\\
    g &: g = m.
    \end{cases}$$ Furthermore, if $g=m$, then this upper bound is realizable.

(ii) If $r\geq 1$, then $n$ is odd.

(iii) If $r=1$, then $n\leq 2(g-m)+1$.

(iv) If $r\geq 2$, then $\displaystyle n \leq \frac{g-m+r-1}{r-1}$.

If ${\operatorname{{\mathcal{C}}}}$ is a separating or a mixed multicurve, $S_g({\operatorname{{\mathcal{C}}}})$ will have multiple connected components. In order to classify roots of $t_{{\operatorname{{\mathcal{C}}}}}$ in this case, we will require multiple actions on the various components of $S_g({\operatorname{{\mathcal{C}}}})$ which can be put together and extended to a root of $t_{{\operatorname{{\mathcal{C}}}}}$ on $S_g$. In Section \[sec:sepmulcurves\], we will show that this extension will require compatibility of orbits across components of $S_g({\operatorname{{\mathcal{C}}}})$ in addition to compatibility of fixed points as in [@KR1]. As an immediate consequence to this theorem, we obtain quadratic bounds for the degree of the root in terms of the genera of the individual components. Furthermore, we obtain a quadratic stable upper bound for the degree of the root as in [@KR1 Theorem 8.14].

In Sections \[sec:classify\_genus3\] and \[sec:classify\_genus4\], we use our theory to obtain a complete classification of roots of $t_{{\operatorname{{\mathcal{C}}}}}$ on surfaces of genus 3 and 4 respectively. We conclude by proving that a root of $t_{{\operatorname{{\mathcal{C}}}}}$ cannot lie in 