In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed orientable surface, $f \colon S \to S$ an automorphism and $c$ is a geodesic 1-submanifold of $S$ such that $f(c) \simeq c$ then $f$ is reducible map.

Suppose $S = T^2 \sharp D^2 \sharp D^2$ (the twice punctured torus) and $\delta$ is a loop around one of the boundary components. Then $\delta$ is non-trivial in $H_1(S, \mathbb{Z})$ but $\forall [\phi] \in \mathcal{MCG}(S)$, $\phi(\delta) \simeq \delta$ or $\phi(\phi(\delta)) \simeq \delta$. Hence this statement doesn't hold for $S$.

Is there a similar result for $S$ (or indeed general surfaces with 2 or more boundary components)?