  The $1/2$-conjecture on the domination game asserts that if $G$ is a
traceable graph, then the game domination number $\gamma_g(G)$ of $G$ is at
most $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a
$1/2$-graph if $\gamma_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ holds.
It is proved that the so-called hatted cycles are $1/2$-graphs and that
unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of
graphs that support the conjecture are determined and computer experiments
related to the conjecture described.
