Filling with separating curves

A pair $(\alpha ,\beta )$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $\alpha$ is separating, then we call it as separating filling pair. In this paper, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\mathrm{\text{Mod}}(S_g)$ on the set of such filling pairs. Furthermore, we construct a Morse function ${{ℱ}}_g$ on the moduli space ${{ℳ}}_g$ which, for a given hyperbolic surface $X$, outputs the length of the shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function ${{ℱ}}_g$ is the same as the number of $\mathrm{\text{Mod}}(S_g)$-orbits of such filling pairs.