$ and $x \in V(G)$, then $\gamma_{\rm cg}(G|x) \geq \left \lceil
\frac12 \gamma_{\rm cg}(G) \right \rceil$ and the bound is sharp. Graphs $G$
and its vertices $x$ for which $\gamma_{\rm cg}'(G|x) = \infty$ holds are also
characterized.
