Filling with Separating Curves

A pair Formula: see text of simple closed curves on a closed and orientable surface Formula: see text of genus Formula: see text is called a filling pair if the complement is a disjoint union of topological disks. If Formula: see text 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 Formula: see text with exactly two complementary disks. We study the combinatorics of the action of the mapping class group Formula: see text on the set of such filling pairs. Furthermore, we construct a Morse function Formula: see text on the moduli space Formula: see text which, for a given hyperbolic surface Formula: see text, outputs the length of the shortest such filling pair with respect to the metric in Formula: see text. We show that the cardinality of the set of global minima of the function Formula: see text is the same as the number of Formula: see text-orbits of such filling pairs.