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Let $F_g$ be a closed orientable surface of genus $g$. A set $\mathrm{\Omega }=\{{\gamma }_1,\dots ,{\gamma }_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a filling system or simply be filling of $F_g$, if $F_g\setminus \mathrm{\Omega }$ is a disjoint union of $b$ topological discs for some $b\ge 1$. A filling system is called minimally intersecting, if the total number of intersection points of the curves is minimum, or equivalently $b=1$. The size of a filling system is defined as the number of its elements. We prove that the maximum possible size of a minimally intersecting filling system is $2g$. Next, we show that for $g\ge 2\text{ and }2\le s\le 2g$ with $(g,s)\ne (2,2)$, there exists a minimally intersecting filling system on $F_g$ of size $s$. Furthermore, we study geometric intersection number of curves in a minimally intersecting filling system. For $g\ge 2$, we show that for a minimally intersecting filling system $\mathrm{\Omega }$ of size $s$, the geometric intersection numbers satisfy $max\{i({\gamma }_i,{\gamma }_j)|i\ne j\}\le 2g-s+1$, and for each such $s$ there exists a minimally intersecting filling system $\mathrm{\Omega }=\{{\gamma }_1,\dots ,{\gamma }_s\}$ such that $max\{i({\gamma }_i,{\gamma }_j)|i\ne j\}=2g-s+1$.

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.

Let $S_g$ be a closed and oriented surface of genus $g\ge 2$. A closed curve $\gamma$ on $S_g$ is said to fill$S_g$ (or simply be filling), if its complement in the surface is a disjoint union of topological discs. It is assumed that the curve $\gamma$ is always in minimal position. To a filling curve, we associate a number $b$, the number of topological discs in its complement. For $b=1$, such a filling curve is called minimally intersecting. We prove that for every $b\ge 1,$ there exists a filling curve ${\gamma }_b$ on $S_g$ whose complement is a disjoint union of $b$ many topological discs. Furthermore, we provide an upper bound on the number of mapping class group orbits of closed curves which fills $S_g$ minimally.

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 $(g,k)\ne (2,1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.