maps of circles.

For a marked ribbon graph with a metric, let $c_i$ be its cycles, let $|c_i|$ be their image in the realization and let $r_i$ be the length of $c_i$. Then there are natural maps $\phi_i:S^1\rightarrow |c_i|$ which map $S^1$ onto the cycle by starting at the vertex $v_i:=\del(\mk(c_i))$ and going around the cycle mapping each point $\theta\in S^1$ to the point at distance $\frac{\theta}{2\pi}r_i$ from $v_i$ along the cycle $c_i$.

### Contracting edges

The contraction $(\bar V_{\Gamma}, \bar F_{\Gamma},\bar
\imath,\bar \del)$ of a graph $(V_{\Gamma},F_{\Gamma},\imath,\del)$ with respect to an edge $e=\{f,\imath(f)\}$ is defined as follows. Let $\sim$ be the equivalence relation induced by $\del(f)\sim\del(\imath(f))$. Then let $\bar V_{\Gamma}:=V_{\Gamma}/\sim$, $\bar
F_{\Gamma}=F_{\Gamma}\setminus\{f,\imath(f)\}$ and $\bar \imath:
\bar F_{\Gamma}\rightarrow \bar F_{\Gamma}, \bar\del: \bar
F_{\Gamma}\rightarrow \bar V_{\Gamma}$ be the induced maps.

For a marked ribbon graph, we define the marking of $(\bar
V_{\Gamma}, \bar F_{\Gamma},\bar \imath,\bar \del)$ to be $\overline{\mk}(\bar c)=\overline{\mk(c)}$ if $\mk(c)\notin\{f,\imath(f)\}$ and $\overline{\mk}(\bar
c)=\overline{N\circ \imath(\mk (c))}$ if $\mk(c)\in\{f,\imath(f)\}$, viz. the image of the next flag in the cycle.

### Labelling graphs

By a labelling of the edges of a graph $\Gamma$ by a set $S$, we simply mean a map $E_{\Gamma}\rightarrow S$. A labelling of a ribbon graph $\Gamma$ by 