This is answered in the paper [Embedding graphs in surfaces](https://doi.org/10.1016/0095-8956(84)90014-5) (_P. Hoffman_ and _B. Richter_, J. Comb. Theory, Ser. B 36, 65–84 (1984; [Zbl 0514.05028](https://zbmath.org/0514.05028))). Quoting from the introduction:
> There are certain foundational results in the overlap between graph theory and the topology of surfaces whose proofs seem both to the [sic] relatively lengthy and not to appear in the literature. In particular an earlier version of this paper was motivated by a query from Jack Edmonds about the proof that any graph contained in a surface is a subcomplex of the $1$-skeleton of a suitably chosen triangulation of that surface. His basic results for the orientable case **[Edm60]**, whose proofs appear in **[You63]**, and many other papers in topological graph theory, depend on this fact. On the other hand, the proof of this "folk theorem" (Theorem 1.3) certainly uses nothing but topological facts known prior to 1925 plus elementary arguments.
>
> We have included a number of other results in topological graph theory. For example, the method of proof of Theorem 1.3, namely Theorem 2.3, is used to obtain a rigorous definition of the "combinatorial boundary" of a face of an embedded graph. The same theorem is useful is [sic] discussing "equivalent" embeddings and the various combinatorial schemes for embeddings which have been developed.
>
> ### References
>
> **[Edm60]** J. Edmonds, A combinatorial representation for polyhedral surfaces, _Notices Amer. Math. Soc._ (1960), 646.
>
> **[You63]** J. W. T. Youngs, Minimal imbeddings and the genus of a graph, _J. Math. Mech._ **12** (1963), 303–315.
The Amplitwist
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2024-05-19T06:51:27Z (6 months ago)