How do I unambiguously define the facial circuits in a $2$-cell embedding of a graph into a surface?
Suppose that $\Gamma$ is a connected, locally finite graph that is embedded into a closed, connected surface $M$. The faces of this embedding are the connected components of $M - \Gamma$ (we choose to denote the image of the embedding also by $\Gamma$). Let us assume that the embedding is such that each face is homeomorphic to a disc (this is called a $2$-cell embedding).
I would like to describe the boundary of a face $F$ by a walk in $\Gamma$. If $F \cup \partial F$ is homeomorphic to a closed disc, then I can do this by restricting such a homeomorphism to the boundary to get a closed path that passes through the vertices and edges incident on $F$ in a specific order. But in general it is possible that $F \cup \partial F$ is not homeomorphic to a closed disc even when we have a $2$-cell embedding. Figure 4b in [Kag37], reproduced below, shows a $2$-cell embedding of $K_{3,3}$ into the torus which has such a face.
So, I want to instead say that if $\varphi \colon B(0,1) \to M$ is a homeomorphism from the unit ball in $\mathbf{R}^2$ onto the face $F$, then there is a surjective continuous function $\Phi \colon B[0,1] \to F \cup \partial F$ from the closed unit ball in $\mathbf{R}^2$ to the union of $F$ and its boundary, which extends $\varphi$ on $B(0,1)$. (Note that if $\Phi$ exists, then it is unique, since $M$ is Hausdorff.) I can then deduce the facial walk of $F$ in $\Gamma$ from the data $\Phi$.
Question: How can I go about this? My primary goal is to be able to unambiguously define the facial circuits of an embedding, though I would be happy to just know how to define $\Phi$ from $\varphi$ for now.
I found a similar assertion made in the [EEK82], where the authors say:
[L]et $D_p$ denote a $p$-gon, that is, a closed disk whose boundary is divided into $p$ edges by $p$ vertices. Given a closed face $\alpha$ of $\Gamma$ there exists a unique positive integer $p$ and a characteristic map $\phi \colon (D_p, \partial D_p) \to (\alpha, \partial \alpha)$ which is an embedding on the interior of $D_p$ and on the interior of each of the $p$ edges along $\partial D_p$.
But they do not prove the existence of such a characteristic map, so perhaps it isn't too difficult?
Some thoughts
We know that $M$ can be embedded into Euclidean space of sufficiently large dimension. So, we can view $\varphi$ as a map of metric spaces. Then, since the closed unit ball is compact, there is a continuous extension $\Phi$ if and only if $\varphi$ is uniformly continuous.
This seems a bit odd to me. Can we assume without loss of generality that the homeomorphism $\varphi$ from the open unit disc to the face $F$ is uniformly continuous?
References
[Kag37] Kagno, I. N. The mapping of graphs on surfaces. J. Math. Phys., Massachusetts, 16, 46–75 (1937). Zbl 0017.42701, JFM 63.0550.02
[EEK82] Edmonds, Allan L.; Ewing, John H.; Kulkarni, Ravi S. Regular tessellations of surfaces and $(p,q,2)$-triangle groups. Ann. Math. (2) 116, 113–132 (1982). Zbl 0497.57001
0 answers
0 comment threads