As I understand it, *rooted* maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A *map* is a graph $G$ embedded on a surface $X$ (such that every face in $X \setminus G$ is homeomorphic to a disc), while a *rooted* map is essentially a map together with a choice of a *dart*, i.e., an edge together with an orientation of that edge. (The "essentially" is because for counting purposes, it is also convenient to consider the vertex map---with one vertex and no edges---to be rooted by convention.) Rooted maps are always considered up to root-preserving homeomorphism. A similar notion of rooting extends to *hypermaps*, which are maps equipped with a two-coloring of vertices.


I'm interested in whether the idea of rooting has any independent motivation for [dessin d'enfants](http://en.wikipedia.org/wiki/Dessin_d'enfant), i.e., when viewing a hypermap as a representation of a Belyi function $f : X \to \bar{\mathbb{C}}$. It seems that a "rooted Belyi function" would correspond to a triple $(X,f,\gamma)$, where $\gamma : [0,1] \to X$ is a path containing no other critical points of $f$ besides $\gamma\_0$ and $\gamma\_1$ and such that $f(\gamma) = [0,1]$. (Is there a better way of putting that?)


Is such a concept natural from the point of view of algebraic geometry, or has something like it already been studied?



How to understand a rooting of a dessin d'enfant?
For dessin d'enfants, I believe the orientation is superfluous - each edge goes between a black vertex and a white vertex, so picking an orientation is just picking one of those, whcih doesn't help uniformize anything.


A dessin d'enfants corresponds to a cover of $\mathbb P^1$ ramified over three points $(0,1,\infty)$. A rooted dessin d'enfants, I think, should be a dessin d'enfants with a marked edge. (Note that the trivial dessin has an edge, so no additional complexities are needed.) It's pretty easy to see that this corresponds to, after fixing a point $\lambda \not\in \{0,1,\infty\}$ in $\mathbb P^1$, taking a ramified cover and fixing a point lying over $\lambda$.


So that sounds pretty natural to me.


This is closely related to some algebraic geometry concepts (curve with marked points, monodromy of a cover), but I'm not sure if this exact concept has been studied, nor do I know whether studying this exact concept will yield new insights. 


