We say a collection of closed curves $\alpha_1,...,\alpha_k$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) fill the surface $S_{g,n}$, if $S_{g,n}-\{\alpha_1,...,\alpha_k\}$ is a collection of disks and annuli.
Let us call the red curve $A$ and the blue curve $B$ in the following picture. It is clear that $A$,$B$ fill the surface $S_{0,4}$ (a disk with 3 holes) in the figure. For an element $\phi\in MCG(S_{0,4})$(=mapping class group of a disk with 3 holes) , let $A'=\phi(A)$ and $B'=\phi(B)$. Is it true that $A'$,$B'$ fill the surface $S_{0,4}$ as well?
Yes, $A'$ and $B'$ fill also, because the criterion you give for filling is obviously invariant under any homeomorphism $\Phi : S \to S$: the map $\Phi$ restricts to a homeomorphism from $S-(A \cup B)$ to $S - (\Phi(A) \cup \Phi(B)) = S - (A' \cup B')$, so if the components of the former are all discs and annuli, so also are the components of the latter.