  Let $\gamma_g(G)$ and $\gamma_{tg}(G)$ be the game domination number and the
total game domination number of a graph $G$, respectively. Then $G$ is
$\gamma_g$-perfect (resp. $\gamma_{tg}$-perfect), if every induced subgraph $F$
of $G$ satisfies $\gamma_g(F)=\gamma(F)$ (resp. $\gamma_{tg}(F)=\gamma_t(F)$).
A recursive characterization of $\gamma_g$-perfect graphs is derived. The
characterization yields a polynomial recognition algorithm for
$\gamma_g$-perfect graphs. It is proved that every minimally
$\gamma_g$-imperfect graph has domination number $2$. All minimally
$\gamma_g$-imperfect triangle-free graphs are determined. It is also proved
that $\gamma_{tg}$-perfect graphs are precisely $\overline{2P_3}$-free
cographs.
