Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), embedded in $S$, and such that the (weighted) shortest path metric in $G$ is quasi-isometric to $m$ (this means that there are constants $\lambda,\epsilon,C$ such that for any vertices $x,y$ in $G$, $\tfrac1\lambda d_G(x,y)-\epsilon \le d_m(f(x),f(y)) \le \lambda d_G(x,y)+\epsilon$, where $f$ denotes the embedding of $G$ in $S$, and every point of $S$ is at distance at most $C$ from some vertex of $G$).

Note that in particular $G$ will be planar. I need that in addition, $G$ is countable and locally finite (meaning that any bounded region of $S$ contains only a finite number of vertices of $G$).

I have seen such statements proved for compact surfaces (in this case $G$ can be taken to be finite), or non-compact surfaces with some uniform bounds on the curvature or strong convexity radius (in both cases the vertex set is just an $\epsilon$-net on the surface, for a sufficiently small $\epsilon>0$), but I suspect the result holds in much greater generality.

I am not an expert in riemannian geometry and I lack some background in the area. Have you seen such a statement proved in a clean way in an article or a textbook? Thank you in advance!

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