Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic differential $\phi$ whose horizontal foliation $\mathcal{F}_{h}(\phi)$ consists of closed leaves foliating one cylinder with core curves in the homotopy class $[\gamma]$. Let $\Gamma$ be the union of leaves both beginning and ending at a critical point of $\phi$ i.e. a zero of $\phi$. This is called the critical graph of $\phi$. For genus $g \geq 2$ one can show that $R - \Gamma$ is a disjoint union of cylinders. $\gamma$ will lie in one of these cylinders. Pick now this cylinder. What is the boundary of this cylinder? Is it made of just one singular leaf (leaf of the foliation which contains zeros of $\phi$)?

Greetings Ben