I don't know much about geometry and topology so forgive me for asking this like that, but is there a theorem that says that every simple graph is pLaNaR (clearly every graph is not planar, I dont't know what to call the property that I am thinking of, so I called it pLaNaR)?
And by pLaNaR I mean something that relates to planarity on a geometric object's surface.
For example if you consider a torus and you discretize it (create a mesh) say you're a computational person and you do so numerical calculations on a torus, you can apply the finite difference or finite element method. The graph resulting from this discretization is by definition pLaNaR with respect to the torus.
Hopefully this gives you an idea of what I mean.
So my question is: is every graph pLaNaR for some smooth geometric object?
Any kind of insight and reference is welcome.
Every time you add an edge $vw$, you can add an extra handle to your surface to make that edge easy to draw without crossing other edges. Delete a disk from the surface somewhere close to $v$, another disk somewhere close to $w$, and join the two holes with a cylinder.
This is incredibly wasteful, but shows that any $n$-vertex graph can be drawn on a surface with genus $\binom n2$: a torus with $\binom n2$ holes. (If we're more careful, then even the complete graph can be drawn on a surface with genus $\lceil \frac16 \binom{n-3}{2}\rceil$, as the link to Wolfram in the comments points out.)