  The Z-domination game is a variant of the domination game in which each newly
selected vertex $u$ in the game must have a not yet dominated neighbor, but
after the move all vertices from the closed neighborhood of $u$ are declared to
be dominated. The Z-domination game is the fastest among the five natural
domination games. The corresponding game Z-domination number of a graph $G$ is
denoted by $\gamma_{Zg}(G)$. It is proved that the game domination number and
the game total domination number of a graph can be expressed as the game
Z-domination number of appropriate lexicographic products. Graphs with a
Z-insensitive property are introduced and it is proved that if $G$ is
Z-insensitive, then $\gamma_{Zg}(G)$ is equal to the game domination number of
$G$. Weakly claw-free graphs are defined and proved to be Z-insensitive. As a
consequence, $\gamma_{Zg}(P_n)$ is determined, thus sharpening an earlier
related approximate result. It is proved that if $\gamma_{Zg}(G)$ is an even
number, then $\gamma_{Zg}(G)$ is strictly smaller than the game L-domination
number. On the other hand, families of graphs are constructed for which all
five game domination numbers coincide. Graphs $G$ with $\gamma_{Zg}(G) =
\gamma(G)$ are also considered and computational results which compare the
studied invariants in the class of trees on at most $16$ vertices reported.
