\rightarrow
F_{\Gamma}$ satisfying the conditions

-   For every cycle $c$ the directed edge $\mk(c)$ belongs to the cycle.

-   All vertices of valence two are in the image of $\mk$, that is $\forall v,\val(v)=2$ implies $v\in Im(\del\circ\mk)$.

Notice that on a marked treelike ribbon graph there is a linear order on each of the cycles $c_i$. This order is defined by upgrading the cyclic order to the linear order $\prec_i$ in which $\mk(c_i)$ is the smallest element.

### Dual b/w tree of a marked ribbon graph

Given a marked treelike ribbon graph $\G$, we define its dual tree to be the colored graph whose black vertices are given by $V_{\G}$ and whose set of white vertices is the set of cycles $c_i$ of $\G$. The set of flags at $c_i$ are the flags $f$ with $f\in c_i$ and the set of flags at $v$ are the flags $\{f:f \in c_0,
\del(f)=v\}$. The involution is given by $\imath_{\t}(f)=N(f)$ if $f\in c_0$ and $\imath_{\t}(f)=N^{-1}(f)$ else.

This graph is a tree and is b/w and bipartite by construction. It is also planar, since the $c_i$ and the sets $F(v)$ have a cyclic order and therefore also $F_v\cap c_0$. It is furthermore rooted by 