Let $S$ be a closed Riemann surface of genus $g>1$. It is well-known that $S$ admits a decomposition into pairs-of-pants. Namely, there exist $3g-3$ disjoint embedded circles $C_{j}\subset S$ such that $S\smallsetminus\bigcup\limits_{j=1}^{3g-3}C_{j}$ is a disjoint union of $2g-2$ copies of the pair-of-pants $P$. The pair-of-pants surface $P$ is homeomorphic to the Riemann sphere ${\mathbb{C}}{\mathbb{P}}^{1}$ punctured in three points.

To such a decomposition we associate a graph $\Gamma$. The vertices of $\Gamma$ correspond to the pairs-of-pants while the edges correspond to the circles $C_{j}$. Each edge joins the vertices corresponding to the adjacent pairs-of-pants.

There exists a fibration $\pi:S\to\Gamma$ such that the circles $C_{j}$ are inverse images of the midpoints of the edges of $\Gamma$. Such fibration is canonically associated to our decomposition into pairs-of-pants. To construct it we fiber each individual pair-of-pants over a tripod graph as pictured on the left-hand-side of Figure 5.