  Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is
${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in
${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a
subgraph. In the ${\mathcal F}$-saturation game on $H$, players Max and Min
alternately add edges of $H$ to $G$, avoiding subgraphs in ${\mathcal F}$,
until $G$ becomes ${\mathcal F}$-saturated relative to $H$. They aim to
maximize or minimize the length of the game, respectively;
$\textrm{sat}_g({\mathcal F};H)$ denotes the length under optimal play (when
Max starts).
  Let ${\mathcal O}$ denote the family of all odd cycles and ${\mathcal T}$ the
family of $n$-vertex trees, and write $F$ for ${\mathcal F}$ when ${\mathcal
F}=\{F\}$. Our results include $\textrm{sat}_g({\mathcal O};K_{2k})=k^2$,
$\textrm{sat}_g({\mathcal T};K_n)=\binom{n-2}{2}+1$ for $n\ge6$,
$\textrm{sat}_g(K_{1,3};K_n)=2\lfloor n/2 \rfloor$ for $n\ge8$,
$\textrm{sat}_g(K_{1,r+1};K_n)=\frac{rn}{2}-\frac{r^2}{8}+O(1)$, and
$|\textrm{sat}_g(P_4;K_n)-(4n-1)/5|\le 1$. We also determine
$\textrm{sat}_g(P_4;K_{m,n})$; with $m\ge n$, it is $n$ when $n$ is even, $m$
when $n$ is odd and $m$ is even, and $m+\lfloor n/2 \rfloor$ when $mn$ is odd.
Finally, we prove the lower bound
$\textrm{sat}_g(C_4;K_{n,n})\ge\frac{1}{10.4}n^{13/12}-O(n^{35/36})$. The
results are very similar when Min plays first, except for the $P_4$-saturation
game on $K_{m,n}$.
