Let $F$ be a compact, oriented surface. A Dehn move, $D_\beta$, on a simple closed curve (scc) $\alpha$ is a Dehn twist applied to $\alpha$ along a scc $\beta$ which intersects $\alpha$ once.

Is there a finite collection of scc's in $F$ such that every non-separating scc $\alpha$ in $F$ can be reduced to one of them by Dehn moves $D_\beta$ as above?

Can such reduction be made by $D_\beta^2$-moves?