  Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple
closed curves is called a filling of $F_g$ if its complement is a disjoint
union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on
the set of fillings of $F_g$. The union of the curves in a filling forms a
graph on the surface which is a so-called decorated fat graph. It is a fact
that two fillings of $F_g$ are in the same $\text{Mod}(F_g)$-orbit if and only
if the corresponding fat graphs are isomorphic. We prove that any filling of
$F_2$ whose complement is a single disc (i.e., a so-called minimal filling) has
either three or four closed curves and in each of these two cases, there is a
unique such filling up to the action of $\text{Mod}(F_2)$.
  We provide a constructive proof to show that the minimum number of discs in
the complement of a filling pair of $F_2$ is two. Finally, given positive
integers $g$ and $k$ with $(g, k)\neq (2, 1)$, we construct a filling pair of
$F_g$ such that the complement is a union of $k$ topological discs.
