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Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. 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 Fg are in the same Mod(Fg)-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of F2 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 Mod(F2).

We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of F2 is two. Finally, given positive integers g and k with (g,k)(2,1), we construct a filling pair of Fg such that the complement is a union of k topological discs.


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